Application of EOS based on machine learning method on CFD study of rapid hydrogen refueling process

Application of EOS based on machine learning method on CFD study of rapid hydrogen refueling process | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Application of EOS based on machine learning method on CFD study of rapid hydrogen refueling process Hyo Min Seo, Byung Heung Park This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5012331/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 17 Apr, 2025 Read the published version in Korean Journal of Chemical Engineering → Version 1 posted 4 You are reading this latest preprint version Abstract Currently, commercially operated hydrogen fuel cell electric vehicles (FECVs) store hydrogen as highly compressed gas form to increase volumetric energy density. To provide a refueling time similar to that of internal combustion engine vehicles (ICEVs), hydrogen refueling stations (HRSs) should supply gaseous hydrogen into FECVs up to high pressure (35 MPa or 70 MPa) in a relatively short time. The refueling process of rapidly filling compressed gas within a confined volume of the storage tank is inevitably accompanied by an increase in temperature. However, the refueling process should be carried out under limited conditions considering the physical safety of the storage tank. Modeling the refueling process under the theoretical basis is useful for understanding the gas filling phenomenon and finding the optimal refueling strategy. In particular, the CFD research method which considers the flow of fluid in a tank offers the local temperature changes inside a storage tank as well as the average temperature. The CFD research is conducted by combining a model representing the fluid properties and a model describing the flow characteristics. Therefore, an appropriate combination of models should be examined before simulating the refueling process of an actual FECVs that requires time and cost that cannot be overlooked. In this study, the hydrogen refueling process is simulated using three equations of state (EOSs) and five turbulent models and, then, the results are compared and quantitively analyzed using experimental data. Experiments of filling type III tank of 74 L up to 35 MPa within 1 min have been chosen to make the assumption of axial symmetry for CFD model valid. Comparing the three EOSs (SRK, PR, ML), it is found that it is possible to improve accuracy and reduce calculation time when using ML EOS which has been developed to describe the behavior of hydrogen. Among the five turbulence models (yPlus, k-ε, realizable k-ε, low Reynolds k-ε, and k-ω) generally used in CFD research, the k-ε and the realizable k-ε model show satisfactory results on the reproduction of mean and local thermal behaviors and calculation time. hydrogen refueling CFD modeling equation of state turbulence model refueling simulation Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 1. Introduction The world has long been heavily dependent on traditional fossil energy sources. Harmful gases and greenhouse gases emitted by excessive use of the fossil fuels have been accumulated in the atmosphere, causing climate change and health problems around the world. The need for alternative energy sources becomes increasingly urgent to alleviate the deepening energy crisis that could result from the limited fossil fuel resources and to sustain international efforts to restrict harmful gas emissions to slow down climate change. Hydrogen is attracting attention as a clean fuel due to its recyclable nature and non-polluting properties and it is considered as a more efficient fuel than hydrocarbon fossil fuels due to its high energy density, high thermal conversion efficiency, and low calorific value. Therefore, hydrogen is expected to play an important role in the energy transition policy of replacing fossil fuels with renewable energy sources [ 1 – 4 ]. Hydrogen can be directly converted into energy by combustion, but the energy efficiency of converting chemical energy of hydrogen into electrical energy using fuel cells is relatively high, so researches on hydrogen energy are closely related to fuel cell technologies [ 5 ]. Fuel cells can produce electric power over a fairly wide range, from a few Watts to hundreds of kilo-Watts, and are already at the stage of commercialization in three applications: small and portable energy markets, transportation, and stationary power systems. The most notable and promising application field for fuel cells is the transportation sector which includes cars, buses, trains, boats, aerial vehicles, and so on [ 6 , 7 ]. Hydrogen storage methods are selected according to the different requirements of each application [ 8 ]. In the transportation sector, the compressed gas storage method is commonly adopted due to its technical simplicity, high reliability, efficiency, and acceptable economics [ 9 , 10 ]. The storage of gaseous hydrogen at high pressure is not only a well-developed technology, but also provides fast hydrogen filling and release rates [ 11 , 12 ]. For a hydrogen storage tank to be mounted on a hydrogen fuel cell vehicle, safety is of utmost importance, and the weight and size of the storage system should be as small as possible. As fuel cell electric vehicles become commercially available, most vehicles use pressure vessels for on-board hydrogen storage. At a given temperature, the density of hydrogen can be increased by rising pressure. Therefore, to achieve higher storage densities, storage vessels are manufactured to withstand very high pressure. Considering the energy required for compression, driving range of vehicles, investment in refueling infrastructure and other factors, the appropriate pressure for an on-board hydrogen system is known to be 35 to 70 MPa [ 13 ] since the density of hydrogen increases fast at 30–40 MPa but changes very little when the pressure is greater than 70 MPa. To achieve these requirements, most vehicle applications use fully-wrapped vessels with a metallic liner (Type III) or a non-metallic liner (Type IV). In the process of fast refueling of hydrogen up to high storage pressure, the temperature rises due to thermodynamic phenomena. The temperature increase is an important issue on the hydrogen refueling process since it would not only reduce SOC (state of charge) but could also bring safety risks to the vessels. The temperature of on-board Type III and IV tanks is restricted to be lower than 85°C [ 14 ] for securing thermal and mechanical safety of tank materials during the refueling process. Experimental studies on high-pressure filling of hydrogen tanks were conducted under 35 MPa [ 15 – 18 ] and 70 MPa [ 9 , 19 – 21 ] conditions, and several rapid refueling tests were carried out on various types and sizes of hydrogen tanks. However, the experiments require a lot of time and cost and involve risks due to high pressure conditions. Various theoretical researches have been conducted to supplement the limited experiments and understand the experimental results. These theoretical approaches can be classified into thermodynamic methods and CFD (computational fluid dynamic) methods. The thermodynamic models [ 22 – 26 ] are mainly based on mathematical formulas which are the energy and mass conservation equations combined with an equation of state for real gases. The thermodynamic models are useful for analyzing the effects of parameters such as initial pressure, filling rate, inlet temperature, and ambient temperature on hydrogen temperature rise during refueling. CFD models [ 15 , 16 , 19 , 27 – 30 ] have been established based on turbulence model, real gas model and heat transfer model to predict the temperature distribution within a storage tank during a refueling process. CFD method is a tool for analyzing thermal-fluid phenomena found during filling and emptying hydrogen tanks and is frequently used to supplement experiments and thermodynamic models. CFD makes it possible to investigate the behavior of all relevant variables such as temperature, density, pressure, and direction of flow at any point in the storage vessel throughout the refueling time. Recently, many CFD studies have been conducted because CFD models can add important information to understand related phenomena and provide insight into the refueling process at given experimental data. Important factors on the hydrogen refueling process can be obtained and analyzed from CFD models; the effect of liner material on temperature [ 29 ], the impact of hydrogen filling strategy on the maximum temperature [ 15 , 16 , 31 ], and the impact of tank shape on the temperature during a filling process [ 32 – 34 ]. CFD models should adopt an appropriate turbulence model and an equation of state to provide reliable prediction results. The Reynolds Averaged Navier-Stokes (RANS) equations have been numerically solved in CFD simulations on hydrogen filling with various turbulence models such as the standard k-ε model [ 19 , 35 , 36 ], the modified k-ε models [ 17 , 21 , 37 , 38 ], and the k-ω SST model [ 27 , 39 ]. Suryan et al. [ 40 ] carried out a comparative study on performance of turbulence models using the realizable k-ε model, the RNG k-ε model, the k-u SST model, and the Reynolds Stress Model and, then, recommended the realizable k-ε model by compromising between accuracy and computational costs. It is also very important to select an appropriate real gas equation of state (EOS) in CFD model describing hydrogen refueling process because the ideal gas EOS could not apply under high-pressure hydrogen filling conditions up to 35–70 MPa due to great compressibility deviations from ideal gas behavior. Redlich-Kwong (RK) EOS was used in many CFD models [ 16 , 17 , 19 , 27 , 32 , 37 ], and a comparative study [ 41 ] of RK, Soave-Redlich-Kwong (SRK), Aungier-Redlich-Kwong (ARK), and Peng-Robinson (PR) EOS confirmed that RK EOS is the most accurate gas model. In the present study, rapid hydrogen filling experimental data are adopted to make the CFD model simple by applying an axial symmetry condition since the one of the main purposes is to find the proper combinations of EOS and turbulence model. Firstly, three EOSs are compared with experimental data on mean temperature and pressure. Then, five turbulence models are systematically analyzed based on the local temperature behaviors. A selected combination of an EOS and a turbulence model is further applied to investigate thermo-flow characteristics inside a tank. 2. Equation of State (EOS): Real Gas Models An equation of state (EOS) is defined as a thermodynamic equation relating state variables such as pressure, (specific) volume, or temperature, which describe the state of matter under a given set of physical conditions. The most well-known EOS is the ideal gas law, which is suitable for representing the behavior of gases at a hypothetical ideal state. The ideal gas law is roughly accurate for weakly polar gases at low pressures and moderate temperatures and becomes increasingly inaccurate at higher pressures and lower temperatures. The cubic EOSs, which originated from the van der Waals (vdW) EOS, have a relatively simple formula and have been widely used to reflect real gas behavior and predict phase equilibrium [ 42 ]. In general, commercial software includes EOSs which are widely used in many different industries handling fluids. Among various cubic EOSs, COMSOL Multiphysics provides SRK and PR equations as built-in functions to calculate thermodynamic properties of gases. SRK EOS is a modification of the original RK EOS and expressed as follows. $$\:P=\frac{RT}{V-{b}_{SRK}}-\frac{{a}_{SRK}}{V(V+{b}_{SRK})}$$ 1 where, P , T , V , and R are pressure, temperature, specific volume, and gas constant, respectively. Two component dependent parameters ( a SRK and b SRK ) are obtained from critical temperature ( T c ), critical pressure ( P c ), and acentric factor ( ω ) of a gas by using the following equations. $$\:{a}_{SRK}=0.42748\frac{{R}^{2}{T}_{c}^{2}}{{P}_{c}}\times\:{\left[1+\left(0.480+1.574\omega\:-0.176{\omega\:}^{2}\right)\times\:\left(1-\sqrt{T/{T}_{c}}\right)\right]}^{2}$$ 2 $$\:{b}_{SRK}=0.08664\frac{R{T}_{c}}{{P}_{c}}$$ 3 PR EOS is also a cubic form similar to SRK EOS. It is frequently used in the simulation and optimization of chemical processes, particularly those involving hydrocarbons. The equation and the correlations of two component parameters are given as follows, $$\:P=\frac{RT}{V-{b}_{PR}}-\frac{{a}_{PR}}{V\left(V+{b}_{PR}\right)+{b}_{PR}(V-{b}_{PR})}$$ 4 $$\:{a}_{PR}=0.45724\frac{{R}^{2}{T}_{c}^{2}}{{P}_{c}}\times\:{\left[1+\left(0.37464+1.54226\omega\:-0.269926{\omega\:}^{2}\right)\times\:\left(1-\sqrt{T/{T}_{c}}\right)\right]}^{2}$$ 5 $$\:{b}_{PR}=0.07780\frac{R{T}_{c}}{{P}_{c}}$$ 6 The critical temperature, the critical pressure, and the acentric factor of hydrogen are reported as 32.98 K, 12.93 bar, and − 0.217, respectively [ 43 ]. The cubic EOSs present the relationship between state variables ( P , T , and V ) in the form of pressure − explicit equations. Therefore, in general, an iterative method is required to find the roots of the equations when calculating specific volume (or density) at a given temperature and pressure. This kind of approach demands a lot of time and cost, especially in CFD calculations. Recently, a generic correlation equation of the n th order has been proposed in a polynomial expansion form with temperature and pressure for gaseous hydrogen [ 44 ]. The equation can be readily applied to estimate accurate values for various properties such as density, enthalpy, internal energy, and so on by selecting a corresponding set of coefficients [ 45 ]. As for density ( ρ ), the equation is expressed as follows, $$\:\rho\:=\sum\:_{i=0}^{n}\sum\:_{j=0}^{n-i}{a}_{ij}{T}^{i}{P}^{j}$$ 7 where, the units of ρ , T , and P are mol/L, K, and MPa, respectively. In Eq. ( 7 ) a ij denotes a coefficient of the product of the i th power of T and the j th power of P . A set of coefficients has been determined to fit literature data [ 46 ] by machine learning (ML) method. Details of the equation such as the number of regressed data and the accuracy can be found elsewhere [ 44 ]. The correlation equation (Eq. ( 7 )) is referred to as ML EOS in this study. The third order ( n = 3) is sufficient and the coefficients are listed in Table 1 , which are applicable in the range of 223.15 K < T < 373.15 K and 0.1 MPa < P < 100.1 MPa. Table 1 Coefficient a ij of ML EOS j i 0 1 2 3 0 1.20463E + 01 7.93448E-01 -4.04683E-03 6.79659E-06 1 -1.02106E-01 -1.54833E-03 5.97575E-06 - 2 2.82873E-04 8.13547E-07 - - 3 -2.56918E-07 - - - The lowest temperature of hydrogen dispensed from a hydrogen refueling station (HRS) is -40°C and the temperature of hydrogen in an on-board tank is limited to be lower than 85°C. The densities of hydrogen calculated by the three EOSs (SRK, PR, and ML) are compared in Fig. 1 at two different temperatures within the acceptable hydrogen refueling temperature range (-40°C < T < 85°C). Compared to literature data [ 46 ], PR EOS overestimates density and SRK EOS slightly underestimates over the entire pressure range. On the other hand, ML EOS accurately reproduces literature data. The accuracies of the three EOS are quantitatively calculated using the following mean absolute percentage error (MAPE) equation with 21 data ( n = 21) at respective temperature and summarized in Table 2 . $$\:MAPE=\frac{100}{n}\sum\:_{i}^{n}\left|\frac{{\rho\:}_{lit,\:i}-{\rho\:}_{EOS,\:i}}{{\rho\:}_{lit,\:i}}\right|$$ 8 Table 2 Mean absolute percentage error (MAPE) [%] of EOSs T [°C] PR EOS SRK EOS ML EOS -40 5.3399 1.0298 0.1006 0 4.7685 0.9058 0.0692 40 4.2528 0.8303 0.0662 80 3.7900 0.7841 0.0570 Average 4.5380 0.8875 0.0733 In the hydrogen refueling temperature range, ML EOS calculates the hydrogen density most accurately, and SRK EOS shows better accuracy than PR EOS in the two cubic EOSs. 3. Turbulence Models Simulation models for hydrogen refueling require the solution of the governing equations describing an unsteady state. Navier-Stokes equations must be applied to mathematically express momentum balance, which are partial differential equations for the motion of viscous fluid substances. However, for turbulent flows the computational effort to solve the complete Navier-Stokes equations is very high, since even the smallest disturbances influence the solution due to the nonlinearity. Therefore, most of CFD programs are based on the Reynolds-averaged Navier-Stokes (RANS) equations, which still represent the flow of fluids accurately enough, but lead to acceptable computational times. The small turbulences are not resolved with RANS equations, but are considered by so-called turbulence models. Turbulence models allow the calculation of the mean flow without first calculating the full time-dependent flow field. Many turbulence models have been developed and some are included in CFD programs. COMSOL Multiphysics also provides different turbulence models. Choosing an appropriate turbulence model is a crucial factor in accurately describing the fluid flow. In the present work, five models, which are usually adopted for engineering purposes, have been applied and compared to simulate hydrogen refueling process; algebraic yPlus, k-ε, realizable k-ε, low Reynolds number k-ε, and k-ω model. The algebraic yPlus turbulence model is a zero-equation turbulence model based on the distance to the nearest wall. The model can be calculated directly from the flow variables since it does not require the solution of any additional equations. Consequently, it may not be able to properly account for history effects on the turbulence, such as convection and diffusion of turbulent energy. The model is too simple for use in general situations, but can be quite useful for initial phases of a computation in which a more complicated turbulence model may have difficulties. The two-equation turbulence models are the most widely applied class of turbulence models. Models such as the k-ε model and k-ω model have become industry standard models and are commonly used to solve most types of engineering problems on flows. The two-equation models are associated with two extra transport equations to describe the turbulent properties of the flow. It allows a two-equation model to represent history effects like convection and diffusion of turbulent energy. One of the transported variables is the turbulent kinetic energy, usually denoted as k. The second transported variable is deferent depending on type of two-equation models. The turbulent dissipation (ε) or the specific turbulence dissipation rate (ω) is commonly taken for the two-equation models. The first variable (k) determines the energy in the turbulence and the second one can be regarded as the variable that determines the length-scale or time-scale of the turbulence. The k-ε model is one of the most widely used and validated turbulence models used in CFD to simulate mean flow characteristics. It includes two extra transport equations to represent the turbulent kinetic (k) and the turbulent dissipation (ε). The standard k-ε model was developed to find an alternative to algebraically prescribing turbulent length scales in moderate to high complexity flows with improvement of the mixing-length model. The model has been known to be useful for free-shear layer flows with relatively small pressure gradients as well as in confined flows where the Reynolds shear stresses are most important. Based on the standard k-ε model, new refined two-equation models have been actively developed for various areas of research. The realizable k-ε turbulence model is an extension to the standard k-ε model which provides improved predictions for flows involving rotation, boundary layers under strong adverse pressure gradients, separation, and recirculation. The realizable k-ɛ model differs from the standard k-ɛ model in two ways. It is associated with a new formulation for the turbulent viscosity and employs a new transport equation for the dissipation rate, which is derived from an exact equation for the transport of the mean-square vorticity fluctuation. The low Reynolds number k-ε model is a basically modification to the standard k-ε model to make it applicable to near-wall conditions. The model integrates through the viscous sub-layer down to the wall and uses turbulent transport equations that are applicable throughout the boundary layer, including the buffer and viscous sublayers. It is still only applicable to high Reynolds number flows where the flow is fully turbulent. In comparison with the wall functions, the low-Reynolds methods have the advantage of resolving the boundary layer details, but the advantage comes at the expense of significantly more computations. The k-ω model is the second most widely applied two-equation model next to the standard k-ε model. The model has been demonstrated to improve the k-ε model for many situations, including turbulent boundary layers with zero or adverse pressure gradients and even near-separation conditions. The model attempts to predict turbulence by two partial differential equations for two variables, k and ω, with the first variable being the turbulence kinetic energy while the second is the specific rate of dissipation. In the present work, the five turbulence models have been applied without further modifications on the formulations provided by COMSOL Multiphysics software. More details of the theoretical background on the turbulence models and derivations of the equations can be found elsewhere [ 47 ]. 4. Computational Model and Implementation 4.1. Governing equations As employed in most of flow field analyses, the hydrogen refueling process is also characterized by balance in mass, momentum, and total energy in the unsteady form described by the continuity equation (Eq. ( 9 )), the Navier-Stokes equations (Eq. ( 10 )), and the total energy equation (Eq. ( 11 )), respectively. $$\:\frac{\partial\:\rho\:}{\partial\:t}+\nabla\:\bullet\:\left(\rho\:\overrightarrow{u}\right)=0$$ 9 $$\:\frac{\partial\:\rho\:\overrightarrow{u}}{\partial\:t}+\nabla\:\bullet\:\left(\rho\:\overrightarrow{u}\overrightarrow{u}\right)=-\nabla\:p+\nabla\:\bullet\:\stackrel{̿}{\tau\:}$$ 10 $$\:\frac{\partial\:}{\partial\:t}\left[\rho\:\left(e+\frac{1}{2}{u}^{2}\right)\right]+\nabla\:\bullet\:\left[\rho\:\overrightarrow{u}\left(e+\frac{1}{2}{u}^{2}\right)\right]=\nabla\:\bullet\:\left({k}_{t}\nabla\:T\right)+\nabla\:\bullet\:\left(-p\overrightarrow{u}+\stackrel{̿}{\tau\:}\bullet\:\overrightarrow{u}\right)$$ 11 where, \(\:\rho\:\) is density of fluid, \(\:\overrightarrow{u}\) is velocity, \(\:\stackrel{̿}{\tau\:}\) is stress tensor, \(\:e\) is specific internal energy of fluid, and \(\:{k}_{t}\) is thermal conductivity. The gravitational force and the potential energy change are ignored in Eqs. ( 10 ) and ( 11 ), respectively, considering the horizontal flow of hydrogen. The solution to the governing equations gives the velocity field ( \(\:\overrightarrow{u}\) ), pressure ( \(\:p\) ), and temperature, ( \(\:T\) ) of the fluid in the modeled domain when combined with the following equations. $$\:e=h-\frac{p}{\rho\:}$$ 12 $$\:\stackrel{̿}{\tau\:}=\mu\:\left[\nabla\:\overrightarrow{u}-\frac{2}{3}\nabla\:\bullet\:\overrightarrow{u}I\right]$$ 13 where, \(\:h\) is specific enthalpy, \(\:\mu\:\) is dynamic viscosity, and \(\:I\) is the unit tensor. 4.2. Geometry and specifications of hydrogen storage tank The objective of the present work is to find out appropriate real gas model and turbulence model. Therefore, we selected experimental data which reported temperature distribution inside a tank under a fast-filling condition. Experimental data [ 15 , 37 ] from a test facility installed with a Type III hydrogen storage cylinder of which internal volume is 74 liters are used to validate CFD models. The dimensions of the tank were 39.9 cm of external diameter and 90 cm of length. A tree-shaped support with 63 thermocouples was inserted into the cylinder to measure the hydrogen gas temperature with respect to radial and axial position as shown in Fig. 2 . The liner is made of aluminum and the laminate is composed of CFRP (carbon fiber reinforced plastic). The specifications of tank materials are summarized in Table 3 . Table 3 Specifications of tank materials Thickness [m] Specific heat [J/kg·K] Thermal conductivity [W/m·K] Density [kg/m 3 ] Liner 0.004 900 167 2730 Laminate 0.015 938 1 1494 4.3. CFD model A dimensional analysis carried out for the experimental conditions [ 37 ] revealed that the Froude number (Fr) is greater than 30 during a process of filling hydrogen into a 74 L tank up to 35 MPa within 40 s. The Froude number is defined as the ratio of the flow inertia to the external field and, thus, the high Fr indicates the motion of fluid is not affected by gravitational force in case of a horizontal filling. For this reason, a CFD model was set up under an assumption that the flow within the tank is axisymmetric with respect to the centerline of the cylinder [ 37 , 40 ]. A structured mesh for a hydrogen storage tank was generated as shown in Fig. 3 by using a mesh creating tool embedded in COMSOL Multiphysics. The quality of grid was set to ‘extra fine’ as predefined by the software. The mesh was denser near the walls and inlet tube inserted into a tank to capture the flow phenomena at fluid boundaries. The total number of cells was 5,229 which was much greater than the minimum value (= 1,200) enough to obtain reliable results for the same system [ 37 ]. The grid presented in Fig. 3 was used to produce all the results provided in this study. As initial conditions, pressure and temperature of the gas were set to be uniform with values of 9.3 MPa and 293.4 K respectively reflecting experimental data. The heat transfer coefficient between the tank outside wall and the ambient was assumed to 10 W/(m 2 ·K). The experiments to be used for the CFD model validation were performed under changing inlet pressure and temperature with hydrogen filling time [ 37 ] because a dispenser controlled the rate of pressure rise within the cylinder through a pressure control valve. Therefore, the experimental inlet pressure and temperature were curve fitted and, then, put into the CFD model for the present work. No-slip condition was applied at the tank inside wall to solve the governing equations. 5. Results and Discussion 5.1. Effect of EOS selection on CFD calculation In this study, three EOSs and five turbulence models are taken into consideration for CFD calculations on hydrogen filling process. The effect of EOS selection was investigated with the realizable k-ɛ turbulence model since it was revealed that the model is robust and gives reasonable accuracy for turbulent internal flow with heat transfer [ 40 ]. An experimental study [ 15 ] presented the volume average gas temperature inside a tank which was calculated by Eq. ( 14 ) using the measured temperature ( \(\:{T}_{k}\) ) at 63 points (see Fig. 2 ). $$\:{T}_{mean}=\frac{{\sum\:}_{k=1}^{63}{y}_{k}{T}_{k}{A}_{k}}{{\sum\:}_{k=1}^{63}{y}_{k}{A}_{k}}$$ 14 where, \(\:{y}_{k}\) is the vertical distance of the thermocouple \(\:k\) to the centerline of the cylinder and \(\:{A}_{k}\) is an area weighting given to each thermocouple. As shown in Fig. 4 , all the three EOSs well describe the temperature increase behavior during a process of hydrogen filling into a confined tank volume. However, compared with experimental data, ML EOS presents the most accurate result throughout the time of refueling. The deviations from the experimental data at the end of fill are calculated to 5.06, 3.02, and 2.11°C for PR EOS, SRK EOS, and ML EOS, respectively. As for pressure inside the tank, all the three EOSs correctly follow the measured data and the pressure against time curves obtained from the simulation results are nearly superimposed as given in Fig. 5 . Comparison of the time required for a simulation is an important consideration in model selection along with the accuracy comparison since the computation time is recognized as a cost in simulation works. The times spent for simulations carried out using the three EOSs are compared in Table 4 which presents the relative times to ML EOS since the actual absolute computation time is determined by computational capability. Table 4 Relative time for simulation with EOSs SRK EOS PR EOS ML EOS Simulation time (Relative) 1.261 1.986 1 ML EOS is shown to simulate the hydrogen refueling process most accurately in the shortest time. Such a result is due to its simple formula compared to other real gas models, which is suitable for simulation. It is expected that the fast and accurate feature can be more important in models of large-capacity tank filling that require more mesh elements and longer simulation times. 5.2. CFD calculations with different turbulence models ML EOS has been proved to be suitable for the simulation of hydrogen filling process in terms of accuracy and computational time. Therefore, in the following study, we compared the results of applying various turbulence models associated with ML EOS to analyze the effect of the turbulence model selection. The temperature distributions in a tank are shown with respect to turbulence models in Fig. 6 . The algebraic yPlus model presents a high temperature at the front of the tank, and the temperature difference between the front and rear is evaluated to be relatively large. The other four models show small temperature deviations inside a tank except for the centerline where pre-cooled hydrogen is rapidly introduced. The overall temperature level is the highest for the low Reynolds model, and quantitative analysis was carried out by comparison with experimental data. Unlike the case of EOS comparison, the simulation results of temperature increase present quite different behavior according to turbulence models as presented in Fig. 7 which is drawn to exhibit deviations of mean temperature ( \(\:{\Delta\:}{T}_{mean}={T}_{mean}^{cal}-{T}_{mean}^{exp}\) ) between calculated values ( \(\:{T}_{mean}^{cal}\) ) and experimental data ( \(\:{T}_{mean}^{exp}\) which can be found in Fig. 4 ) during the filling process. The algebraic yPlus model initially shows lower temperatures than the experimental values (negative \(\:{\Delta\:}{T}_{mean}\) ), but after about 10 s, it tends to show higher temperatures than the experimental data. The other models predict higher temperatures than the literature data throughout the process. Based on the final temperature, the errors are decreasing in the order of the low Reynolds k-ɛ, the algebraic yPlus, the k-ɛ, the k-ω, and the realizable k-ɛ model. The realizable k-ε model was found to most accurately fit the experimental data, with the deviations stably remaining within 2°C throughout the process after the initial approximately 5 s. The accuracies of calculated hydrogen pressures in the vessel are compared in Fig. 8 where \(\:{\Delta\:}P\) is defined as \(\:{P}^{cal}-{P}^{exp}\) where \(\:{P}^{exp}\) behavior has been given in Fig. 5 . All the models except for the algebraic yPlus model show satisfactory accordance with the experimental data and no significant differences could be found between the models after 10 s. The pressure behavior predicted by the algebraic yPlus model is lower than the other models at the early stage, but the deviation diminishes toward the end of the filling. All models fit the experimental data within 0.5 MPa of the final pressure. However, the algebraic yPlus model still shows a large error compared to the other models at the end of the experiment. The measured temperatures at the cylinder axis (denoted as H0 in Fig. 2 ) are co-plotted with simulation results in Fig. 9 . The x-axis of Fig. 9 means relative position from the inlet along the centerline and the temperatures are presented at the y-axis as deviations from the mean temperature. The changes over time are presented as Fig. 9 (a), (b), and (c), which show the temperatures at 13 s, 26 s, and 39 s, respectively. In practice, the temperature of the gas is measured at the centerline of a vessel for commercial hydrogen storage tank. The experimental data in Fig. 9 proposed that the accurate temperature could be measured when a sensor is extended to a position of 70% of the tank length. However, the proper position could be different according to the dimensions of a vessel and the process conditions such as a mass flow rate. The temperature at the front of the tank where the pre-cooled hydrogen flows in was the lowest, and the temperatures along the centerline were found to be lower than the mean temperature throughout the filling process. Figure 9 shows that the k-ε model reproduces the most similar results to the experimental data throughout the entire position along the axis during the filling time. The low Reynolds k-ε model also presents good agreements with the experimental data. However, considering the accuracy on the mean temperature behavior (see Fig. 7 ), the realizable k-ε model is quite compatible with the k-ε model. The hydrogen rapid filling investigation [ 15 , 16 , 19 , 27 – 30 ] has reported temperature measurement data with respect to positions in a vessel. The data were used to compare the turbulence model results on radial temperature distribution at different axial positions which are denoted as V1 (x/L ≅ 0.2), V2 (x/L ≅ 0.45), and V3 (x/L ≅ 0.85) in Fig. 10 (a), (b), and (c), respectively. Temperatures are presented as relative values to \(\:{T}_{mean}\) at the time of the measurement (t = 20 s) in Fig. 10 . When analyzing the experimental data, it was found that the front of the tank where pre-cooled hydrogen flows in at the centerline has a low temperature and temperature difference on temperature distribution has been observed between the upper and lower regions of a tank. On the other hand, as the fluid flows deep into the tank, the temperature distribution between the upper and lower regions at V2 and V3 became symmetric with respect to the centerline. The model calculations at V1 were not satisfactory since the temperature behavior changes very rapidly at near centerline and the CFD calculations have been carried out by using an axial symmetry model. However, the calculation results at V2 and V3 reproduce the experimental values accurately. Among the adopted turbulence models, the k-ε and the realizable k-ε models show better performance than the other models over the entire region. The two-equation models (k-ε, realizable k-ε, low Reynolds k-ε, and k-ω) commonly adopted the turbulent kinetic energy (k). The average values of k with time are compared in Fig. 11 . The low Reynolds k-ε model exhibits very low values of k. It is understood that the model is not suitable since the mathematical formulation of the model is capable to resolve the turbulence flow down to the wall, but the hydrogen filling process associated with a circulating flow inside a tank. The computational time for a simulation has been summarized in Table 5 as relative values to the time spent applying the realizable k-ε model. The algebraic yPlus model offers calculation result very fast. The k-ε model requires the shortest time among the two-equation models and the realizable k-ε model takes compatible time with the k-ε model. Table 5 Relative time for simulation with turbulence models Algebraic yPlus k-ε Realizable k-ε Low Reynolds k-ε k-ω Simulation time (Relative) 0.066 0.910 1 2.037 5.036 5.3. Thermo-flow analysis inside a tank The flow inside a tank is closely related to the temperature rise of the fluid. However, it is very difficult to measure flow properties such as the velocity of fluid in the tank, and experimental data generally present only the temperature of a fluid. The CFD model is used to obtain properties that are difficult to measure experimentally because it can simulate thermo-flow phenomena. The most significant data to be monitored during the hydrogen filling process is the average temperature of the tank because the maximum temperature is limited to ensure the safety of the tank storing highly compressed gas. Therefore, the realizable k-ε model is used to analyze the flow inside a tank, which shows the least deviation from the experimental data (see Fig. 7 ). The streamline and the turbulent kinetic energy inside a tank are given in Fig. 12 with time. The introduced flow directly touches the back of the tank since the tank is rapidly filled by a high-speed fluid, and then the flow moves along the tank wall to form a rotational motion. Throughout the filling process, the center of rotation is located in the upper part of the back. The turbulent kinetic energy is found to be the greatest on the back side of the tank center, where the fluid flows upward after hitting the tank wall, and observes large in the area where the rotational flow occurs along the streamline. The magnitude of the turbulent kinetic energy decreases with time, which can be recognized by the range shown in the scale bar in Fig. 12 . Therefore, it is found that the contribution of energy due to turbulent dissipation to temperature rise becomes smaller as the process proceeds. The temperature increases along an axis at different radial positions are shown in Fig. 13 . The temperature rises with time at all locations, but, instantly, the temperature does not rise monotonically due to fluctuations of local flow. The temperature differences in the axial direction are larger with increasing distance from the central axis (longer radial position) since the extent of fluid mixing is not instant as a consequence of the decreased flow velocity near the tank wall. It is observed that the maximum temperature is found at the upper region of tank backside throughout the filling process. Therefore, the temperature during commercial tank filling processes should be controlled considering the deviation of the monitored temperature and the actual maximum temperature, which could be different according to filling conditions such as initial tank pressure, process time, and ambient temperature. Conclusions The temperature increases of the gas during the high-pressure hydrogen filling process is a thermo-physical phenomenon that inevitably takes place for rapid refueling. However, the temperature rise of hydrogen should be limited to maintain the physical safety of tanks which store highly-compressed gas for usually fuel cell electric vehicles (FCEVs). Since hydrogen refueling can be carried out under various conditions such as ambient temperature, initial pressure, and pre-cooling level, the safety of the hydrogen storage tank must be ensured under all conditions. CFD research is an important tool that can complement experimental researches. It could be used to theoretically interpret experimental results and reduce costs by reducing the number of experiments. The equation of state (EOS) which can explain the volumetric physical properties of a fluid and the turbulence model which simulates flow characteristics are the basic equations that make up a CFD model, and the reliability of CFD results can be secured only when they are properly combined. In the present study, an appropriate combination of EOS and turbulence model is explored, and the thermos-flow behavior of the hydrogen storage tank is investigated by a CFD applying a selected combination. As a result of comparing the three EOSs (SRK, PR, and ML) using the experimental results that fill the hydrogen up to 35 MPa, the ML EOS shows the best accuracy and significantly reduced calculation time. The ML EOS, which is in the form of a polynomial equation, has been shown to be suitable for accuracy and computational cost reduction in CFD model studies that require a lot of iterative computation. The performance of five turbulence models (one zero-equation model and four two-equation models) are compared for the reproduction of local temperatures as well as mean temperature behavior. As for the mean temperature behavior, the realizable k-ε model is found to be the most accurate. The k-ε model presents the best accuracy for local temperatures and the realizable k-ε model also shows satisfactory results compatible with the results of the k-ε model. In terms of calculation time, the yPlus model which is the zero-equation model takes the least time, but it presents the lowest accuracy. Therefore, the yPlus model is considered to be suitable only for initial approach, such as examine on the quality of the structured mesh of a CFD model. In the four two-equation models, the low Reynolds k-ε model is found to show very low average turbulent kinetic energy unlike the other models. It is understood that the mathematical formulation of the low Reynolds k-ε model is not suitable for the tank filling process associated with a circulating flow inside a tank. A CFD model composed of the ML EOS and the realizable k-ε turbulence model is used to analyze the inside flow and corresponding local temperature behavior. It is found that the maximum temperature tanks place at the upper region of tank backside due to the turbulence energy dissipation. In this study, experimental data on rapid refueling of hydrogen into a storage tank are used to simplify comparative studies by applying axial symmetry assumption. Therefore, further study would be required for extended refueling experiments in which axial symmetry assumptions are not valid due to temperature stratification. Declarations Acknowledgment This research was supported by the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Ministry of Trade, Industry & Energy (MOTIE) of the Republic of Korea (Project No. 20227310100060). It was also supported by the Korea Agency for Infrastructure Technology Advancement (KAIA) and the Ministry of Land, Infrastructure and Transport (MOLIT) (Project No. 21OHTI-C163280-01). References S. Sharma, S. Agarwal and A. Jain, Energies, 14 , 7389 (2021). N. Mac Dowell, N. Sunny, N. Brandon, H. Herzog, A. Y. Ku, W. Maas, A. Ramirez, D. M. Reiner, G. N. Sant and N. Shah, Joule, 5 , 2524 (2021). A. Kovač, M. Paranos and D. Marciuš, International Journal of Hydrogen Energy, 46 , 10016 (2021). J. Cader, R. Koneczna and P. Olczak, Energies, 14 , 4811 (2021). M. K. Singla, P. Nijhawan and A. S. Oberoi, Environmental Science and Pollution Research, 28 , 15607 (2021). A. Alaswad, A. Baroutaji, H. Achour, J. Carton, A. Al Makky and A.-G. Olabi, International Journal of Hydrogen Energy, 41 , 16499 (2016). H. Yang, Y. J. Han, J. Yu, S. Kim, S. Lee, G. Kim and C. Lee, Sustainability, 14 , 917 (2022). M. R. Usman, Renewable Sustainable Energy Rev., 167 , 112743 (2022). J. Zheng, X. Liu, P. Xu, P. Liu, Y. Zhao and J. Yang, Int. J. Hydrogen Energy, 37 , 1048 (2012). E. Rivard, M. Trudeau and K. Zaghib, Materials, 12 , 1973 (2019). D. Durbin and C. Malardier-Jugroot, Int. J. Hydrogen Energy, 38 , 14595 (2013). S. W. Jorgensen, Curr. Opin. Solid State Mater. Sci., 15 , 39 (2011). R. Von Helmolt and U. Eberle, J. Power Sources, 165 , 833 (2007). SAE, Fueling protocols for light duty gaseous hydrogen surface vehicles , SAE international (2020). C. Dicken and W. Merida, J. Power Sources, 165 , 324 (2007). L. Zhao, Y. Liu, J. Yang, Y. Zhao, J. Zheng, H. Bie and X. Liu, Int. J. Hydrogen Energy, 35 , 8092 (2010). J. Zheng, J. Guo, J. Yang, Y. Zhao, L. Zhao, X. Pan, J. Ma and L. Zhang, Int. J. Hydrogen Energy, 38 , 10956 (2013). M. Monde, Y. Mitsutake, P. L. Woodfield and S. Maruyama, Heat Transfer—Asian Research: Co‐sponsored by the Society of Chemical Engineers of Japan and the Heat Transfer Division of ASME, 36 , 13 (2007). S. C. Kim, S. H. Lee and K. B. Yoon, Int. J. Hydrogen Energy, 35 , 6830 (2010). R. O. Cebolla, B. Acosta, N. De Miguel and P. Moretto, Int. J. Hydrogen Energy, 40 , 4698 (2015). M. C. Galassi, D. Baraldi, B. A. Iborra and P. Moretto, Int. J. Hydrogen Energy, 37 , 6886 (2012). H. Tun, K. Reddi, A. Elgowainy and S. Poudel, International Journal of Hydrogen Energy, 48 , 28869 (2023). J.-Q. Li, J.-C. Li, K. Park, S.-J. Jang and J.-T. Kwon, Energies, 14 , 2635 (2021). T. Kuroki, K. Nagasawa, M. Peters, D. Leighton, J. Kurtz, N. Sakoda, M. Monde and Y. Takata, Int. J. Hydrogen Energy, 46 , 22004 (2021). R. Caponi, A. M. Ferrario, E. Bocci, G. Valenti and M. Della Pietra, Int. J. Hydrogen Energy, 46 , 18630 (2021). L. Zhao, F. Li, Z. Li, L. Zhang, G. He, Q. Zhao, J. Yuan, J. Di and C. Zhou, Int. J. Hydrogen Energy, 44 , 3993 (2019). M. Heitsch, D. Baraldi and P. Moretto, Int. J. Hydrogen Energy, 36 , 2606 (2011). D. Melideo, D. Baraldi, B. Acosta-Iborra, R. O. Cebolla and P. Moretto, Int. J. Hydrogen Energy, 42 , 7304 (2017). J. Liu, S. Zheng, Z. Zhang, J. Zheng and Y. Zhao, International Journal of Hydrogen Energy, 45 , 9241 (2020). J. Martin, Q. Nouvelot, V. Ren, G. Lodier, E. Vyazmina, F. Ammouri and P. Carrere, International Journal of Hydrogen Energy, 54 , 562 (2023). M. Monde and M. Kosaka, SAE International Journal of Alternative Powertrains, 2 , 61 (2013). Q. Li, J. Zhou, Q. Chang and W. Xing, Int. J. Hydrogen Energy, 37 , 6043 (2012). T. Bourgeois, T. Brachmann, F. Barth, F. Ammouri, D. Baraldi, D. Melideo, B. Acosta-Iborra, D. Zaepffel, D. Saury and D. Lemonnier, Int. J. Hydrogen Energy, 42 , 13789 (2017). V. Ramasamy and E. Richardson, Int. J. Heat Mass Transfer, 160 , 120179 (2020). R. Immel and A. Mack-Gardner, SAE International Journal of Engines, 4 , 1850 (2011). T. Johnson, R. Bozinoski, J. Ye, G. Sartor, J. Zheng and J. Yang, Int. J. Hydrogen Energy, 40 , 9803 (2015). C. Dicken and W. Merida, Numerical Heat Transfer, Part A: Applications, 53 , 685 (2008). Y. Zhao, G. Liu, Y. Liu, J. Zheng, Y. Chen, L. Zhao, J. Guo and Y. He, Int. J. Hydrogen Energy, 37 , 17517 (2012). T. Setoguchi, M. Alam, M. Monde and H. Kim, International Journal of Aeroacoustics, 12 , 455 (2013). A. Suryan, H. D. Kim and T. Setoguchi, Int. J. Hydrogen Energy, 38 , 9562 (2013). A. Suryan, H. D. Kim and T. Setoguchi, Int. J. Hydrogen Energy, 37 , 7600 (2012). J. O. Valderrama, Industrial & engineering chemistry research, 42 , 1603 (2003). B. E. Poling, The properties of gases and liquids , (2004). B. H. Park and C. K. Chae, Int. J. Hydrogen Energy, 47 , 4185 (2022). B. H. Park and C. H. Joe, Int. J. Hydrogen Energy, 49 , 1140 (2024). E. W. Lemmon, M. L. Huber and M. O. McLinden, NIST standard reference database, 23 , v7 (2002). D. C. Wilcox, Turbulence modeling for CFD , DCW industries La Canada, CA (1998). Cite Share Download PDF Status: Published Journal Publication published 17 Apr, 2025 Read the published version in Korean Journal of Chemical Engineering → Version 1 posted Reviewers agreed at journal 30 Oct, 2024 Reviewers invited by journal 29 Oct, 2024 Editor assigned by journal 04 Sep, 2024 First submitted to journal 02 Sep, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-5012331","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":371782597,"identity":"43cbd0f9-72cd-4be8-929e-8003c47ec186","order_by":0,"name":"Hyo Min Seo","email":"","orcid":"","institution":"Korea National University of Transportation","correspondingAuthor":false,"prefix":"","firstName":"Hyo","middleName":"Min","lastName":"Seo","suffix":""},{"id":371782598,"identity":"525771c6-8a9b-4589-898e-76d763ca224d","order_by":1,"name":"Byung Heung Park","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAwUlEQVRIiWNgGAWjYBACAwYeCM0PFyBai2QDM6laDA4Qq8WcgfcAc0HNPWPjG/kHGH7UMBibNxDQYtnAl8A841ixmdmNZAbGnmMMZjIHCDnsAI8BMw9bgg1ICwNvA4ONBEG/gLX8S7AxngG05S/RWnjbEswMJJIZmIG2mBHUYtnMY3B4Zl+CscSZxwaHZY5JGBPUYs7eY/i44FuCYX974sOHb2psDGcQ0sIAjI3DMPYBBgaCdsB1jYJRMApGwSjAAwDBqzKLxIKpqAAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0002-4808-5811","institution":"Korea National University of Transportation","correspondingAuthor":true,"prefix":"","firstName":"Byung","middleName":"Heung","lastName":"Park","suffix":""}],"badges":[],"createdAt":"2024-09-01 09:59:10","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5012331/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5012331/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11814-025-00460-x","type":"published","date":"2025-04-17T15:57:53+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":68556714,"identity":"c0477b0c-0e97-40c9-a03d-28cd5fd89ff0","added_by":"auto","created_at":"2024-11-08 13:38:42","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":24133,"visible":true,"origin":"","legend":"\u003cp\u003eHydrogen density with pressure calculated by three EOSs and comparison with literature data [46] at two different temperatures.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-5012331/v1/a6a70ba535c16b3355d7de6f.png"},{"id":68556709,"identity":"ce0696c5-1a27-46fe-a30d-2c93a5f8e431","added_by":"auto","created_at":"2024-11-08 13:38:41","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":36665,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic structure of hydrogen storage tank used for experiments [15].\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-5012331/v1/8c648f3280a90e1a877871e3.png"},{"id":68556719,"identity":"6cf2bad5-4e39-40b6-a389-40405da74739","added_by":"auto","created_at":"2024-11-08 13:38:43","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":84855,"visible":true,"origin":"","legend":"\u003cp\u003eStructured mesh for computational model.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-5012331/v1/e053dcf2be820b1b66d998fd.png"},{"id":68556711,"identity":"4e343538-ff30-4dbb-82c0-6a230280e611","added_by":"auto","created_at":"2024-11-08 13:38:41","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":24341,"visible":true,"origin":"","legend":"\u003cp\u003eMean temperature rises calculated by three EOSs coupled to realizable k-ɛ turbulence model and comparison with literature data [15, 16, 19, 27-30].\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-5012331/v1/0c96636f4f390d18eb65ec7c.png"},{"id":68556716,"identity":"650d4958-bc25-4cbf-b127-784b8a2e70aa","added_by":"auto","created_at":"2024-11-08 13:38:42","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":16652,"visible":true,"origin":"","legend":"\u003cp\u003ePressure increase calculated by three EOSs coupled to realizable k-ɛ turbulence model and comparison with literature data [15, 16, 19, 27-30].\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-5012331/v1/6edd762009350a2ab7e0fec2.png"},{"id":68556712,"identity":"ac1c2071-a50c-4a68-8d5f-71579bf8b274","added_by":"auto","created_at":"2024-11-08 13:38:41","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":138983,"visible":true,"origin":"","legend":"\u003cp\u003eTemperature distribution with respect to turbulence models coupled to ML EOS at 30 s.\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-5012331/v1/81c30191373f2e4505a7f27f.png"},{"id":68556710,"identity":"3229b87f-6ff9-46df-aa24-9b889cfe0ea7","added_by":"auto","created_at":"2024-11-08 13:38:41","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":25038,"visible":true,"origin":"","legend":"\u003cp\u003eMean temperature deviations with respect to turbulence models coupled to ML EOS.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-5012331/v1/371e923cc2717bfdab458852.png"},{"id":68556707,"identity":"8ae56868-f6d0-4fd0-9f88-a5a9042e6918","added_by":"auto","created_at":"2024-11-08 13:38:40","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":24360,"visible":true,"origin":"","legend":"\u003cp\u003ePressure deviations with respect to turbulence models coupled to ML EOS.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-5012331/v1/7a618a97cc0d7b9caafeffe4.png"},{"id":68556703,"identity":"00edfdf0-9964-4152-8a00-6b23b54f4beb","added_by":"auto","created_at":"2024-11-08 13:38:40","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":48452,"visible":true,"origin":"","legend":"\u003cp\u003eTemperatures along the centerline with relative length of the axis at (a) 13 s, (b) 26 s, and (c) 39 s.\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-5012331/v1/282f2b7da1613f690976c97f.png"},{"id":68556721,"identity":"06090cca-d6f4-42c5-9310-718ceaccd399","added_by":"auto","created_at":"2024-11-08 13:38:43","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":66055,"visible":true,"origin":"","legend":"\u003cp\u003eRadial temperature distribution comparison of turbulence models with experimental data [15, 16, 19, 27-30] at 20 s; (a) V1 (x/L @ 0.2), (b) V2 (x/L @ 0.45), and (c) V3 (x/L @ 0.85) axial position (see Fig. 2).\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-5012331/v1/0340dae1408d58262b2244f6.png"},{"id":68556713,"identity":"4f2e2b02-672a-44f3-a7b4-74bb15d393aa","added_by":"auto","created_at":"2024-11-08 13:38:41","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":27008,"visible":true,"origin":"","legend":"\u003cp\u003eAverage turbulent kinetic energy of the two-equation models.\u003c/p\u003e","description":"","filename":"11.png","url":"https://assets-eu.researchsquare.com/files/rs-5012331/v1/7f51ba2984f27bdeac63f18f.png"},{"id":68556715,"identity":"9fb31615-28f4-4d6a-93fa-5cd54e550b32","added_by":"auto","created_at":"2024-11-08 13:38:42","extension":"png","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":202653,"visible":true,"origin":"","legend":"\u003cp\u003eStreamline and turbulent kinetic energy inside a tank.\u003c/p\u003e","description":"","filename":"12.png","url":"https://assets-eu.researchsquare.com/files/rs-5012331/v1/e0bac72e016e52834f892f27.png"},{"id":68557789,"identity":"5dcfbc75-8dd9-428b-ad27-6e5d0aed6419","added_by":"auto","created_at":"2024-11-08 13:46:40","extension":"png","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":257860,"visible":true,"origin":"","legend":"\u003cp\u003eTemperature rises inside a tank at different radial positions; (a) y/r = 0.25, (b) y/r = 0.5, and (c) y/r = 0.75.\u003c/p\u003e","description":"","filename":"13.png","url":"https://assets-eu.researchsquare.com/files/rs-5012331/v1/2ce0005b0b867437a06df75e.png"},{"id":81051011,"identity":"0aecafb0-fedd-4f0f-82fb-409f48819783","added_by":"auto","created_at":"2025-04-21 16:09:46","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1704004,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5012331/v1/9709078c-d6d7-418a-976c-16ff41e8fafb.pdf"}],"financialInterests":"","formattedTitle":"Application of EOS based on machine learning method on CFD study of rapid hydrogen refueling process","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eThe world has long been heavily dependent on traditional fossil energy sources. Harmful gases and greenhouse gases emitted by excessive use of the fossil fuels have been accumulated in the atmosphere, causing climate change and health problems around the world. The need for alternative energy sources becomes increasingly urgent to alleviate the deepening energy crisis that could result from the limited fossil fuel resources and to sustain international efforts to restrict harmful gas emissions to slow down climate change. Hydrogen is attracting attention as a clean fuel due to its recyclable nature and non-polluting properties and it is considered as a more efficient fuel than hydrocarbon fossil fuels due to its high energy density, high thermal conversion efficiency, and low calorific value. Therefore, hydrogen is expected to play an important role in the energy transition policy of replacing fossil fuels with renewable energy sources [\u003cspan additionalcitationids=\"CR2 CR3\" citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eHydrogen can be directly converted into energy by combustion, but the energy efficiency of converting chemical energy of hydrogen into electrical energy using fuel cells is relatively high, so researches on hydrogen energy are closely related to fuel cell technologies [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. Fuel cells can produce electric power over a fairly wide range, from a few Watts to hundreds of kilo-Watts, and are already at the stage of commercialization in three applications: small and portable energy markets, transportation, and stationary power systems. The most notable and promising application field for fuel cells is the transportation sector which includes cars, buses, trains, boats, aerial vehicles, and so on [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eHydrogen storage methods are selected according to the different requirements of each application [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]. In the transportation sector, the compressed gas storage method is commonly adopted due to its technical simplicity, high reliability, efficiency, and acceptable economics [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. The storage of gaseous hydrogen at high pressure is not only a well-developed technology, but also provides fast hydrogen filling and release rates [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]. For a hydrogen storage tank to be mounted on a hydrogen fuel cell vehicle, safety is of utmost importance, and the weight and size of the storage system should be as small as possible. As fuel cell electric vehicles become commercially available, most vehicles use pressure vessels for on-board hydrogen storage. At a given temperature, the density of hydrogen can be increased by rising pressure. Therefore, to achieve higher storage densities, storage vessels are manufactured to withstand very high pressure. Considering the energy required for compression, driving range of vehicles, investment in refueling infrastructure and other factors, the appropriate pressure for an on-board hydrogen system is known to be 35 to 70 MPa [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e] since the density of hydrogen increases fast at 30\u0026ndash;40 MPa but changes very little when the pressure is greater than 70 MPa. To achieve these requirements, most vehicle applications use fully-wrapped vessels with a metallic liner (Type III) or a non-metallic liner (Type IV). In the process of fast refueling of hydrogen up to high storage pressure, the temperature rises due to thermodynamic phenomena. The temperature increase is an important issue on the hydrogen refueling process since it would not only reduce SOC (state of charge) but could also bring safety risks to the vessels. The temperature of on-board Type III and IV tanks is restricted to be lower than 85\u0026deg;C [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e] for securing thermal and mechanical safety of tank materials during the refueling process.\u003c/p\u003e \u003cp\u003eExperimental studies on high-pressure filling of hydrogen tanks were conducted under 35 MPa [\u003cspan additionalcitationids=\"CR16 CR17\" citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e] and 70 MPa [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e, \u003cspan additionalcitationids=\"CR20\" citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e] conditions, and several rapid refueling tests were carried out on various types and sizes of hydrogen tanks. However, the experiments require a lot of time and cost and involve risks due to high pressure conditions. Various theoretical researches have been conducted to supplement the limited experiments and understand the experimental results. These theoretical approaches can be classified into thermodynamic methods and CFD (computational fluid dynamic) methods. The thermodynamic models [\u003cspan additionalcitationids=\"CR23 CR24 CR25\" citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e] are mainly based on mathematical formulas which are the energy and mass conservation equations combined with an equation of state for real gases. The thermodynamic models are useful for analyzing the effects of parameters such as initial pressure, filling rate, inlet temperature, and ambient temperature on hydrogen temperature rise during refueling. CFD models [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e, \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e, \u003cspan additionalcitationids=\"CR28 CR29\" citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e] have been established based on turbulence model, real gas model and heat transfer model to predict the temperature distribution within a storage tank during a refueling process.\u003c/p\u003e \u003cp\u003eCFD method is a tool for analyzing thermal-fluid phenomena found during filling and emptying hydrogen tanks and is frequently used to supplement experiments and thermodynamic models. CFD makes it possible to investigate the behavior of all relevant variables such as temperature, density, pressure, and direction of flow at any point in the storage vessel throughout the refueling time. Recently, many CFD studies have been conducted because CFD models can add important information to understand related phenomena and provide insight into the refueling process at given experimental data. Important factors on the hydrogen refueling process can be obtained and analyzed from CFD models; the effect of liner material on temperature [\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e], the impact of hydrogen filling strategy on the maximum temperature [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e, \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e], and the impact of tank shape on the temperature during a filling process [\u003cspan additionalcitationids=\"CR33\" citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eCFD models should adopt an appropriate turbulence model and an equation of state to provide reliable prediction results. The Reynolds Averaged Navier-Stokes (RANS) equations have been numerically solved in CFD simulations on hydrogen filling with various turbulence models such as the standard k-ε model [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e, \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e, \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e], the modified k-ε models [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e, \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e, \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e, \u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e], and the k-ω SST model [\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e, \u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e39\u003c/span\u003e]. Suryan et al. [\u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e] carried out a comparative study on performance of turbulence models using the realizable k-ε model, the RNG k-ε model, the k-u SST model, and the Reynolds Stress Model and, then, recommended the realizable k-ε model by compromising between accuracy and computational costs.\u003c/p\u003e \u003cp\u003eIt is also very important to select an appropriate real gas equation of state (EOS) in CFD model describing hydrogen refueling process because the ideal gas EOS could not apply under high-pressure hydrogen filling conditions up to 35\u0026ndash;70 MPa due to great compressibility deviations from ideal gas behavior. Redlich-Kwong (RK) EOS was used in many CFD models [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e, \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e, \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e, \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e], and a comparative study [\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e41\u003c/span\u003e] of RK, Soave-Redlich-Kwong (SRK), Aungier-Redlich-Kwong (ARK), and Peng-Robinson (PR) EOS confirmed that RK EOS is the most accurate gas model.\u003c/p\u003e \u003cp\u003eIn the present study, rapid hydrogen filling experimental data are adopted to make the CFD model simple by applying an axial symmetry condition since the one of the main purposes is to find the proper combinations of EOS and turbulence model. Firstly, three EOSs are compared with experimental data on mean temperature and pressure. Then, five turbulence models are systematically analyzed based on the local temperature behaviors. A selected combination of an EOS and a turbulence model is further applied to investigate thermo-flow characteristics inside a tank.\u003c/p\u003e"},{"header":"2. Equation of State (EOS): Real Gas Models","content":"\u003cp\u003eAn equation of state (EOS) is defined as a thermodynamic equation relating state variables such as pressure, (specific) volume, or temperature, which describe the state of matter under a given set of physical conditions. The most well-known EOS is the ideal gas law, which is suitable for representing the behavior of gases at a hypothetical ideal state. The ideal gas law is roughly accurate for weakly polar gases at low pressures and moderate temperatures and becomes increasingly inaccurate at higher pressures and lower temperatures. The cubic EOSs, which originated from the van der Waals (vdW) EOS, have a relatively simple formula and have been widely used to reflect real gas behavior and predict phase equilibrium [\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e]. In general, commercial software includes EOSs which are widely used in many different industries handling fluids. Among various cubic EOSs, COMSOL Multiphysics provides SRK and PR equations as built-in functions to calculate thermodynamic properties of gases.\u003c/p\u003e \u003cp\u003eSRK EOS is a modification of the original RK EOS and expressed as follows.\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:P=\\frac{RT}{V-{b}_{SRK}}-\\frac{{a}_{SRK}}{V(V+{b}_{SRK})}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere, \u003cem\u003eP\u003c/em\u003e, \u003cem\u003eT\u003c/em\u003e, \u003cem\u003eV\u003c/em\u003e, and \u003cem\u003eR\u003c/em\u003e are pressure, temperature, specific volume, and gas constant, respectively. Two component dependent parameters (\u003cem\u003ea\u003c/em\u003e\u003csub\u003e\u003cem\u003eSRK\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003eSRK\u003c/em\u003e\u003c/sub\u003e) are obtained from critical temperature (\u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003ec\u003c/em\u003e\u003c/sub\u003e), critical pressure (\u003cem\u003eP\u003c/em\u003e\u003csub\u003e\u003cem\u003ec\u003c/em\u003e\u003c/sub\u003e), and acentric factor (\u003cem\u003eω\u003c/em\u003e) of a gas by using the following equations.\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:{a}_{SRK}=0.42748\\frac{{R}^{2}{T}_{c}^{2}}{{P}_{c}}\\times\\:{\\left[1+\\left(0.480+1.574\\omega\\:-0.176{\\omega\\:}^{2}\\right)\\times\\:\\left(1-\\sqrt{T/{T}_{c}}\\right)\\right]}^{2}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:{b}_{SRK}=0.08664\\frac{R{T}_{c}}{{P}_{c}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ePR EOS is also a cubic form similar to SRK EOS. It is frequently used in the simulation and optimization of chemical processes, particularly those involving hydrocarbons. The equation and the correlations of two component parameters are given as follows,\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\:P=\\frac{RT}{V-{b}_{PR}}-\\frac{{a}_{PR}}{V\\left(V+{b}_{PR}\\right)+{b}_{PR}(V-{b}_{PR})}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$\\:{a}_{PR}=0.45724\\frac{{R}^{2}{T}_{c}^{2}}{{P}_{c}}\\times\\:{\\left[1+\\left(0.37464+1.54226\\omega\\:-0.269926{\\omega\\:}^{2}\\right)\\times\\:\\left(1-\\sqrt{T/{T}_{c}}\\right)\\right]}^{2}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$\\:{b}_{PR}=0.07780\\frac{R{T}_{c}}{{P}_{c}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe critical temperature, the critical pressure, and the acentric factor of hydrogen are reported as 32.98 K, 12.93 bar, and \u0026minus;\u0026thinsp;0.217, respectively [\u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e43\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eThe cubic EOSs present the relationship between state variables (\u003cem\u003eP\u003c/em\u003e, \u003cem\u003eT\u003c/em\u003e, and \u003cem\u003eV\u003c/em\u003e) in the form of pressure\u0026thinsp;\u0026minus;\u0026thinsp;explicit equations. Therefore, in general, an iterative method is required to find the roots of the equations when calculating specific volume (or density) at a given temperature and pressure. This kind of approach demands a lot of time and cost, especially in CFD calculations.\u003c/p\u003e \u003cp\u003eRecently, a generic correlation equation of the \u003cem\u003en\u003c/em\u003e\u003csup\u003eth\u003c/sup\u003e order has been proposed in a polynomial expansion form with temperature and pressure for gaseous hydrogen [\u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e44\u003c/span\u003e]. The equation can be readily applied to estimate accurate values for various properties such as density, enthalpy, internal energy, and so on by selecting a corresponding set of coefficients [\u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e45\u003c/span\u003e]. As for density (\u003cem\u003eρ\u003c/em\u003e), the equation is expressed as follows,\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$\\:\\rho\\:=\\sum\\:_{i=0}^{n}\\sum\\:_{j=0}^{n-i}{a}_{ij}{T}^{i}{P}^{j}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere, the units of \u003cem\u003eρ\u003c/em\u003e, \u003cem\u003eT\u003c/em\u003e, and \u003cem\u003eP\u003c/em\u003e are mol/L, K, and MPa, respectively. In Eq.\u0026nbsp;(\u003cspan refid=\"Equ7\" class=\"InternalRef\"\u003e7\u003c/span\u003e) \u003cem\u003ea\u003c/em\u003e\u003csub\u003e\u003cem\u003eij\u003c/em\u003e\u003c/sub\u003e denotes a coefficient of the product of the \u003cem\u003ei\u003c/em\u003e\u003csup\u003eth\u003c/sup\u003e power of \u003cem\u003eT\u003c/em\u003e and the \u003cem\u003ej\u003c/em\u003e\u003csup\u003eth\u003c/sup\u003e power of \u003cem\u003eP\u003c/em\u003e. A set of coefficients has been determined to fit literature data [\u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e46\u003c/span\u003e] by machine learning (ML) method. Details of the equation such as the number of regressed data and the accuracy can be found elsewhere [\u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e44\u003c/span\u003e]. The correlation equation (Eq.\u0026nbsp;(\u003cspan refid=\"Equ7\" class=\"InternalRef\"\u003e7\u003c/span\u003e)) is referred to as ML EOS in this study. The third order (\u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;3) is sufficient and the coefficients are listed in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, which are applicable in the range of 223.15 K\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;373.15 K and 0.1 MPa\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eP\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;100.1 MPa.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eCoefficient \u003cem\u003ea\u003c/em\u003e\u003csub\u003e\u003cem\u003eij\u003c/em\u003e\u003c/sub\u003e of ML EOS\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003ej\u003c/em\u003e\u003c/p\u003e \u003cp\u003e\u003cem\u003ei\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.20463E\u0026thinsp;+\u0026thinsp;01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e7.93448E-01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-4.04683E-03\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e6.79659E-06\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-1.02106E-01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-1.54833E-03\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e5.97575E-06\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e2.82873E-04\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e8.13547E-07\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-2.56918E-07\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe lowest temperature of hydrogen dispensed from a hydrogen refueling station (HRS) is -40\u0026deg;C and the temperature of hydrogen in an on-board tank is limited to be lower than 85\u0026deg;C. The densities of hydrogen calculated by the three EOSs (SRK, PR, and ML) are compared in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e at two different temperatures within the acceptable hydrogen refueling temperature range (-40\u0026deg;C\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;85\u0026deg;C).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eCompared to literature data [\u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e46\u003c/span\u003e], PR EOS overestimates density and SRK EOS slightly underestimates over the entire pressure range. On the other hand, ML EOS accurately reproduces literature data. The accuracies of the three EOS are quantitatively calculated using the following mean absolute percentage error (MAPE) equation with 21 data (\u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;21) at respective temperature and summarized in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$$\\:MAPE=\\frac{100}{n}\\sum\\:_{i}^{n}\\left|\\frac{{\\rho\\:}_{lit,\\:i}-{\\rho\\:}_{EOS,\\:i}}{{\\rho\\:}_{lit,\\:i}}\\right|$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eMean absolute percentage error (MAPE) [%] of EOSs\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003e\u003cem\u003eT\u003c/em\u003e [\u0026deg;C]\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003ePR EOS\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eSRK EOS\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eML EOS\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e-40\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e5.3399\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.0298\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.1006\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c5\" namest=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e4.7685\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.9058\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.0692\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c5\" namest=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e40\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e4.2528\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.8303\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.0662\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c5\" namest=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e80\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.7900\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.7841\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.0570\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c5\" namest=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAverage\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e4.5380\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.8875\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.0733\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c5\" namest=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eIn the hydrogen refueling temperature range, ML EOS calculates the hydrogen density most accurately, and SRK EOS shows better accuracy than PR EOS in the two cubic EOSs.\u003c/p\u003e"},{"header":"3. Turbulence Models","content":"\u003cp\u003eSimulation models for hydrogen refueling require the solution of the governing equations describing an unsteady state. Navier-Stokes equations must be applied to mathematically express momentum balance, which are partial differential equations for the motion of viscous fluid substances. However, for turbulent flows the computational effort to solve the complete Navier-Stokes equations is very high, since even the smallest disturbances influence the solution due to the nonlinearity. Therefore, most of CFD programs are based on the Reynolds-averaged Navier-Stokes (RANS) equations, which still represent the flow of fluids accurately enough, but lead to acceptable computational times. The small turbulences are not resolved with RANS equations, but are considered by so-called turbulence models. Turbulence models allow the calculation of the mean flow without first calculating the full time-dependent flow field.\u003c/p\u003e \u003cp\u003eMany turbulence models have been developed and some are included in CFD programs. COMSOL Multiphysics also provides different turbulence models. Choosing an appropriate turbulence model is a crucial factor in accurately describing the fluid flow. In the present work, five models, which are usually adopted for engineering purposes, have been applied and compared to simulate hydrogen refueling process; algebraic yPlus, k-ε, realizable k-ε, low Reynolds number k-ε, and k-ω model.\u003c/p\u003e \u003cp\u003eThe algebraic yPlus turbulence model is a zero-equation turbulence model based on the distance to the nearest wall. The model can be calculated directly from the flow variables since it does not require the solution of any additional equations. Consequently, it may not be able to properly account for history effects on the turbulence, such as convection and diffusion of turbulent energy. The model is too simple for use in general situations, but can be quite useful for initial phases of a computation in which a more complicated turbulence model may have difficulties.\u003c/p\u003e \u003cp\u003eThe two-equation turbulence models are the most widely applied class of turbulence models. Models such as the k-ε model and k-ω model have become industry standard models and are commonly used to solve most types of engineering problems on flows. The two-equation models are associated with two extra transport equations to describe the turbulent properties of the flow. It allows a two-equation model to represent history effects like convection and diffusion of turbulent energy. One of the transported variables is the turbulent kinetic energy, usually denoted as k. The second transported variable is deferent depending on type of two-equation models. The turbulent dissipation (ε) or the specific turbulence dissipation rate (ω) is commonly taken for the two-equation models. The first variable (k) determines the energy in the turbulence and the second one can be regarded as the variable that determines the length-scale or time-scale of the turbulence.\u003c/p\u003e \u003cp\u003eThe k-ε model is one of the most widely used and validated turbulence models used in CFD to simulate mean flow characteristics. It includes two extra transport equations to represent the turbulent kinetic (k) and the turbulent dissipation (ε). The standard k-ε model was developed to find an alternative to algebraically prescribing turbulent length scales in moderate to high complexity flows with improvement of the mixing-length model. The model has been known to be useful for free-shear layer flows with relatively small pressure gradients as well as in confined flows where the Reynolds shear stresses are most important.\u003c/p\u003e \u003cp\u003eBased on the standard k-ε model, new refined two-equation models have been actively developed for various areas of research. The realizable k-ε turbulence model is an extension to the standard k-ε model which provides improved predictions for flows involving rotation, boundary layers under strong adverse pressure gradients, separation, and recirculation. The realizable k-ɛ model differs from the standard k-ɛ model in two ways. It is associated with a new formulation for the turbulent viscosity and employs a new transport equation for the dissipation rate, which is derived from an exact equation for the transport of the mean-square vorticity fluctuation. The low Reynolds number k-ε model is a basically modification to the standard k-ε model to make it applicable to near-wall conditions. The model integrates through the viscous sub-layer down to the wall and uses turbulent transport equations that are applicable throughout the boundary layer, including the buffer and viscous sublayers. It is still only applicable to high Reynolds number flows where the flow is fully turbulent. In comparison with the wall functions, the low-Reynolds methods have the advantage of resolving the boundary layer details, but the advantage comes at the expense of significantly more computations.\u003c/p\u003e \u003cp\u003eThe k-ω model is the second most widely applied two-equation model next to the standard k-ε model. The model has been demonstrated to improve the k-ε model for many situations, including turbulent boundary layers with zero or adverse pressure gradients and even near-separation conditions. The model attempts to predict turbulence by two partial differential equations for two variables, k and ω, with the first variable being the turbulence kinetic energy while the second is the specific rate of dissipation.\u003c/p\u003e \u003cp\u003eIn the present work, the five turbulence models have been applied without further modifications on the formulations provided by COMSOL Multiphysics software. More details of the theoretical background on the turbulence models and derivations of the equations can be found elsewhere [\u003cspan citationid=\"CR47\" class=\"CitationRef\"\u003e47\u003c/span\u003e].\u003c/p\u003e"},{"header":"4. Computational Model and Implementation","content":"\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e4.1. Governing equations\u003c/h2\u003e \u003cp\u003eAs employed in most of flow field analyses, the hydrogen refueling process is also characterized by balance in mass, momentum, and total energy in the unsteady form described by the continuity equation (Eq.\u0026nbsp;(\u003cspan refid=\"Equ9\" class=\"InternalRef\"\u003e9\u003c/span\u003e)), the Navier-Stokes equations (Eq.\u0026nbsp;(\u003cspan refid=\"Equ10\" class=\"InternalRef\"\u003e10\u003c/span\u003e)), and the total energy equation (Eq.\u0026nbsp;(\u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e11\u003c/span\u003e)), respectively.\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$$\\:\\frac{\\partial\\:\\rho\\:}{\\partial\\:t}+\\nabla\\:\\bullet\\:\\left(\\rho\\:\\overrightarrow{u}\\right)=0$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ10\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ10\" name=\"EquationSource\"\u003e\n$$\\:\\frac{\\partial\\:\\rho\\:\\overrightarrow{u}}{\\partial\\:t}+\\nabla\\:\\bullet\\:\\left(\\rho\\:\\overrightarrow{u}\\overrightarrow{u}\\right)=-\\nabla\\:p+\\nabla\\:\\bullet\\:\\stackrel{̿}{\\tau\\:}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ11\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ11\" name=\"EquationSource\"\u003e\n$$\\:\\frac{\\partial\\:}{\\partial\\:t}\\left[\\rho\\:\\left(e+\\frac{1}{2}{u}^{2}\\right)\\right]+\\nabla\\:\\bullet\\:\\left[\\rho\\:\\overrightarrow{u}\\left(e+\\frac{1}{2}{u}^{2}\\right)\\right]=\\nabla\\:\\bullet\\:\\left({k}_{t}\\nabla\\:T\\right)+\\nabla\\:\\bullet\\:\\left(-p\\overrightarrow{u}+\\stackrel{̿}{\\tau\\:}\\bullet\\:\\overrightarrow{u}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\rho\\:\\)\u003c/span\u003e\u003c/span\u003e is density of fluid, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\overrightarrow{u}\\)\u003c/span\u003e\u003c/span\u003e is velocity, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\stackrel{̿}{\\tau\\:}\\)\u003c/span\u003e\u003c/span\u003e is stress tensor, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:e\\)\u003c/span\u003e\u003c/span\u003e is specific internal energy of fluid, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{k}_{t}\\)\u003c/span\u003e\u003c/span\u003e is thermal conductivity. The gravitational force and the potential energy change are ignored in Eqs.\u0026nbsp;(\u003cspan refid=\"Equ10\" class=\"InternalRef\"\u003e10\u003c/span\u003e) and (\u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e11\u003c/span\u003e), respectively, considering the horizontal flow of hydrogen.\u003c/p\u003e \u003cp\u003eThe solution to the governing equations gives the velocity field (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\overrightarrow{u}\\)\u003c/span\u003e\u003c/span\u003e), pressure (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:p\\)\u003c/span\u003e\u003c/span\u003e), and temperature, (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:T\\)\u003c/span\u003e\u003c/span\u003e) of the fluid in the modeled domain when combined with the following equations.\u003cdiv id=\"Equ12\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ12\" name=\"EquationSource\"\u003e\n$$\\:e=h-\\frac{p}{\\rho\\:}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e12\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ13\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ13\" name=\"EquationSource\"\u003e\n$$\\:\\stackrel{̿}{\\tau\\:}=\\mu\\:\\left[\\nabla\\:\\overrightarrow{u}-\\frac{2}{3}\\nabla\\:\\bullet\\:\\overrightarrow{u}I\\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e13\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:h\\)\u003c/span\u003e\u003c/span\u003e is specific enthalpy, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mu\\:\\)\u003c/span\u003e\u003c/span\u003e is dynamic viscosity, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:I\\)\u003c/span\u003e\u003c/span\u003e is the unit tensor.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e4.2. Geometry and specifications of hydrogen storage tank\u003c/h2\u003e \u003cp\u003eThe objective of the present work is to find out appropriate real gas model and turbulence model. Therefore, we selected experimental data which reported temperature distribution inside a tank under a fast-filling condition. Experimental data [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e, \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e] from a test facility installed with a Type III hydrogen storage cylinder of which internal volume is 74 liters are used to validate CFD models. The dimensions of the tank were 39.9 cm of external diameter and 90 cm of length. A tree-shaped support with 63 thermocouples was inserted into the cylinder to measure the hydrogen gas temperature with respect to radial and axial position as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe liner is made of aluminum and the laminate is composed of CFRP (carbon fiber reinforced plastic). The specifications of tank materials are summarized in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eSpecifications of tank materials\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eThickness\u003c/p\u003e \u003cp\u003e[m]\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eSpecific heat [J/kg\u0026middot;K]\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eThermal conductivity [W/m\u0026middot;K]\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eDensity [kg/m\u003csup\u003e3\u003c/sup\u003e]\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLiner\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.004\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e900\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e167\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2730\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLaminate\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.015\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e938\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1494\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e4.3. CFD model\u003c/h2\u003e \u003cp\u003eA dimensional analysis carried out for the experimental conditions [\u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e] revealed that the Froude number (Fr) is greater than 30 during a process of filling hydrogen into a 74 L tank up to 35 MPa within 40 s. The Froude number is defined as the ratio of the flow inertia to the external field and, thus, the high Fr indicates the motion of fluid is not affected by gravitational force in case of a horizontal filling. For this reason, a CFD model was set up under an assumption that the flow within the tank is axisymmetric with respect to the centerline of the cylinder [\u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e, \u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eA structured mesh for a hydrogen storage tank was generated as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e by using a mesh creating tool embedded in COMSOL Multiphysics. The quality of grid was set to \u0026lsquo;extra fine\u0026rsquo; as predefined by the software. The mesh was denser near the walls and inlet tube inserted into a tank to capture the flow phenomena at fluid boundaries. The total number of cells was 5,229 which was much greater than the minimum value (=\u0026thinsp;1,200) enough to obtain reliable results for the same system [\u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e]. The grid presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e was used to produce all the results provided in this study.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAs initial conditions, pressure and temperature of the gas were set to be uniform with values of 9.3 MPa and 293.4 K respectively reflecting experimental data. The heat transfer coefficient between the tank outside wall and the ambient was assumed to 10 W/(m\u003csup\u003e2\u003c/sup\u003e\u0026middot;K). The experiments to be used for the CFD model validation were performed under changing inlet pressure and temperature with hydrogen filling time [\u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e] because a dispenser controlled the rate of pressure rise within the cylinder through a pressure control valve. Therefore, the experimental inlet pressure and temperature were curve fitted and, then, put into the CFD model for the present work. No-slip condition was applied at the tank inside wall to solve the governing equations.\u003c/p\u003e \u003c/div\u003e"},{"header":"5. Results and Discussion","content":"\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e5.1. Effect of EOS selection on CFD calculation\u003c/h2\u003e \u003cp\u003eIn this study, three EOSs and five turbulence models are taken into consideration for CFD calculations on hydrogen filling process. The effect of EOS selection was investigated with the realizable k-ɛ turbulence model since it was revealed that the model is robust and gives reasonable accuracy for turbulent internal flow with heat transfer [\u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eAn experimental study [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e] presented the volume average gas temperature inside a tank which was calculated by Eq.\u0026nbsp;(\u003cspan refid=\"Equ14\" class=\"InternalRef\"\u003e14\u003c/span\u003e) using the measured temperature (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{T}_{k}\\)\u003c/span\u003e\u003c/span\u003e) at 63 points (see Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e).\u003c/p\u003e\u003cdiv id=\"Equ14\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ14\" name=\"EquationSource\"\u003e\n$$\\:{T}_{mean}=\\frac{{\\sum\\:}_{k=1}^{63}{y}_{k}{T}_{k}{A}_{k}}{{\\sum\\:}_{k=1}^{63}{y}_{k}{A}_{k}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e14\u003c/div\u003e\u003c/div\u003e\u003cp\u003e\u003c/p\u003e \u003cp\u003ewhere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{y}_{k}\\)\u003c/span\u003e\u003c/span\u003e is the vertical distance of the thermocouple \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:k\\)\u003c/span\u003e\u003c/span\u003e to the centerline of the cylinder and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{A}_{k}\\)\u003c/span\u003e\u003c/span\u003e is an area weighting given to each thermocouple.\u003c/p\u003e \u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e, all the three EOSs well describe the temperature increase behavior during a process of hydrogen filling into a confined tank volume. However, compared with experimental data, ML EOS presents the most accurate result throughout the time of refueling. The deviations from the experimental data at the end of fill are calculated to 5.06, 3.02, and 2.11°C for PR EOS, SRK EOS, and ML EOS, respectively. As for pressure inside the tank, all the three EOSs correctly follow the measured data and the pressure against time curves obtained from the simulation results are nearly superimposed as given in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eComparison of the time required for a simulation is an important consideration in model selection along with the accuracy comparison since the computation time is recognized as a cost in simulation works. The times spent for simulations carried out using the three EOSs are compared in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e which presents the relative times to ML EOS since the actual absolute computation time is determined by computational capability.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eRelative time for simulation with EOSs\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e\u003ccolgroup cols=\"4\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSRK EOS\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003ePR EOS\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eML EOS\u003c/p\u003e \u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSimulation time (Relative)\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.261\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.986\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e \u003cp\u003e\u003c/p\u003e \u003cp\u003eML EOS is shown to simulate the hydrogen refueling process most accurately in the shortest time. Such a result is due to its simple formula compared to other real gas models, which is suitable for simulation. It is expected that the fast and accurate feature can be more important in models of large-capacity tank filling that require more mesh elements and longer simulation times.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e5.2. CFD calculations with different turbulence models\u003c/h2\u003e \u003cp\u003eML EOS has been proved to be suitable for the simulation of hydrogen filling process in terms of accuracy and computational time. Therefore, in the following study, we compared the results of applying various turbulence models associated with ML EOS to analyze the effect of the turbulence model selection.\u003c/p\u003e \u003cp\u003eThe temperature distributions in a tank are shown with respect to turbulence models in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e. The algebraic yPlus model presents a high temperature at the front of the tank, and the temperature difference between the front and rear is evaluated to be relatively large. The other four models show small temperature deviations inside a tank except for the centerline where pre-cooled hydrogen is rapidly introduced. The overall temperature level is the highest for the low Reynolds model, and quantitative analysis was carried out by comparison with experimental data.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eUnlike the case of EOS comparison, the simulation results of temperature increase present quite different behavior according to turbulence models as presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e which is drawn to exhibit deviations of mean temperature (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\Delta\\:}{T}_{mean}={T}_{mean}^{cal}-{T}_{mean}^{exp}\\)\u003c/span\u003e\u003c/span\u003e) between calculated values (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{T}_{mean}^{cal}\\)\u003c/span\u003e\u003c/span\u003e) and experimental data (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{T}_{mean}^{exp}\\)\u003c/span\u003e\u003c/span\u003e which can be found in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e) during the filling process. The algebraic yPlus model initially shows lower temperatures than the experimental values (negative \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\Delta\\:}{T}_{mean}\\)\u003c/span\u003e\u003c/span\u003e), but after about 10 s, it tends to show higher temperatures than the experimental data. The other models predict higher temperatures than the literature data throughout the process. Based on the final temperature, the errors are decreasing in the order of the low Reynolds k-ɛ, the algebraic yPlus, the k-ɛ, the k-ω, and the realizable k-ɛ model. The realizable k-ε model was found to most accurately fit the experimental data, with the deviations stably remaining within 2°C throughout the process after the initial approximately 5 s.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe accuracies of calculated hydrogen pressures in the vessel are compared in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\Delta\\:}P\\)\u003c/span\u003e\u003c/span\u003e is defined as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{P}^{cal}-{P}^{exp}\\)\u003c/span\u003e\u003c/span\u003e where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{P}^{exp}\\)\u003c/span\u003e\u003c/span\u003e behavior has been given in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e. All the models except for the algebraic yPlus model show satisfactory accordance with the experimental data and no significant differences could be found between the models after 10 s. The pressure behavior predicted by the algebraic yPlus model is lower than the other models at the early stage, but the deviation diminishes toward the end of the filling. All models fit the experimental data within 0.5 MPa of the final pressure. However, the algebraic yPlus model still shows a large error compared to the other models at the end of the experiment.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe measured temperatures at the cylinder axis (denoted as H0 in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) are co-plotted with simulation results in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e. The x-axis of Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e means relative position from the inlet along the centerline and the temperatures are presented at the y-axis as deviations from the mean temperature. The changes over time are presented as Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e(a), (b), and (c), which show the temperatures at 13 s, 26 s, and 39 s, respectively.\u003c/p\u003e \u003cp\u003eIn practice, the temperature of the gas is measured at the centerline of a vessel for commercial hydrogen storage tank. The experimental data in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e proposed that the accurate temperature could be measured when a sensor is extended to a position of 70% of the tank length. However, the proper position could be different according to the dimensions of a vessel and the process conditions such as a mass flow rate. The temperature at the front of the tank where the pre-cooled hydrogen flows in was the lowest, and the temperatures along the centerline were found to be lower than the mean temperature throughout the filling process.\u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e shows that the k-ε model reproduces the most similar results to the experimental data throughout the entire position along the axis during the filling time. The low Reynolds k-ε model also presents good agreements with the experimental data. However, considering the accuracy on the mean temperature behavior (see Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e), the realizable k-ε model is quite compatible with the k-ε model.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe hydrogen rapid filling investigation [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e, \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e, \u003cspan additionalcitationids=\"CR28 CR29\" citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e–\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e] has reported temperature measurement data with respect to positions in a vessel. The data were used to compare the turbulence model results on radial temperature distribution at different axial positions which are denoted as V1 (x/L ≅ 0.2), V2 (x/L ≅ 0.45), and V3 (x/L ≅ 0.85) in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e(a), (b), and (c), respectively. Temperatures are presented as relative values to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{T}_{mean}\\)\u003c/span\u003e\u003c/span\u003e at the time of the measurement (t = 20 s) in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eWhen analyzing the experimental data, it was found that the front of the tank where pre-cooled hydrogen flows in at the centerline has a low temperature and temperature difference on temperature distribution has been observed between the upper and lower regions of a tank. On the other hand, as the fluid flows deep into the tank, the temperature distribution between the upper and lower regions at V2 and V3 became symmetric with respect to the centerline. The model calculations at V1 were not satisfactory since the temperature behavior changes very rapidly at near centerline and the CFD calculations have been carried out by using an axial symmetry model. However, the calculation results at V2 and V3 reproduce the experimental values accurately. Among the adopted turbulence models, the k-ε and the realizable k-ε models show better performance than the other models over the entire region.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe two-equation models (k-ε, realizable k-ε, low Reynolds k-ε, and k-ω) commonly adopted the turbulent kinetic energy (k). The average values of k with time are compared in Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e11\u003c/span\u003e. The low Reynolds k-ε model exhibits very low values of k. It is understood that the model is not suitable since the mathematical formulation of the model is capable to resolve the turbulence flow down to the wall, but the hydrogen filling process associated with a circulating flow inside a tank.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe computational time for a simulation has been summarized in Table\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e as relative values to the time spent applying the realizable k-ε model. The algebraic yPlus model offers calculation result very fast. The k-ε model requires the shortest time among the two-equation models and the realizable k-ε model takes compatible time with the k-ε model.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab5\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eRelative time for simulation with turbulence models\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e\u003ccolgroup cols=\"6\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eAlgebraic\u003c/p\u003e \u003cp\u003eyPlus\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003ek-ε\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eRealizable\u003c/p\u003e \u003cp\u003ek-ε\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eLow Reynolds\u003c/p\u003e \u003cp\u003ek-ε\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003ek-ω\u003c/p\u003e \u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSimulation time (Relative)\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.066\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.910\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e2.037\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e5.036\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e \u003cp\u003e\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e5.3. Thermo-flow analysis inside a tank\u003c/h2\u003e \u003cp\u003eThe flow inside a tank is closely related to the temperature rise of the fluid. However, it is very difficult to measure flow properties such as the velocity of fluid in the tank, and experimental data generally present only the temperature of a fluid. The CFD model is used to obtain properties that are difficult to measure experimentally because it can simulate thermo-flow phenomena.\u003c/p\u003e \u003cp\u003eThe most significant data to be monitored during the hydrogen filling process is the average temperature of the tank because the maximum temperature is limited to ensure the safety of the tank storing highly compressed gas. Therefore, the realizable k-ε model is used to analyze the flow inside a tank, which shows the least deviation from the experimental data (see Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThe streamline and the turbulent kinetic energy inside a tank are given in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e with time. The introduced flow directly touches the back of the tank since the tank is rapidly filled by a high-speed fluid, and then the flow moves along the tank wall to form a rotational motion. Throughout the filling process, the center of rotation is located in the upper part of the back. The turbulent kinetic energy is found to be the greatest on the back side of the tank center, where the fluid flows upward after hitting the tank wall, and observes large in the area where the rotational flow occurs along the streamline. The magnitude of the turbulent kinetic energy decreases with time, which can be recognized by the range shown in the scale bar in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e. Therefore, it is found that the contribution of energy due to turbulent dissipation to temperature rise becomes smaller as the process proceeds.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe temperature increases along an axis at different radial positions are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e. The temperature rises with time at all locations, but, instantly, the temperature does not rise monotonically due to fluctuations of local flow. The temperature differences in the axial direction are larger with increasing distance from the central axis (longer radial position) since the extent of fluid mixing is not instant as a consequence of the decreased flow velocity near the tank wall. It is observed that the maximum temperature is found at the upper region of tank backside throughout the filling process. Therefore, the temperature during commercial tank filling processes should be controlled considering the deviation of the monitored temperature and the actual maximum temperature, which could be different according to filling conditions such as initial tank pressure, process time, and ambient temperature.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"Conclusions","content":"\u003cp\u003eThe temperature increases of the gas during the high-pressure hydrogen filling process is a thermo-physical phenomenon that inevitably takes place for rapid refueling. However, the temperature rise of hydrogen should be limited to maintain the physical safety of tanks which store highly-compressed gas for usually fuel cell electric vehicles (FCEVs). Since hydrogen refueling can be carried out under various conditions such as ambient temperature, initial pressure, and pre-cooling level, the safety of the hydrogen storage tank must be ensured under all conditions.\u003c/p\u003e\u003cp\u003eCFD research is an important tool that can complement experimental researches. It could be used to theoretically interpret experimental results and reduce costs by reducing the number of experiments. The equation of state (EOS) which can explain the volumetric physical properties of a fluid and the turbulence model which simulates flow characteristics are the basic equations that make up a CFD model, and the reliability of CFD results can be secured only when they are properly combined. In the present study, an appropriate combination of EOS and turbulence model is explored, and the thermos-flow behavior of the hydrogen storage tank is investigated by a CFD applying a selected combination.\u003c/p\u003e\u003cp\u003eAs a result of comparing the three EOSs (SRK, PR, and ML) using the experimental results that fill the hydrogen up to 35 MPa, the ML EOS shows the best accuracy and significantly reduced calculation time. The ML EOS, which is in the form of a polynomial equation, has been shown to be suitable for accuracy and computational cost reduction in CFD model studies that require a lot of iterative computation.\u003c/p\u003e\u003cp\u003eThe performance of five turbulence models (one zero-equation model and four two-equation models) are compared for the reproduction of local temperatures as well as mean temperature behavior. As for the mean temperature behavior, the realizable k-ε model is found to be the most accurate. The k-ε model presents the best accuracy for local temperatures and the realizable k-ε model also shows satisfactory results compatible with the results of the k-ε model. In terms of calculation time, the yPlus model which is the zero-equation model takes the least time, but it presents the lowest accuracy. Therefore, the yPlus model is considered to be suitable only for initial approach, such as examine on the quality of the structured mesh of a CFD model. In the four two-equation models, the low Reynolds k-ε model is found to show very low average turbulent kinetic energy unlike the other models. It is understood that the mathematical formulation of the low Reynolds k-ε model is not suitable for the tank filling process associated with a circulating flow inside a tank.\u003c/p\u003e\u003cp\u003eA CFD model composed of the ML EOS and the realizable k-ε turbulence model is used to analyze the inside flow and corresponding local temperature behavior. It is found that the maximum temperature tanks place at the upper region of tank backside due to the turbulence energy dissipation.\u003c/p\u003e\u003cp\u003eIn this study, experimental data on rapid refueling of hydrogen into a storage tank are used to simplify comparative studies by applying axial symmetry assumption. Therefore, further study would be required for extended refueling experiments in which axial symmetry assumptions are not valid due to temperature stratification.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAcknowledgment\u003c/h2\u003e \u003cp\u003eThis research was supported by the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Ministry of Trade, Industry \u0026amp; Energy (MOTIE) of the Republic of Korea (Project No. 20227310100060). It was also supported by the Korea Agency for Infrastructure Technology Advancement (KAIA) and the Ministry of Land, Infrastructure and Transport (MOLIT) (Project No. 21OHTI-C163280-01).\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eS. Sharma, S. Agarwal and A. Jain, Energies, \u003cstrong\u003e14\u003c/strong\u003e, 7389 (2021).\u003c/li\u003e\n\u003cli\u003eN. Mac Dowell, N. Sunny, N. Brandon, H. Herzog, A. Y. Ku, W. Maas, A. Ramirez, D. M. Reiner, G. N. Sant and N. Shah, Joule, \u003cstrong\u003e5\u003c/strong\u003e, 2524 (2021).\u003c/li\u003e\n\u003cli\u003eA. Kovač, M. Paranos and D. Marciu\u0026scaron;, International Journal of Hydrogen Energy, \u003cstrong\u003e46\u003c/strong\u003e, 10016 (2021).\u003c/li\u003e\n\u003cli\u003eJ. Cader, R. Koneczna and P. Olczak, Energies, \u003cstrong\u003e14\u003c/strong\u003e, 4811 (2021).\u003c/li\u003e\n\u003cli\u003eM. K. Singla, P. Nijhawan and A. S. Oberoi, Environmental Science and Pollution Research, \u003cstrong\u003e28\u003c/strong\u003e, 15607 (2021).\u003c/li\u003e\n\u003cli\u003eA. Alaswad, A. Baroutaji, H. Achour, J. Carton, A. Al Makky and A.-G. Olabi, International Journal of Hydrogen Energy, \u003cstrong\u003e41\u003c/strong\u003e, 16499 (2016).\u003c/li\u003e\n\u003cli\u003eH. Yang, Y. J. Han, J. Yu, S. Kim, S. Lee, G. Kim and C. Lee, Sustainability, \u003cstrong\u003e14\u003c/strong\u003e, 917 (2022).\u003c/li\u003e\n\u003cli\u003eM. R. Usman, Renewable Sustainable Energy Rev., \u003cstrong\u003e167\u003c/strong\u003e, 112743 (2022).\u003c/li\u003e\n\u003cli\u003eJ. Zheng, X. Liu, P. Xu, P. Liu, Y. Zhao and J. Yang, Int. J. Hydrogen Energy, \u003cstrong\u003e37\u003c/strong\u003e, 1048 (2012).\u003c/li\u003e\n\u003cli\u003eE. Rivard, M. Trudeau and K. Zaghib, Materials, \u003cstrong\u003e12\u003c/strong\u003e, 1973 (2019).\u003c/li\u003e\n\u003cli\u003eD. Durbin and C. Malardier-Jugroot, Int. J. Hydrogen Energy, \u003cstrong\u003e38\u003c/strong\u003e, 14595 (2013).\u003c/li\u003e\n\u003cli\u003eS. W. Jorgensen, Curr. Opin. Solid State Mater. Sci., \u003cstrong\u003e15\u003c/strong\u003e, 39 (2011).\u003c/li\u003e\n\u003cli\u003eR. Von Helmolt and U. Eberle, J. Power Sources, \u003cstrong\u003e165\u003c/strong\u003e, 833 (2007).\u003c/li\u003e\n\u003cli\u003eSAE, \u003cem\u003eFueling protocols for light duty gaseous hydrogen surface vehicles\u003c/em\u003e, SAE international (2020).\u003c/li\u003e\n\u003cli\u003eC. Dicken and W. Merida, J. Power Sources, \u003cstrong\u003e165\u003c/strong\u003e, 324 (2007).\u003c/li\u003e\n\u003cli\u003eL. Zhao, Y. Liu, J. Yang, Y. Zhao, J. Zheng, H. Bie and X. Liu, Int. J. Hydrogen Energy, \u003cstrong\u003e35\u003c/strong\u003e, 8092 (2010).\u003c/li\u003e\n\u003cli\u003eJ. Zheng, J. Guo, J. Yang, Y. Zhao, L. Zhao, X. Pan, J. Ma and L. Zhang, Int. J. Hydrogen Energy, \u003cstrong\u003e38\u003c/strong\u003e, 10956 (2013).\u003c/li\u003e\n\u003cli\u003eM. Monde, Y. Mitsutake, P. L. Woodfield and S. Maruyama, Heat Transfer\u0026mdash;Asian Research: Co‐sponsored by the Society of Chemical Engineers of Japan and the Heat Transfer Division of ASME, \u003cstrong\u003e36\u003c/strong\u003e, 13 (2007).\u003c/li\u003e\n\u003cli\u003eS. C. Kim, S. H. Lee and K. B. Yoon, Int. J. Hydrogen Energy, \u003cstrong\u003e35\u003c/strong\u003e, 6830 (2010).\u003c/li\u003e\n\u003cli\u003eR. O. Cebolla, B. Acosta, N. De Miguel and P. Moretto, Int. J. Hydrogen Energy, \u003cstrong\u003e40\u003c/strong\u003e, 4698 (2015).\u003c/li\u003e\n\u003cli\u003eM. C. Galassi, D. Baraldi, B. A. Iborra and P. Moretto, Int. J. Hydrogen Energy, \u003cstrong\u003e37\u003c/strong\u003e, 6886 (2012).\u003c/li\u003e\n\u003cli\u003eH. Tun, K. Reddi, A. Elgowainy and S. Poudel, International Journal of Hydrogen Energy, \u003cstrong\u003e48\u003c/strong\u003e, 28869 (2023).\u003c/li\u003e\n\u003cli\u003eJ.-Q. Li, J.-C. Li, K. Park, S.-J. Jang and J.-T. Kwon, Energies, \u003cstrong\u003e14\u003c/strong\u003e, 2635 (2021).\u003c/li\u003e\n\u003cli\u003eT. Kuroki, K. Nagasawa, M. Peters, D. Leighton, J. Kurtz, N. Sakoda, M. Monde and Y. Takata, Int. J. Hydrogen Energy, \u003cstrong\u003e46\u003c/strong\u003e, 22004 (2021).\u003c/li\u003e\n\u003cli\u003eR. Caponi, A. M. Ferrario, E. Bocci, G. Valenti and M. Della Pietra, Int. J. Hydrogen Energy, \u003cstrong\u003e46\u003c/strong\u003e, 18630 (2021).\u003c/li\u003e\n\u003cli\u003eL. Zhao, F. Li, Z. Li, L. Zhang, G. He, Q. Zhao, J. Yuan, J. Di and C. Zhou, Int. J. Hydrogen Energy, \u003cstrong\u003e44\u003c/strong\u003e, 3993 (2019).\u003c/li\u003e\n\u003cli\u003eM. Heitsch, D. Baraldi and P. Moretto, Int. J. Hydrogen Energy, \u003cstrong\u003e36\u003c/strong\u003e, 2606 (2011).\u003c/li\u003e\n\u003cli\u003eD. Melideo, D. Baraldi, B. Acosta-Iborra, R. O. Cebolla and P. Moretto, Int. J. Hydrogen Energy, \u003cstrong\u003e42\u003c/strong\u003e, 7304 (2017).\u003c/li\u003e\n\u003cli\u003eJ. Liu, S. Zheng, Z. Zhang, J. Zheng and Y. Zhao, International Journal of Hydrogen Energy, \u003cstrong\u003e45\u003c/strong\u003e, 9241 (2020).\u003c/li\u003e\n\u003cli\u003eJ. Martin, Q. Nouvelot, V. Ren, G. Lodier, E. Vyazmina, F. Ammouri and P. Carrere, International Journal of Hydrogen Energy, \u003cstrong\u003e54\u003c/strong\u003e, 562 (2023).\u003c/li\u003e\n\u003cli\u003eM. Monde and M. Kosaka, SAE International Journal of Alternative Powertrains, \u003cstrong\u003e2\u003c/strong\u003e, 61 (2013).\u003c/li\u003e\n\u003cli\u003eQ. Li, J. Zhou, Q. Chang and W. Xing, Int. J. Hydrogen Energy, \u003cstrong\u003e37\u003c/strong\u003e, 6043 (2012).\u003c/li\u003e\n\u003cli\u003eT. Bourgeois, T. Brachmann, F. Barth, F. Ammouri, D. Baraldi, D. Melideo, B. Acosta-Iborra, D. Zaepffel, D. Saury and D. Lemonnier, Int. J. Hydrogen Energy, \u003cstrong\u003e42\u003c/strong\u003e, 13789 (2017).\u003c/li\u003e\n\u003cli\u003eV. Ramasamy and E. Richardson, Int. J. Heat Mass Transfer, \u003cstrong\u003e160\u003c/strong\u003e, 120179 (2020).\u003c/li\u003e\n\u003cli\u003eR. Immel and A. Mack-Gardner, SAE International Journal of Engines, \u003cstrong\u003e4\u003c/strong\u003e, 1850 (2011).\u003c/li\u003e\n\u003cli\u003eT. Johnson, R. Bozinoski, J. Ye, G. Sartor, J. Zheng and J. Yang, Int. J. Hydrogen Energy, \u003cstrong\u003e40\u003c/strong\u003e, 9803 (2015).\u003c/li\u003e\n\u003cli\u003eC. Dicken and W. Merida, Numerical Heat Transfer, Part A: Applications, \u003cstrong\u003e53\u003c/strong\u003e, 685 (2008).\u003c/li\u003e\n\u003cli\u003eY. Zhao, G. Liu, Y. Liu, J. Zheng, Y. Chen, L. Zhao, J. Guo and Y. He, Int. J. Hydrogen Energy, \u003cstrong\u003e37\u003c/strong\u003e, 17517 (2012).\u003c/li\u003e\n\u003cli\u003eT. Setoguchi, M. Alam, M. Monde and H. Kim, International Journal of Aeroacoustics, \u003cstrong\u003e12\u003c/strong\u003e, 455 (2013).\u003c/li\u003e\n\u003cli\u003eA. Suryan, H. D. Kim and T. Setoguchi, Int. J. Hydrogen Energy, \u003cstrong\u003e38\u003c/strong\u003e, 9562 (2013).\u003c/li\u003e\n\u003cli\u003eA. Suryan, H. D. Kim and T. Setoguchi, Int. J. Hydrogen Energy, \u003cstrong\u003e37\u003c/strong\u003e, 7600 (2012).\u003c/li\u003e\n\u003cli\u003eJ. O. Valderrama, Industrial \u0026amp; engineering chemistry research, \u003cstrong\u003e42\u003c/strong\u003e, 1603 (2003).\u003c/li\u003e\n\u003cli\u003eB. E. Poling, \u003cem\u003eThe properties of gases and liquids\u003c/em\u003e, (2004).\u003c/li\u003e\n\u003cli\u003eB. H. Park and C. K. Chae, Int. J. Hydrogen Energy, \u003cstrong\u003e47\u003c/strong\u003e, 4185 (2022).\u003c/li\u003e\n\u003cli\u003eB. H. Park and C. H. Joe, Int. J. Hydrogen Energy, \u003cstrong\u003e49\u003c/strong\u003e, 1140 (2024).\u003c/li\u003e\n\u003cli\u003eE. W. Lemmon, M. L. Huber and M. O. McLinden, NIST standard reference database, \u003cstrong\u003e23\u003c/strong\u003e, v7 (2002).\u003c/li\u003e\n\u003cli\u003eD. C. Wilcox, \u003cem\u003eTurbulence modeling for CFD\u003c/em\u003e, DCW industries La Canada, CA (1998).\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"korean-journal-of-chemical-engineering","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"kjce","sideBox":"Learn more about [Korean Journal of Chemical Engineering](http://link.springer.com/journal/11814)","snPcode":"11814","submissionUrl":"https://www.editorialmanager.com/kjce/default2.aspx","title":"Korean Journal of Chemical Engineering","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Subscription","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"hydrogen refueling, CFD modeling, equation of state, turbulence model, refueling simulation","lastPublishedDoi":"10.21203/rs.3.rs-5012331/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5012331/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eCurrently, commercially operated hydrogen fuel cell electric vehicles (FECVs) store hydrogen as highly compressed gas form to increase volumetric energy density. To provide a refueling time similar to that of internal combustion engine vehicles (ICEVs), hydrogen refueling stations (HRSs) should supply gaseous hydrogen into FECVs up to high pressure (35 MPa or 70 MPa) in a relatively short time. The refueling process of rapidly filling compressed gas within a confined volume of the storage tank is inevitably accompanied by an increase in temperature. However, the refueling process should be carried out under limited conditions considering the physical safety of the storage tank.\u003c/p\u003e \u003cp\u003eModeling the refueling process under the theoretical basis is useful for understanding the gas filling phenomenon and finding the optimal refueling strategy. In particular, the CFD research method which considers the flow of fluid in a tank offers the local temperature changes inside a storage tank as well as the average temperature.\u003c/p\u003e \u003cp\u003eThe CFD research is conducted by combining a model representing the fluid properties and a model describing the flow characteristics. Therefore, an appropriate combination of models should be examined before simulating the refueling process of an actual FECVs that requires time and cost that cannot be overlooked. In this study, the hydrogen refueling process is simulated using three equations of state (EOSs) and five turbulent models and, then, the results are compared and quantitively analyzed using experimental data. Experiments of filling type III tank of 74 L up to 35 MPa within 1 min have been chosen to make the assumption of axial symmetry for CFD model valid.\u003c/p\u003e \u003cp\u003eComparing the three EOSs (SRK, PR, ML), it is found that it is possible to improve accuracy and reduce calculation time when using ML EOS which has been developed to describe the behavior of hydrogen. Among the five turbulence models (yPlus, k-ε, realizable k-ε, low Reynolds k-ε, and k-ω) generally used in CFD research, the k-ε and the realizable k-ε model show satisfactory results on the reproduction of mean and local thermal behaviors and calculation time.\u003c/p\u003e","manuscriptTitle":"Application of EOS based on machine learning method on CFD study of rapid hydrogen refueling process","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-11-08 13:38:14","doi":"10.21203/rs.3.rs-5012331/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"reviewerAgreed","content":"","date":"2024-10-30T08:23:39+00:00","index":0,"fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-10-29T13:41:07+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-09-04T12:58:12+00:00","index":"","fulltext":""},{"type":"submitted","content":"Korean Journal of Chemical Engineering","date":"2024-09-02T19:39:03+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"korean-journal-of-chemical-engineering","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"kjce","sideBox":"Learn more about [Korean Journal of Chemical Engineering](http://link.springer.com/journal/11814)","snPcode":"11814","submissionUrl":"https://www.editorialmanager.com/kjce/default2.aspx","title":"Korean Journal of Chemical Engineering","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Subscription","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"82dc778e-f205-4ef0-b6ed-16a7f68ff06c","owner":[],"postedDate":"November 8th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2025-04-21T16:04:37+00:00","versionOfRecord":{"articleIdentity":"rs-5012331","link":"https://doi.org/10.1007/s11814-025-00460-x","journal":{"identity":"korean-journal-of-chemical-engineering","isVorOnly":false,"title":"Korean Journal of Chemical Engineering"},"publishedOn":"2025-04-17 15:57:53","publishedOnDateReadable":"April 17th, 2025"},"versionCreatedAt":"2024-11-08 13:38:14","video":"","vorDoi":"10.1007/s11814-025-00460-x","vorDoiUrl":"https://doi.org/10.1007/s11814-025-00460-x","workflowStages":[]},"version":"v1","identity":"rs-5012331","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-5012331","identity":"rs-5012331","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}