Euler–Lagrange simulations coupled with the multiphase particleincell (MPPIC) approach for considering interparticulate collisions have been performed to simulate a nonreacting fluidized bed at laboratoryscale. The objective of this work is to assess dynamic properties of the fluidized bed in terms of the specific kinetic energy of the bed material k_{S} in J/kg and the bubble frequency f_{B} in Hz, which represent suitable measures for the efficiency of the multiphase momentum exchange and the characteristic timescale of the fluidized bed system. The simulations have reproduced the bubbling fluidization regime observed in the experiments, and the calculated pressure drop $ \Delta p$ in Pa has shown a reasonably good agreement with measured data. While varying the bed inventory m_{S} in kg and the superficial gas velocity u_{G} in m/s, k_{S} increases with u_{G} due to the increased momentum of the gas flow, which leads to a reinforced gastosolid momentum transfer. In contrast, f_{B} decreases with m_{S}, which is attributed to the increased bed height h_{B} in m at larger m_{S}. An increased gas temperature T_{G} from 20 to 500 °C has led to an increase in k_{S} by approximately 50%, whereas $ \Delta p$, h_{B}, and f_{B} are not sensitive to T_{G}. This is due to the increased gas viscosity with T_{G}, which results in an increased drag force exerted by the gas on the solid phase. While upscaling the reactor to increase the bed inventory, bubble formation is enhanced significantly. This has led to an increased f_{B}, whereas k_{S}, h_{B}, and $ \Delta p$ remain almost unchanged during the scaleup process. The results reveal that the general parameters such as h_{B} and $ \Delta p$ are not sufficient for assessing the hydrodynamic behavior of a fluidized bed while varying the operating temperatures and upscaling the reactor dimension. In these cases, the dynamic properties k_{S} and f_{B} can be used as more suitable parameters for characterizing the hydrodynamics of fluidized beds.
I. INTRODUCTION
Gas–solid fluidized beds are characterized by a large gas–solid contact surface and an intensive gas–solid mixing, which promote heat and mass transfer. Due to these advantages, the fluidized bed technology is widely employed to environmental, chemical, pharmaceutical, energy and process industries, which can benefit significantly to higher efficiencies and lower emissions of these processes. For example, fluidized beds are used as a key technology in the petroleum refining industry (fluid catalytic cracking), drying of powder and granules, reduction of iron ore, biomass/coal combustion, and gasification.^{1–3} The operational performance of fluidized beds relies strongly on the hydrodynamics of the gas–solid flow in terms of bubble formation and particle dynamics, which represent the fundamental mechanism for the mass and heat transfer, and also influence the product yield. As the harsh environment caused by the dense particulate flow limits the experimental assessment of the gas–solid flow system within the fluidized bed, numerical simulations have been used extensively in the last few decades to gain an indepth understanding of the underlying fluidization dynamics.
There are two fundamental approaches for modeling gas–solid flows: the Euler–Euler and the Euler–Lagrange approach.^{4} The twofluid model (TFM) uses the Euler–Euler approach, and both gas and solid are regarded as penetrating continuous phases. Therewith, the particle properties such as density or diameter are calculated using the kinetic theory of granular flow (KTGF).^{5} The TFM is widely used due to its low computational cost. However, the effect of particle size cannot be considered properly by TFM, which can have significant influence on prediction performance of fluidized beds.^{6} In the Euler–Lagrange method, a large number of Lagrange particles (LPs) are tracked, which interact with the continuous gas phase. Both phases are coupled through source terms concerning mass, momentum, and heat transfer. Compared with TFM, the Euler–Lagrange approach takes the effect of particle size distribution (PSD) into account^{7} and it provides information on the trajectories as well as transient forces acting on the particles.^{8} The Euler–Lagrange approach can be further subdivided to the Discrete Particle method (DPM), MultiphaseParticleinCell method (MPPIC), Dense Discrete Phase Model incorporated with Kinetic Theory of Granular Flow (DDPMKTGF), and CFDDiscrete Element Method (CFDDEM), which differ in their treatment of the particle–particle interaction.^{9}
Due to the advantages of the Euler–Lagrange approach with regard to computational efficiency and accuracy, it has been widely used for the simulation of gas–solid flows. For instance, Wang et al.^{10} applied the Euler–Lagrange method along with the MPPIC approach for a cold circulating fluidized bed (CFB) with a loop seal and studied the effects of grid resolution, drag model, and particle size distribution (PSD) on the hydrodynamic behavior of a circulating fluidized bed (CFB). The model setups were then used to study the influence of operating parameters, including loop seal aeration rate, fluidized air velocity in the riser and total bed inventory, on the solid circulation characteristics.^{11} In Ref. 12, a CFB combustor under cold flow condition has been calculated using the CPFD method coupled with different drag models, and the results showed good agreement with experiments for varied bed inventory. Zafiryadis et al.^{13} applied the Euler–Lagrange method along with the MPPIC approach for a cold flow CFB, where the pressure constant and the exponent of the MPPIC particle stress model were found to have the largest influence on correctly reproducing the fluidization behavior. A biomass gasification plant in cold flow operation has been simulated by Lunzer et al.^{14} with the Euler–Lagrange/MPPIC method and different drag models, which revealed a significant impact of the particle size distribution. Furthermore, the model constant for the particle stress used by the MPPIC approach was found to play an important role in closepacked regions. Wu et al.^{15} employed the dense discrete phase model (DDPM) to investigate flow dynamics in a swirling gas–solid fluidized bed, which results in a decreased bed height and pressure drop compared with general setups without using a swirling gas flow. Moreover, different operation regimes have been identified with increasing superficial velocity and an increase in the operation velocity was found to be more beneficial in terms of particle mixing in the swirling fluidized bed. In Ref. 16, two models for considering heat transfer close to the walls of fluidized beds have been introduced in the framework of the CFDDEM (discrete element method) approach and were validated with experimental results. A review on the simulation of cold flow fluidized beds has been provided in Ref. 17, and the influence of some key models such as interphase drag model has been highlighted.
Another class of modeling gas–solid flows is given by the particleresolved direct numerical simulation (PRDNS),^{18} which represents the most accurate, but also the most computationally expensive method due to the resolution of the flow field around the particle, including its boundary layer. By using PRDNS, Moriche et al.^{19} have studied the clustering of oblate spheroids settling in ambient fluid. Tenneti et al.^{20} has applied PRDNS to study the acceleration of particles due to gas–solid and interparticle interactions by means of the particle velocity variance (granular temperature). Esteghamatian et al.^{21} have performed particleresolved simulations for liquid/solid and gas/solid fluidization, which revealed nonisotropic mechanisms for driving the particle motion and the dominance of diffusive and convective mechanisms.
In addition, a large number of works have focused on improving the accuracy of the Euler–Lagrange method along with its submodels. In particular, great efforts have been made to develop more sophisticated drag model as well as the quantification of uncertainties induced by the drag model.^{22,24–28} Moreover, a computationally efficient particle cloud tracer method is presented in Ref. 23 for tracing a large number of particles and modeling statistic moments of particle groups in Euler–Lagrange formulations. Patel et al.^{29} have compared the performance of TFM and Euler–Lagrange methods, where the convergence of the methods under grid refinement is found to depend on the simulation method and the specific case of concern. The problem of large particlesize to meshspacing ratio in dilute particleladen flows has been studied by Evrard et al.^{30} Similar work has been conducted in Ref. 31, where a strategy for simulating particleladen flows using cell sizes smaller than the particle diameter has been proposed. A comprehensive review on the development of mathematical models for gas–solid flows as well as their applications to fluidized beds has been presented in Ref. 9, which confirms the capability of numerical models for designing industrial plants based on the fluidized bed technology.
As the gas–solid and solid–solid interactions occur over a wide range of length and time scales and are affected by a large number of operational, dimensional, and design parameters, the functional dependence of fluidized bed characteristics on these design parameters still have large uncertainties. Therefore, despite the achieved progress in recent works, detailed knowledge with regard to accurate prediction of the hydrodynamic behavior of fluidized beds is missing. In particular, most of the previous numerical works have focused on the validation of specific submodels or model parameters and studying the influence of operational parameters on the overall behavior of the considered fluidized bed. However, in general, it is not sufficient to solely use general features such as bed height, pressure drop, or solid circulation rate for characterizing the temporally developing, multiscale gas–solid system within the fluidized bed, as these are not directly related to the performance of fluidized beds in terms of an efficient mixing or heating. In order to gain more detailed knowledge of the hydrodynamic process of the fluidized bed, highly resolved numerical simulations have been conducted for a laboratoryscale fluidized bed at isothermal condition. Objective of this work is to evaluate its unsteady dynamic properties in terms of the specific kinetic energy of the bed material and the bubble frequency. The specific kinetic energy is equivalent with the averaged moving velocity of the bed material within the fluidized bed, which can be regarded as a measure for the efficiency of the multiphase momentum exchange from the gas to the solid phase. On the other hand, the bubble frequency represents a measure for the characteristic dominating timescale of the bubbles within fluidized bed system. The correlations of these properties with the operating parameters have been quantitatively evaluated, which reveals their usefulness for a more detailed assessment of the hydrodynamics of fluidized beds.
II. SIMULATION METHODS
A. Euler–Lagrange approach for simulation of gassolid flows
As a detailed resolution of each solid particle including its boundary layer in a fluidized bed is computationally too expensive, a hybrid Euler–Lagrange approach is used in the present work for modeling the multiphase interactions.^{32} In this method, the gas flow is regarded as a continuous phase, which is modeled by means of the Naiver–Stokes equations. The solid particles are treated as dispersed, and their trajectories are calculated based on a balance of forces acting on the particle, along with the equation of motion. Both sets of Euler and Lagrange equations concerning the gas and solid phase are coupled via source terms for considering the transfer of momentum between the different phases.
B. Modeling of interparticle collisions
III. SIMULATION SETUPS
A. Operating conditions
The laboratoryscale fluidized bed reactor considered in this work has a cylindrical geometry with a diameter of 5 cm and a length of 100 cm. A porous sintered metal plate is used at the bottom of the reactor as gas distributor, which is permeable to the gas and generates homogeneous regular incoming gas flow toward the bed materials. Quasispherical silica sand particles with a Gaussian size distribution and an arithmetic mean diameter of 212 $\mu $m have been used as carrier material, along with nitrogen as fluidizing agent. Figure 1 on the left shows the experimental test rig along with a snapshot of the fluidized bed in coldmode operation. The bulk gas flow velocity u_{G} has been varied from 13.6 to 29.7 cm/s and the bed inventory m_{s} from 195 to 586 g, as shown on the right of Fig. 1.
The operating parameters have been designed according to the Gracediagram^{38,40} to generate a bubbling fluidized bed considering an efficient mixing. In this way, the dimensionless superficial velocity $ u G * = u G \rho g 2 \mu g ( \rho s \u2212 \rho g ) g 3$ is within the range of $ 0.1 < u * < 0.5$ (for $ 0.136 < u G < 0.297$ m/s) along with a dimensionless particle diameter at $ d P * = d P \rho g ( \rho s \u2212 \rho g ) g \mu g 2 3 = 10.24$ by using a mean particle diameter of $ d P = 0.21$ mm (see Fig. 2). By doing so, the operating conditions in terms of the dimensionless parameters can be applied to upscaled fluidized beds for engineering applications, too.
B. Computational domain and resolutions
The computational domain used for the simulation is given by a cylinder, which has a length of 60 cm and a diameter of 5 cm. The dimensions of the domain have been selected to be sufficiently large to cover the whole fluidized bed at all considered operating conditions (the maximum bed height is approximately 30 cm, see Fig. 7). It is shorter than the tube used in the experiment (with 100 cm) in order to save computational cost. As shown in Fig. 1 on the right, the topology of the computational grid for the crosssectional area is built with an Otype grid, which creates uniformly distributed grid cells from the center to the sidewall with an almost equidistant grid length of ca. 1 mm in the radial direction. The grid resolution in the axial direction is 1 mm at the ground of the tube, which increases with a small expansion factor in the streamwise direction. Overall, the length of the cylindrical domain has been resolved by 180 cells and the diameter of the tube with approximately 40 cells.
For the Euler–Lagrange simulation, a number of Lagrange particles (LPs) is injected from given locations, which are then tracked during the simulation. These LPs represent collections of spherical particles with the same characteristics, e.g., diameters and velocities. In this work, the LPs are initialized uniformly in space along the whole domain, which fall down due to gravity and interact with the incoming gas flow. A quasi steadystate solution with a fluidized sand bed is generated after about 1 s. The sizes of the particles are set according to the measured particle size distribution (PSD), which is shown in Fig. 2. The number of tracked LPs increases proportionally with the bed inventory, i.e., with $ n P \u2248 2 \xd7 10 6 , \u2009 3 \xd7 10 6 , \u2009 4 \xd7 10 6 , \u2009 5 \xd7 10 6 , \u2009 6 \xd7 10 6$ for $ m G = 195 , \u2009 293 , \u2009 390 , \u2009 488 , \u2009 586$ g, leading to a constant number of particle per parcel for all cases. The grid resolution and the number of LPs have been selected based on previous gridindependence studies and a compromise between simulation accuracy and computational cost, where a further refinement of the grid or increase in the number of LPs does not lead to a clear improvement with regard to comparison with the measured pressure drop.
C. Boundary conditions
The boundaries of the computational domain are indicated in Fig. 1 on the right, where nitrogen gas enters the domain from the inlet and leaves at the outlet. A nonslip condition is used for the reactor wall. The flow velocity at the inlet is calculated from $ u inlet = u G / \epsilon $ with ε being the void or gas volume fraction. In the case of $ \epsilon = 1$ or without sand particles, the inlet flow velocity is equal to u_{G}; if sand particles are available or $ \epsilon < 1$, the local flow velocity at the inlet is larger than u_{G} to preserve continuity. The pressure at the outlet is fixed at ambient pressure, whereas its gradient at the reactor wall and at the inlet is set to zero. The simulations have been conducted under isothermal condition at 20 °C and 1 atm. The densities of the gas and the sand are set to $ \rho G = 1.14$ and $ \rho S = 2660$ kg/m^{3}.
The opensource code OpenFOAMv2206^{41} has been used to perform the numerical simulations, and the standard solver MPPICFoam has been applied to simulate the gas–solid flow in the fluidized bed. The main reason for using MPPICFoam in OpenFOAM is attributed to the fact that, compared with other commercial or opensource CFD codes, the solver can be extended more easily to model heat transfer and heterogeneous reactions in fluidized bed, which represent future work of the present study for simulation of plastic pyrolysis in fluidized beds. The balance equations are solved in an incompressible formulation, employing a secondorder interpolation scheme for discretization of the convection and diffusion terms, along with an implicit scheme (Euler) for time integration. The time step was set to 0.1 ms, ensuring a maximum CFL (Courant–Friedrichs–Lewy) number smaller than unity. The simulations have been run for a physical time of 4 s, where statistical averaging of the flow has been performed for a physical time of 3 s after initialization of the fluidized bed.
IV. SIMULATION RESULTS
A. Morphology of the fluidized bed
Figure 3 depicts time series of the contours of the calculated void or gas fraction ε on a cutting plane passing through the centerline axis for the reference case with m_{S} = 390 g and u_{G} = 21 cm/s. The sand particles are indicated by the black dots, which are shown for a disk across the symmetry axis with a thickness of 1 mm. The time interval between the snapshots is 30 ms. Only a subset of the total particles is shown and the particles are scaled up by a factor of 5 for better visualization. The bubbles are illustrated by the red zones with large ε. The bubbling fluidization behavior observed in the experiment has been reproduced by the simulation. While the gas flow passes through the sand bed, small bubbles are first generated close to the bottom of the reactor and rise in streamwise direction due to buoyancy forces. With increased axial distance, the size of bubbles increases due to coalescence of the small bubbles until they reach the upper surface of the fluidized bed.
As the bubbles rise along the centerline axis, the sand particles are driven to the wall side. Because the gas flow velocity is low close to the wall due to the nonslip condition, the particles fall down along the wall. Near to the base of the reactor, the sand particles interact with the incoming gas flow and are pushed upwardly by the initial small bubbles. As shown in Fig. 4 by the contours of timeaveraged solid fraction $ \alpha \xaf P$ and streamwise flow velocity $ u \xaf$, the circulation of the sand particles leads to a core–annulus flow pattern, where the gas bubbles dominate the core region and the particles are concentrated close to the wall. In addition, a negative correlation between the $ \alpha \xaf P$ and $ u \xaf$ can be identified. In the core region, the share of particles is lowest and the gas flow velocity is largest, whereas the reversed trend is found in the nearwall region. At the upper surface of the fluidized bed, a small portion of sand particles is ejected from the fluidized bed due to bursting of the bubbles.
Figure 5 shows instantaneous contours of ε on a meridian cutting plane passing through the centerline axis for varied u_{G} from 14 to 30 cm/s panel 5(a) and for varied m_{S} from 195 to 586 g panel 5(b). The case with the smallest u_{G} at 14 cm/s shows a smooth or closetominimum fluidization behavior, where the sand bed is only weakly fluidized without forming clear bubbles. A further increase in u_{G} results in the formation of bubbles, corresponding to the bubbling fluidization regime. These bubbles rise along the axial direction due to buoyancy and collapse at the upper boundary of the fluidized bed. The size of the generated bubbles increases with u_{G} and becomes as large as the reactor diameter at large u_{G}. The enhanced bubble formation at larger u_{G} is attributed to the increased momentum of the gas flow, leading to a reinforced gastosolid momentum transfer and recirculation of the sand particles. The onset of a slugging type fluidization can be observed at u_{G} = 30 cm/s, with a large number of particles thrown away by the fluidized bed.
As shown in Fig. 5(b), the volume of the fluidized bed expands while increasing m_{S} at constant u_{G} = 21 cm/s, which indicates an increase in the bed height h_{B}. As the volume of the fluidized bed increases with m_{S}, the small bubbles generated near the ground of the reactor have more space to develop and to coalesce with each other so that the size of the bubbles increases with m_{S}.
The operating parameters used in the experiment are designed to establish the bubbling fluidization regime, which is confirmed in the numerical simulations. However, a direct comparison of the bubble formation and the particle circulation between experiment and simulation is not possible due to limitations given by the lineofsight measurement techniques. The desired bubbling fluidization regime can be achieved for a moderate range of u_{G} and m_{S}, which is beneficial with regard to an efficient mixing and heat/mass transfer.
B. Pressure drop and bed height
Figures 7(a) and 7(b) depicts profiles of timeaveraged particle volume fraction $ \alpha \xaf P$ along the centerline axis for constant m_{S} at 390 g with varied u_{G} and for constant u_{G} at 21 cm/s with varied m_{S}. For the case with u_{G} = 13.6 cm/s, the fluidized bed is in the closetominimum fluidization regime, without clear formation of bubbles, as shown in Fig. 5(a) on the left. Therefore, $ \alpha \xaf p$ yields an almost constant distribution over the whole bed height at $ \alpha \xaf p \u2248 0.53$ for u_{G} = 13.6 cm/s, which decreases rapidly to 0 while approaching the upper surface of the bed. With further increased u_{G}, the fluidized bed is characterized by the bubbling fluidization regime. In this case, $ \alpha \xaf p$ is largest at the base of the reactor and decreases gradually to 0 at the upper surface of the sand bed. Figure 7(b) shows profiles of $ \alpha \xaf p$ for varied m_{S}, which are similar in the lower part of the fluidized bed with x < 70 mm. Therefore, the behavior of initial bubble formation near to the bottom of the fluidized bed is similar for all cases due to the use of a constant u_{G}, as shown in Fig. 5(b). However, $ \alpha \xaf p$ decreases more slowly further downstream at larger m_{S}, indicating an increase in the bed height h_{B} with m_{S}. Due to the largescale bubbles generated at the centerline axis, $ \alpha \xaf p$ increases in the radial direction and reaches its maximum in a region close to the wall.
C. Specific kinetic energy of bed material
Figure 8(a) depicts the temporal evolution of the calculated k_{S} at m_{S} = 390 g and with varied u_{G}, which fluctuates over time. Figure 8(b) shows the timeaveraged k_{S} at varied m_{S} and u_{G}. k_{S} increases with u_{G}, which is attributed to the increased momentum flux of the gas flow, leading to a reinforced momentum transfer from the gas to the solid phase. Moreover, $ k \xaf S$ increases with m_{S}, as shown in Fig. 8(b), which indicates a stronger momentum exchange between the gas and solid phases in the case of increased bed inventory. This could be attributable to the increased contact surface area between the gas and solid phases.
D. Bubble frequency
The periodically rising bubbles and the collapse of these bubbles at the upper side of the fluidized bed trigger the whole system into a pulsating mode. The dominating bubble frequency f_{B} corresponds to the number of repetitions of the recurring bubbles within one second and represents a measure for the averaged moving speed or dynamics of the gas bubbles. The higher the bubble frequency, the more intense is the mixing process. Figure 9(a) shows the calculated temporal evolution of k_{S} for a constant gas velocity u_{G} = 21 cm/s and different sand mass m_{S}, which exhibit distinct periodical fluctuations. The number of repetitions of k_{S} decreases with m_{S}, which indicates a decrease in f_{B} with m_{S}. This is due to the fact that the fluidized sand bed expands with increased bed inventory, as shown by the instantaneous contour of ε in Fig. 5(b). In the cases with small m_{S}, the height of fluidized bed is low and dominated by a number of small bubbles, which travel along a shorter distance up to the upper surface of the fluidized bed. This results in a shorter residence time of the bubbles or a higher bubble frequency, respectively. As the bed height increases with m_{S}, the small bubbles coalesce with each other while rising to the top of the fluidized bed. Therefore, the distance or time required for the bubbles to move through the bed volume is increased with m_{S}, leading to a decreased f_{B}. The bubble frequency f_{B} has been evaluated from spectral analysis (Fourier transformation) of the temporal development of k_{S} and plotted against m_{S} in Fig. 9(b), which yields a decrease with m_{S}. Under the current conditions, the fluidized bed is dominated by f_{B} in a relatively low frequency range between 2 and 7 Hz, which is attributed to the low gas flow velocity.
In summary, the bed height h_{B} and pressure drop $ \Delta p$ increase with m_{S}, whereas u_{G} has a subordinate effect on h_{B} and $ \Delta p$. However, u_{G} has a strong impact on the fluidization behavior and, the specific kinetic energy of sand k_{S} increases with u_{G}, as shown in Fig. 8. In addition, the bubble frequency f_{B} decreases with m_{S} due to the increased bed volume, whereas f_{B} is not sensitive to u_{G}. The results reveal strong correlations of k_{S} and f_{B} with the general operating parameters m_{S} and u_{G}, which can be used for characterizing the hydrodynamics of fluidized beds in addition to h_{B} and $ \Delta p$.
E. Effect of gas temperature
Fluidized beds are often operated at hightemperature condition like for drying, combustion, or gasification. The current fluidized bed has been designed for recycling of plastic wastes via pyrolysis process in the range of 400–600 °C.^{44} Therefore, in order to study the behavior of k_{S} and f_{B} at elevated operating temperatures, an additional simulation has been conducted at a gas–solid temperature of $ T G = 500 \u2009 \xb0$ C, while the bed inventory and superficial velocity have been kept constant at m_{S} = 390 g and u_{G} = 21 cm/s. In this way, the density of the nitrogen gas decreases from $ \rho G = 1.14$ kg/m^{3} at 20 °C to $ \rho G = 0.44$ kg/m^{3} at $ T G = 500 \u2009 \xb0$ C and the kinetic viscosity of the gas increases from $ 1.5 \xd7 10 \u2212 6$ to $ 8.0 \xd7 10 \u2212 6$ m^{2}/s.
Figure 10(a) compares instantaneous contours of the void fraction ε on a meridian cutting plane passing through the symmetry axis. As m_{S} and u_{G} are kept constant, the bubbling fluidization regime remains almost unchanged at elevated temperature. Figures 10(b) and 10(c) show the timeaveraged contours of ε and streamwise velocity of the gas phase on the same cutting plane, which reveal similar distributions at different T_{G}. Moreover, the bed height h_{B} is increased slightly with T_{G}, which leads to a slight increase in the pressure drop at elevated reactor temperature. This is attributed to the increased density difference or buoyancy force, respectively.
Figure 11(a) compares the temporal developments of the specific kinetic energy k_{S} at different T_{G}, where an increase in k_{S} with T_{G} can be detected. The timeaveraged k_{S} is increased from $ k \xaf S = 13.6$ mJ/kg at 20 °C to $ k \xaf S = 19.2$ mJ/kg at 500 °C, which is more than 40%. The results reveal that the particles move with a higher velocity on average in the case of elevated temperature. The reason is given by the strongly increased kinetic viscosity of the gas phase, which causes a higher drag force exerted by the gas flow on the particles. As shown in Fig. 11(b), the volumespecific drag force calculated from Eq. (8) at given particle volume fractions $ \alpha P = 0.2$–0.4 and using a particle diameter of 0.2 mm increases with T_{G}, indicating a reinforced multiphase momentum exchange. The bubble frequency remains almost unchanged with T_{G}, which can be detected from the number of repetitions of k_{S} in the time evolution of k_{S} shown in Fig. 11(a).
F. Effect of upscaling
The scaleup has often proven to be a significant obstacle in the development of new fluidized bed processes in the past, as reactors designed based on measurements from small laboratory apparatuses did not achieve the expected reaction rate at the operational scale. The cause of this wellknown effect, which in its magnitude was never calculable in the past, is ultimately characterized by changes in fluid mechanics with increasing fluidized bed diameter. To study the effect of upscaling on the hydrodynamic properties of fluidized beds, the reactor diameter d_{R} has been scaled up to 3, 5, and 10 cm. The pressure drop and the bed height have been kept constant while upscaling the fluidized bed, leading to an increased bed inventory. At the same time, the superficial velocity has been kept constant so that the same operating point within the bubbling fluidization regime from the Gracediagram^{38} can be achieved, as shown in Fig. 12 with the dimensionless superficial velocity of $ u G * = 0.3$ and the dimensionless particle diameter of $ d P * = 10.2$. In this way, the bed inventory increases with the reactor diameter by $ m S \u221d d R 2$, whereas the bubbling fluidization regime is retained. For the numerical simulation, the height of the domain as well as the grid resolution in the radial direction was kept constant at 60 cm and 1 mm while upscaling the reactor. This leads to a proportional increase in the total number of grid cells n_{C} with m_{S}. In addition, the number of Lagrange parcels n_{P} increases linearly with m_{S}. The parameters used for the current study of upscaling are listed in Table I.
d_{R} (cm)  3  5  10 
m_{S} (g)  140  390  1600 
u_{G} (m/s)  21  21  21 
n_{C} (−)  57.600  152 100  562.500 
n_{P} (×10^{6})  2.9  8.0  32.0 
d_{R} (cm)  3  5  10 
m_{S} (g)  140  390  1600 
u_{G} (m/s)  21  21  21 
n_{C} (−)  57.600  152 100  562.500 
n_{P} (×10^{6})  2.9  8.0  32.0 
Figures 13(a) and 13(b) depict snapshots of isocontour of $ \epsilon = 0.66$ for different reactor sizes, which illustrate 3D structures of the bubbles. The solid particles are shown additionally in the second row. For the cases with d_{R} = 3 and 5 cm, the hydrodynamics of the fluidized bed is dominated by largescale bubbles with sizes similar to the reactor diameter. In contrast, the bubbles rise along multiple offcentered columns in the case of d_{R} = 10 cm. The bubbles coalesce with each other so that with increasing height above the gas distributor base, the average bubble size rapidly increases. In narrow fluidized bed vessels with small d_{R}, the bubbles fill quickly the entire cross section. The formation of bubbles is significantly stronger while using d_{R} = 10 cm, and the bubbles generally do not rise evenly distributed in the fluidized bed. Close to the base of fluidized bed, a nearwall zone with intensified bubble formation develops, which shifts toward the center of the pipe with increasing height above the distributor base. In the small fluidized bed, this shift leads quickly to large bubbles preferentially rising along the centerline axis of the reactor. The cause of this characteristic flow profile is the proximity of the reactor wall, which influences the coalescence process.
Figure 13(c) depicts calculated instantaneous contours of ε on a meridian cutting plane across the centerline axis for different reactor sizes, where the sand particles are illustrated by the filled circles. For the smallest reactor with d_{R} = 3 cm, the fluidized bed is dominated by large bubbles rising along the centerline axis, whose size is similar to the diameter of the reactor. This is attributed to the narrow domain bounded by the reactor wall, which leads to a more intense coalescence of the initial small bubbles. At increased reactor size with d_{R} = 5 cm, there is more space between the large bubbles and the reactor wall, which allows formation of a lowspeed region close to the wall and a recirculation of sand particles there. While further increasing the reactor size to d_{R} = 10 cm, the number of bubbles is increased significantly, and the bubbles rise along multiple columns without coalescing with each other further downstream. In this case, the gas flow recirculates additionally toward the symmetry axis, leading to an accumulation of sand particles with increased share of solid phase around the centerline axis.
The same behavior can be detected from Fig. 14(a), which shows the timemean contours of ε on a meridian cutting plane across the symmetry axis. For d_{R} = 3 and 5 cm, $ \epsilon \xaf$ is at largest along the centerline axis due to the largescale bubbles, which dominate the reactor volume. On the contrary, $ \epsilon \xaf$ is lower at the centerline axis for the case with d_{R} = 10 cm, which indicates a higher concentration of sand particles there. This is attributed to the bubbles rising along multiple offcentered axes, resulting in a recirculation of the flow toward the centerline axis. For all cases, the timeaveraged flow velocity $ u \xaf$ shown in Fig. 14(b) yields a positive correlation with $ \epsilon \xaf$, which is small in the regions close to the wall and large in the core regions for d_{R} = 3 and 5 cm. For d_{R} = 10 cm, $ u \xaf$ is small in the core region due to recirculation of the flow or the particles toward the symmetry axis. The zones with the largest $ \epsilon \xaf$ or $ u \xaf$ can be traced back to the rising bubbles.
Figure 15 shows the temporal developments of the specific kinetic energy of sand k_{S} calculated by using different reactor diameters. The table on the right summarizes the calculated timemean bed height h_{B} (distance along the centerline axis from the reactor base to the position with $ \epsilon \xaf = 0.99$), pressure drop $ \Delta p , \u2009 k \xaf S$, and bubble frequency f_{B}. h_{B} is slightly decreased with increased d_{R}, which can also be detected from the contour plots of ε shown in Fig. 14(a). In accordance with Eq. (13), the decrease in h_{B} with d_{R} leads to a slight decrease in $ \Delta p$. The same behavior is found for $ k \xaf S$, which is insensitive to the reactor size. As shown in Fig. 15, although k_{S} fluctuates in time at different d_{R}, the timeaveraged values of k_{S} remain almost constant. However, the enhanced bubble formation at increased d_{R} has led to a clear increase in the bubble frequency. At larger reactor diameters, the occurrence of bubble chains accompanies with the development of the largescale solid circulation, where the bubbles rise in close succession at high speed, leading to an increase in f_{B}. The results indicate that the kinetic energy or the averaged moving velocity of the sand particles remain almost constant while upscaling the fluidized bed reactor. The dominant timescale with regard to the rising bubbles and the circulating bed materials is, however, decreased.
In summary, similar to the bed height h_{B} and pressure drop $ \Delta p$, the specific kinetic energy k_{S} and the bubble frequency f_{B} represent integral properties of fluidized beds, which depend on the operating conditions or the dimensionless parameters like $ u G *$ or Re. Compared with h_{B} and $ \Delta p$, k_{S} and f_{B} can be used for a more detailed assessment of fluidized beds, which reveal dynamic behaviors of the particles and the bubbles in terms of their moving velocities. It has been shown in this work that, in the bubbling fluidization regime, k_{S} increases with the superficial velocity u_{G} due to the increased momentum flux of the gas flow and it increases with the operating temperature T_{G} due to the increased drag force, corresponding to a more intensive mixing or heating; on the contrary, $ \Delta p$ remains almost constant with u_{G}. In addition, f_{B} increases while upscaling the fluidized bed, whereas h_{B} and $ \Delta p$ remain unaffected. These highlight the need of introducing additional performancerelated parameters like k_{S} and f_{B}.
V. CONCLUSION
A laboratoryscale, cylindrical fluidized bed reactor has been studied numerically in coldmode operation. The objective of this work is to assess the dynamic characteristics of the fluidized bed in terms of the total kinetic energy of the bed materials k_{S} and the bubble frequency f_{B}, which represent measures for the efficiency of multiphase momentum transfer and the dominant time scales prevailing the gas–solid system. The main findings are summarized below:

The bubbling fluidization regime of the fluidized bed observed in experiments has been reproduced well by the simulations, where the calculated pressure drop has shown a good agreement with measured data.

While varying the superficial flow velocity u_{G} and the bed inventory for a fixed geometry of the fluidized bed, k_{S} has found to increase with u_{G}. This is due to the increased momentum flux of the gas flow, leading to an enhanced aerodynamic forces exerted on the particles. The same behavior has been confirmed for the correlation of k_{S} with m_{S}.

f_{B} decreases with m_{S} at constant u_{G}, which is attributed to the increased bed volume with m_{S}.

At constant m_{S} and u_{G}, k_{S} increases with the reactor temperature T_{G}. This is caused by the increased kinetic viscosity of the gas, which leads to an increased drag force or enhanced gastosolid momentum transfer.

An increase of the bed inventory via upscaling results in enhanced formation of bubbles and an increased f_{B}. However, the averaged moving speed of the bed materials in terms of k_{S}, as well as the pressure drop and bed height remain almost unchanged.
The results reveal strong correlations of k_{S} and f_{B} with the operating parameters, which can be used to characterize the hydrodynamic behavior of fluidized beds. In particular, the commonly used properties such as $ \Delta p$ and h_{B} are not sufficient for studying effects related to scaleup or elevated temperature, as $ \Delta p$ and h_{B} are not sensitive to these conditions. In these cases, the proposed dynamic properties k_{S} and f_{B} represent suitable measures for a detailed assessment of the flow behaviors in fluidized beds.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the financial support by the Helmholtz Association of German Research Centers (HGF), within the research field Energy, program Materials and Technologies for the Energy Transition (MTET), topic Resource and Energy Efficiency. This work utilized computing resources provided by the High Performance Computing Center Stuttgart (HLRS) at the University of Stuttgart and the Steinbuch Centre for Computing (SCC) at the Karlsruhe Institute of Technology.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Feichi Zhang: Conceptualization (lead); Investigation (lead); Methodology (lead); Supervision (equal); Writing – original draft (lead); Writing – review & editing (lead). Dieter Stapf: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal). Salar Tavakkol: Conceptualization (supporting); Investigation (supporting); Project administration (equal); Supervision (supporting). Stefan Dercho: Methodology (equal); Software (equal); Visualization (equal). Jialing Zhou: Investigation (equal); Validation (lead); Visualization (lead). Thorsten Zirwes: Methodology (equal); Software (equal); Writing – review & editing (equal). Michael Zeller: Investigation (equal); Resources (equal); Writing – review & editing (equal). Jonas Vogt: Methodology (equal); Resources (equal); Writing – review & editing (equal). Rui Zhang: Supervision (equal). Henning Bockhorn: Supervision (equal).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.