Parametric study of the time-averaged gas–solid drag force in circulating fluidized bed conditions
Graphical abstract
Introduction
The flow patterns in a circulating fluidized bed (CFB) are characterized by large spatial and temporal fluctuations in solid concentration as well as in gas and solid velocities. Length scales of these variations can be very small, down to a few particle diameters, which constitutes a difficult challenge for computational fluid dynamic (CFD) simulation of industrial scale CFBs utilized in energy production and conversion processes. Similar computational mesh related spatial resolution challenges are encountered both with Eulerian–Eulerian two-fluid models and with the multi-phase particle-in-cell methods [1]. Resolving the smallest flow structures would require a very fine mesh that would be computationally unfeasible for simulation of typical industrial scale processes. As a result, coarser meshes that filter out the fine sub-grid scale flow structures are commonly used. To account for the filtered information, corrections or sub-grid scale models for the terms in the transport equations should be applied. Such closure models have been proposed in the literature for different terms in the gas and solid phase momentum equations on the basis of theoretical considerations and analysis of results from transient simulations of fairly small geometries with high spatial resolution. As the filtering scale depends on the resolution of the simulation, these closure models should have the resolution as a parameter which complicates derivation of the closure laws. No general closures that would cover all typical conditions in circulating fluidized beds and all terms in the equations have yet been proposed.
To avoid mesh dependence in the closures and to speed up the simulations, steady state modeling has been suggested as an alternative for the transient simulations [2], [3]. Transient coarse mesh simulations filter out spatial fluctuations smaller than the mesh spacing and temporal fluctuations shorter than the time step. Steady-state simulation models are derived by time-averaging the transient equations, which filters out all temporal variations in the flow properties. Closures need to be developed to account for the effects of the filtered variations. These closures are of a similar character as the sub-grid closures required for transient coarse mesh simulations. In both cases, the closures describe clustering of particles and fluctuations in flow properties. A closure for a very coarse mesh should approach the closure for a time-averaged equation, since a volume average over a distance longer than the longest length-scales of the fluctuations should produce the same result as a time average in a single point, when the statistical characteristics over the averaging space and time frame are the same. That would be the case if there would be no spatial gradients in the time-averaged flow field. Although this condition is never exactly met, the gradients in the time-averaged volume fraction and velocities are often small compared to the fluctuations, for example in the center of a large CFB riser.
To evaluate the need for equation closures, a study of the terms in the time-averaged momentum equations in CFB conditions was carried out by Kallio et al. [4]. One of the largest terms in the momentum equations was the gas–solid drag force, for which several closures have been proposed to describe the effects of clustering of particles. Closures can be derived on experimental basis like what was done by Kallio et al. [5] who applied a drag law that was based on the empirical equations suggested by Matsen [6] for bed expansion in CFB and BFB conditions and on the Ergun [7] equation at the packing limit. In recent years, instead of using empirical correlations as basis, it has been more common to develop models based on transient simulations in fine meshes.
Agrawal et al. [8], Andrews et al. [9], and Igci et al. [10], [11] developed closures for the average drag and stress terms through 2D simulations in small domains with periodic boundary conditions and, by volume-averaging the results, they derived closures for the sub-grid scales. In Igci and Sundaresan [12] and Igci et al. [13], wall effects were included in the analysis of a case of Geldart A particles and in the closures that were derived by volume-averaging 2D transient simulation results for the drag force and the normal and shear stresses of the solid phase. Milioli et al. [14] further developed the closures by including the slip velocity between the phases in the equations. Shah et al. [15] studied wall effects on the gas–solid drag force in a case of Geldart B particles through volume-averaging 2D transient simulation results for a small CFB. Zhang and VanderHeyden [16] discussed the effects of mesh spacing on Reynolds stresses and the drag force. In Zhang and VanderHeyden [17], an added-mass force closure was suggested for the correlation between fluctuations of the pressure gradient of the continuous phase and fluctuations of solid volume fraction. De Wilde [18] analyzed the same term from simulations and accounted also for the average drag force in the derivation of new closure models that were applied in De Wilde et al. [19] for steady state simulation of a riser. In addition to models based on measurement data and transient simulation results, closures for the gas–solid drag force have been suggested on the basis of theoretical analysis of clustering flow in dense gas–solid suspensions [20], [21]. Common to all the suggested sub-grid and steady-state drag models is that the predicted drag force is reduced from what the homogeneous flow drag correlations would predict in the same flow conditions.
The models presented in the literature often take into account only a small number of factors that can affect the magnitude of fluctuations in flow properties. To improve our understanding of the requirements for a comprehensive filtered drag law, the present study analyzes the effects of material properties on the average drag force applicable to steady state modeling. In addition, the effects of process conditions on the required drag closures are evaluated. The analysis is based on time-averaging the results from transient Eulerian–Eulerian simulations of circulating fluidized beds of Geldart B particles. Since 3D simulations are very time consuming, development of sub-grid closure laws has commonly been carried out by averaging results from 2D simulations. For the same reason, this parametric study is carried out in 2D. Although the observed quantitative effects of the studied parameters cannot be applied to 3D simulations with high accuracy, some implications for further 3D studies and closure development can be derived, since the parameters required for drag correction in 2D should be significant also in 3D geometries. Still, as long as a similar dataset from 3D simulations is not available, the data collected in the present study can be used to derive models for the time-averaged drag force. Furthermore, the results of the present study provide an indication which parameters should be included in the drag correction functions applicable to coarse mesh simulations.
Section snippets
Transient simulations
A large number of 2D simulations were carried out for analysis of the required closures for the time-averaged momentum equations. A selection of the simulations with varying gas viscosities, gas densities, solid densities, particle sizes, riser size and mesh spacings was chosen for the present analysis. The geometry is a simple riser with straight walls. A uniform gas inflow is introduced at the bottom and gas and solids exit the riser through the top edge of the simulated domain. Solids that
Effect of solid volume fraction
Volume fraction of solid particles, αs, is the most important parameter affecting the amount of correction that needs to be applied in clustering flow on the customary drag laws derived for homogeneous conditions. For that reason, the drag correction coefficient Cdrag determined from the transient simulations is in the following plotted as a function αs with other flow properties as parameters. In the comparisons, one parameter at a time has been varied in the simulations to allow easy
Conclusions
In the present work, parameters affecting the reduction of the average gas–solid drag force in dense gas–solid suspensions due to particle clustering were studied by analyzing time-averaged results from a number of transient simulations carried out in fairly fine meshes. The analysis was limited to Geldart B particles and CFB conditions. The simulations were carried out in 2D, which limits direct quantitative applicability of the results to 2D simulations. Qualitatively, it can be expected that
Notation
- d
diameter
- h
height
- g
gravitational acceleration
- p
pressure
- Re
Reynolds number
- u
velocity
- U
time-averaged velocity
- U0
superficial velocity
- vt
terminal velocity
- w
riser width
- x
distance
- Δxmesh
mesh spacing
- Δxwall
distance to wall
- Kqs
inter-phase momentum transfer coefficient
- α
volume fraction
- δqs
Kronecker delta
- μ
viscosity
- ρ
density
- τ
stress
Greek symbols
- bot
bottom
- k
coordinate k
- i
coordinate i
- g
gas
- p
particle
- q
phase q
- s
solid
- t
related to terminal velocity
Subscripts
- M
turbulent
- *
calculated from time-averaged values
Superscripts
Acknowledgment
The authors gratefully acknowledge the financial support of Tekes, VTT Technical Research Centre of Finland, Etelä-Savon Energia Oy, Fortum, Metso Power Oy and Numerola Oy, and the support from Saarijärven Kaukolämpö Oy.
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