Author's Accepted Manuscript
3D CFD simulation of a circulating fluidized BEd with on-line adjustment of mechanical valve Cenfan Liu, Mingzhao Zhao, Wei Wang, Jinghai Li
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S0009-2509(15)00491-1 http://dx.doi.org/10.1016/j.ces.2015.07.006 CES12478
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Chemical Engineering Science
Received date: 23 April 2015 Revised date: 5 July 2015 Accepted date: 8 July 2015 Cite this article as: Cenfan Liu, Mingzhao Zhao, Wei Wang, Jinghai Li, 3D CFD simulation of a circulating fluidized BEd with on-line adjustment of mechanical valve, Chemical Engineering Science, http://dx.doi.org/10.1016/j. ces.2015.07.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
3D CFD simulation of a circulating fluidized bed with on-line adjustment of mechanical valve Cenfan Liu1,2, Mingzhao Zhao3, Wei Wang1*, Jinghai Li1 1
State Key Laboratory of Multiphase Complex Systems, Institute of Process Engineering,
Chinese Academy of Sciences, Beijing 100190, China 2
University of Chinese Academy of Sciences, Beijing 100049, China
School of Chemical Engineering, Sichuan University, Chengdu 610065, Sichuan, China
*Email: [email protected]
Past computational fluid dynamics (CFD) simulations on circulating fluidized bed (CFB) are all based on static geometries. To better understand the real operating performance of CFBs and make the simulation closer to virtual experiment, this work is to introduce on-line adjustment of solids flow rate via a mechanical valve in simulation. The two-fluid model (TFM) is used as the governing equations, for which the solids stress is closed by using the kinetic theory of granular flow and the drag is closed by using the EMMS/matrix scheme. The sliding mesh model is used to mimic 1
the dynamic change of the opening or closing of the valve. And its effects on the flow behavior are studied by investigating the variation of axial voidage profile and solids flux in the riser under different solids inventories. The simulation results are in qualitative agreement with experimental observation, as detailed in the literature (Li et al., 1998. Chem. Eng. Sci. 53, 3367-3379). This work can be expected to be a starting point for the virtual process engineering, the next paradigm of CFD simulation.
A CFB with on-line adjustment of valve is simulated with CFD.
The dynamic response of flow of CFB agrees with the experiment.
This is a first step toward the virtual process engineering, an emerging CFD paradigm.
Keywords CFD, simulation, fluidization, meso-scale, EMMS, sliding mesh
A typical circulating fluidized bed (CFB) includes the riser, gas-solid separator, downcomer and solids flow control device (Grace et al., 2003; Jin et al., 2001). Of all 2
these components, the riser has received much research as it is normally the place where reactions happen. The smooth operation of a CFB, which can be characterized by a steady-state distribution of solids, depends not only on the superficial gas velocity and solids flow rate in the riser, but also on the status of solids flow control device (Knowlton, 2003). A solids flow control device can be mechanical or nonmechanical and, in general, it has two functions: preventing the high pressure gas in the riser from leaking into the downcomer and adjusting the solids flow rate to the riser. Typical mechanical devices include e.g. the rotary, screw, butterfly and slide valves, whereas non-mechanical devices include the L-valve, J-valve, U-valve and siphon and so on. The nonmechanical valves are used extensively in circulating fluidized bed combustion (CFBC) systems, whereas in fluid catalytic cracking (FCC) systems, the mechanical slide valve is widely used. Simulating the effects of such valves on the reactor behavior is certainly important to industrial applications. The solids flux can be adjusted by changing the rate of aeration in the nonmechanical valve or the opening of the mechanical valve. Yang and Knowlton (1993) introduced a set of equations to relate the solids flux, the aeration rate with the pressure drop of the L-valve. Bai et al. (1992) found that, when starting a CFB with the smallest opening of the mechanical valve, gradual increase of the opening ratio weakens the inlet restriction, so that the solids flow rate increases until reaching the saturated entrainment capacity and the axial voidage distribution changes accordingly from the exponential profile to the typical S-shaped profile. Li et al. (1998) indicated 3
that the S-shaped voidage profile in the riser corresponds to the “choking” phenomenon in terms of coexistence of two stable flow states, and its variation is closely related with the opening status of the mechanical valve and the solids inventory inside the whole loop of CFB. The choking was initially introduced as a flow instability which was characterized by abrupt rise of pressure drop per unit length of pipe and apparent collapse to a slugging state, when decreasing superficial gas velocity under a constant solids flux (Zenz, 1949). Later it was employed in the context of the so-called “fast fluidization” of CFB and related with the clustering of particles (Bi et al., 1993; Yang et al., 2004; Wang et al., 2007). The choking phenomenon is often discussed with using mechanical valve, such that the solids flow rate is expected to be fixed. When a thin and long riser is used, the size of meso-scale clusters inside the riser may be comparable with the tube diameter, thus causing slugging and choking. When a CFBC with nonmechanical siphon and large cross-sectional furnace is encountered, however, solids are recycled without delay and the flow rate cannot be fixed directly. The smooth transition from the dilute flow to the S-shaped dilute-dense coexisting flow can be achieved and easily adjusted by changing the solids inventory without apparent flow instability (Reh, 1996; Reh, 2003). In all, to understand the flow behavior in a CFB, the effects of geometric factors and solids flow control device should be taken into account besides the normally recognized factors of gas and solids flow rates. That requires a systematic viewpoint of the whole loop of CFB instead of only the riser 4
part. With the rapid development of computational fluid dynamics (CFD) in recent decades, more and more simulations have been done to understand the flow behavior in CFBs. Most of these efforts are focused on simplified geometry of two-dimensional (2D) domain of the riser part (Andrews IV et al., 2005; Benyahia et al., 2003; Dong et al., 2008; Hong et al., 2012; Li et al., 2012; Soundararajan et al., 2001; Tsuji et al., 1998; Yang et al., 2003; Yang et al., 2004). 3D and more vividly 3D, full-loop simulation of the whole CFB system becomes feasible only in recent years (Chu and Yu, 2008; Lu et al., 2013; Xie et al., 2008a, b; Zhang et al., 2008, 2010). It is expected that the 3D, full-loop simulation will pave the way for better understanding of global phenomena such as loop instability (Sundaresan, 2011). However, all these simulations of CFBs are performed for static equipments with fixed domains. In practice, the sliding mesh method can be used to simulate dynamic motion of geometries and it has been applied successfully in CFD simulation of e.g. the rotating screw within a bubbling fluidized bed-moving bed reactor assembly (Schneiderbauer et al., 2015) and the blade propeller in stirred tanks (Luo et al., 1993). To our knowledge, however, there is no CFD simulation of CFBs with mechanically adjustable valve in literature yet. And that is a critical step toward the next paradigm of simulation or the virtual process engineering (VPE) (Ge et al., 2011), through which we hope to be able to virtually adjust the operation of a CFB reactor and get response simultaneously online on a computer screen. And the simulation is exactly 5
like doing experiment on a computer. The global flow phenomenon related with dynamic adjustment, such as the choking instability, can be thus better understood. In this work, we try to implement our first step toward the VPE by carrying out a time-dependent, 3D, full-loop, CFD simulation of a CFB with a mechanical valve. The mechanical valve is modeled by using a sliding mesh zone and it can be partially or fully closed online during simulation. The variation of flow behavior with valve status is discussed. And the effects of solids inventory are also investigated.
2.1 Governing equations The simulation is performed using Ansys Fluent 13 parallel solver with 16 cores, in which the multiphase Eulerian granular model is used. The drag coefficient is closed by using the EMMS/matrix model (Wang and Li, 2007), which has been validated in our previous simulations of the same riser (Hong et al., 2013; Lu et al., 2009; Wang and Li, 2007). The detailed fitting functions of the drag are given in Table 1 and the relevant governing equations are given in Table A1. The kinetic theory of granular flow (KTGF) (Gidaspow, 1994) is used to close the solid-phase pressure and viscosities. For the sake of numerical convergence, the algebraic approximation of KTGF in Fluent is adopted in computation of granular temperature. We do not include turbulence model in this work, because the influence of the turbulence decays with the 6
increase of solids concentration. For example, as indicated in Elgobashi (2006), the influence of turbulence on the gas-solid velocity field with solids concentration higher than 0.001 is weak, whereas in a CFB, the solids concentration is normally much higher than 0.01. Furthermore, the use of turbulence models is usually related to much more empirical parameters and hence additional run time. As the major interest of this research is on the dynamic response of riser hydrodynamics to the valve adjustment, we would like to leave more elaborate research on cyclone separator with consideration of turbulence to future efforts.
2.2 Geometry and settings The main objective of this article is to show how the on-line adjustment of a valve for controlling solids flux can affect the flow behavior of a CFB. Such dynamic response is important to understand the loop instability frequently encountered in practice, e.g., the choking phenomenon. The experimental setup of Li and Kwauk (1994) well fits this objective, because it is the only one to our knowledge that has records of the dynamic variation of axial profile of voidage along with the valve closing/adjustment. Indeed because of its importance in understanding the choking phenomenon, this experiment was even repeated as documented in Li et al. (1998). To realize the objective of this article, we tried our best to build the 3D geometry according to the diagram in literature (Li and Kwauk, 1994) and our own record.
Based on this geometry, we can reproduce the experimental phenomena when the valve is fully open or closed, as will be detailed in Sections 3.1 and 3.3. However, when the valve is partially closed, we have no record of the valve opening ratio. Thus, it will be hard to reproduce quantitatively the axial displacement of the inflection point of the S-shaped curve, as will be discussed in Section 3.2. As shown in Fig. 1, the CFB consists of a riser of 90 mm I.D. and 10.5 m in height, whose exit is connected to a gas-solid cyclone separator. The collected particles flow downward through a downcomer of 120 mm I.D.. The detailed geometric parameters are listed in Table 2. To keep constant solids inventory inside this CFB, the particles entrained out of the vortex finder of cyclone, where the atmospheric pressure outlet is prescribed, are recycled into the riser through the bottom aeration inlet. The mechanical valve mounted on the inclined pipe, which is between the loop seal and the riser, is modeled by using a sliding mesh zone. The detailed configuration of valve is provided in the next section. Ansys ICEM 13.0 is used to discretize the fluidized bed with hexahedral meshes. The grid number totals about 400,000. According to our simulation experience and grid-dependency test in similar or larger units (Zhang et al., 2008; Zhang et al., 2010; Lu et al, 2013; Liu et al., 2015), such resolution is sufficient to predict the inhomogeneous gas-solids flow behaviors in this CFB, because sub-grid, meso-scale modeling of the drag force allows reasonable prediction with using coarse grid (Wang et al., 2010). The primary gas inlet and the aeration inlet are prescribed as the velocity inlets, 8
where the superficial gas velocity of the primary one is set in line with the experimental data, 1.52m/s. The aeration inlet is specified with a constant of 0.005m/s (~Umf) to keep the particles fluidized in the loop seal. The no-slip boundary is set for the gas-phase at the wall and the partial slip boundary is set for the solid phase. The detailed settings are summarized in Table 3.
2.3 Sliding mesh for mechanical valve To mimic the closing of the mechanical valve, we apply the sliding mesh method in Fluent. When using the sliding meshes, the cells belonging to the sliding zone move rigidly without deforming (Fluent, 2006). The static and sliding zones are associated with one another to form a “grid interface”, along which the two cell zones move relative to each other. And any part of the interface zones that does not contact with the adjacent interface is treated as the wall. As shown in Fig. 2, here the mechanical valve is represented with a sliding mesh zone, and the cell faces connecting the valve and the inclined pipe are grid interfaces. Initially, the valve is open and the interfaces are totally associated with one another to form a full cross section for flow. When the sliding mesh zone moves in a direction perpendicular to the inclined pipe, the connecting area between the pipe and sliding valve decreases, thus the flow resistance increases and the valve is partially closed. Once the sliding zone moves out of the inclined pipe and the connecting area of the
interfaces vanishes, the solids circulation from the downcomer to the riser is cut off and the valve is completely closed. Three typical positions of the sliding valve relative to the static loop seal are schematically given in Fig. 3 to highlight the whole process of valve closing. The solids flux at the cyclone dipleg and the mean solids concentration in the riser are continuously monitored during simulation. To investigate the effects of the valve opening, the mechanical valve is partially closed or fully closed after the gas-solid flow in the CFB reaches steady state. The time span of the uniform, partial/full closing is 0.6s. The cross-sectionally averaged solids fraction in the riser at different elevations will be recorded to show the variation of axial voidage profiles against valve adjustment.
3.1 Valve open When the valve is open, the sliding zone of the mechanical valve is thoroughly connected with the other static parts of the CFB. The simulation is hence similar to what we have reported in Zhang et al. (2008). Fig. 4 gives a snapshot of how the solids are distributed in the CFB loop and several cross-sectional views of solids volume fraction and axial solids velocity at different elevations of the riser. A dense bottom and a dilute top can be distinguished 10
in the riser, where the dense clusters are more likely formed in the lower section and near the wall. A densely fluidized bed with clear surface can be found below the falling solids in the downcomer. And the fluidized particles overflow into the riser bottom through the mechanical valve smoothly as shown in the close-up of the loop seal. In the riser, the solids in the dense region near the wall tend to flow downwards and the flow direction reverses in the dilute core area. Fig. 5 gives a close-up of the contour of solids distribution in the cyclone and two typical gas streamlines to describe the gas-solid separation. The solids descend in a swirl near the wall, finally discharged to the dipleg and downcomer. The gas streams swirl downward first near the wall (the outer vortex) together with the solids until reaching near the end of the cone, where the so-called vortex stabilizer can be installed (Chen, 2011), then the gas flow reverses and swirls upward out of the outlet of the vortex finder (the inner vortex). This result is in accordance with the gas flow behavior observed in cyclones (Peng et al., 2002; Hoffmann and Stein, 2002). Fig. 6 shows the simulated results of pressure balance in the CFB. It is worth noting that all the data hereafter are from simulation instead of experimental measurement. The pressure data are collected by cross-sectional averaging across the whole loop. Large pressure gradient is found in the lower parts of both the riser and downcomer. Here we do not provide quantitative comparison between experiments and simulation, because the exact data of solids inventory of the whole CFB is not available. Anyway, the qualitative behavior revealed in this simulation agrees with our 11
experience as discussed in Zhang et al. (2008). That is the basis of our further investigation on how the valve opening affects the flow behavior in the CFB. The solids flux and mean solids volume fraction in the riser are collected to see when the steady state is reached. The quantitative analysis is performed between 20s and 30s when the valve is open and the solids flux and mean solids volume fraction oscillate around certain constant values. When the valve is adjusted during simulation, 10 more seconds will be performed to ensure reaching steady state. Fig. 7 shows how the time-averaged solids flux and mean solids volume fraction in the riser change with the overall solids inventory. The mean solids volume fraction in the riser increases almost linearly with the solids inventory. The solids flux behaves differently, showing an almost constant solids flux of 23.5kg/m2s between solids inventory of 27kg and 29 kg. Such a plateau corresponds to the saturation carrying capacity (K*), which is the solids flow rate at the so-called choking in engineering. And it was found closely related to the axially S-shaped profile of voidage, as revealed in CFB experiments (Li and Kwauk, 1994) and simulations (Wang et al., 2007, Wang et al., 2008). The prediction of K* (23.5kg/m2s) is higher than the experimental data (14.3kg/m2s) at superficial gas velocity of 1.52m/s. However, compared to TFM simulation of the same riser with homogeneous drag and much finer grid size (1mm in both axial and lateral directions) (Benyahia, 2012), which predicted solids flux of 130kg/m2s, almost ten times higher than experimental data, the current simulation result is acceptable. Further refining grid may help improve the 12
quantitative agreement with experiment. However, it is easy to see that doubling the grid number in each direction will lead to cell count of over millions. The transient simulation with such resolution will take several months for even one testing case, which is obviously beyond the available capability of computing resources. In addition, it should be noted that qualitative trend of the flow distribution and choking transition is well predicted in this work. That provides a reasonable base for further investigating the effects of valve on flow behavior. After all, we are still short of the physical understanding of the loop instability. The experimental work is few on the dependence of valve adjustment on the dynamic behavior of riser flow, and the available result remains largely a qualitative description as indicated in Li et al. (1998). As a result, qualitative comparison is of major concern of this work at the current status of development. After the plateau of the choking transition, the gas-solid flow is quite dense from the bottom to top sections of the riser. In this situation, the dense gas-solid flow is much affected by the outlet and hence the profile may manifest certain different change. We think that is probably the reason for the phenomenon at the inventories of 34-36kg. More definite explanation needs more specific simulations by varying the design of the outlet and operating conditions, which is however beyond the objective of this work. Fig. 8 shows the axial profiles of cross-sectionally averaged voidage at different solids inventories. The simulation results are given on the left while the experimental data are given on the right. At I=20kg, the whole riser is mainly occupied by the dilute 13
flow with a dense bottom near the loop seal, above which the voidage increases nearly exponentially with height. When the solids inventory increases to 27kg, the voidage at the dense bottom varies little whereas the voidage at the top gets smaller and the distribution manifests somewhat S-shaped profile. At between 27 and 30kg, corresponding to the plateau in Fig. 7, the voidages at both the top and the bottom change little. The increase of solids inventory mainly results in a higher inflection point of the S-shaped curve. Further increase of the solids inventory to more than 34kg leads to an all-dense flow, where voidages of both the top and the bottom shift to the denser side. It should be noted that the ideally vertical curve with respect to both all-dilute and all-dense flows cannot be found in this figure because of the strong limitation of the inlet/outlet region. Moreover, the predicted voidage in the dense bottom section is around 0.8, which is in reasonable agreement with experiments. Probably due to overpredicted saturation carrying capacity as discussed above, the solids concentration in the upper section is generally denser in simulation than in experiment as shown on the right hand side of Fig. 8. The higher saturation carrying capacity also leads to different range of solids inventory where the choking transition happens. Generally, the qualitative trend of simulation results, as shown by the choking transition with respect to the S-shaped profile, is similar to the experiment observed by Li and Kwauk (1994).
3.2 Valve partially closed Li and kwauk (1994) found that, when the axial voidage profile is S-shaped with the co-existence of dilute top and dense bottom regions and the solids flux is equal to the saturation carrying capacity (K*), then, “if the opening is decreased, the input solid rate becomes smaller than K* temporarily. However, the output solid rate still remains at K*, thus depleting the fast bed of solids. Meanwhile the voidage of the dilute region at the top also remains constant. As a result, the dilute-phase region extends toward the bottom of the bed, and the new position of the inflection point will be lower”. Such an online adjustment is typical of CFB operation. To simulate that process of closing valve, we turn off partially the valve by moving the sliding zone 90% out of the pipe at No. 45s after reaching steady state of flow. The process of closing takes 0.5s. The solids inventory is set to be 28kg, which corresponds to a state at the plateau. From Fig. 9 we can see that the solids flux of the riser has almost no change after the valve is partially closed, so does the amplitude of the solids flux fluctuation. In contrast, the mean solids volume fraction in the riser drops apparently, and the amplitude of fluctuation becomes smaller. This phenomenon is closely related with the choking. As indicated by Li and Kwauk (1994), at Gs = K*, a change of the opening of the valve can only upset the initial equilibrium temporarily. When the
valve is partially closed, the input solids rate becomes smaller than K* temporarily. However, the output solid rate still remains at K*, thus, depleting the solids in the riser. Meanwhile the voidage of the dilute region at the top remains unchanged. As a result, the dilute-phase region extends toward the bottom of the bed. The excess solids which have thus far been removed from the riser are accumulated in the bottom of downcomer, thus increasing the imposed pressure, forcing more solids flow from the bottom of the downcomer into the riser. This dynamic process results in Gs = K* again, but at a new equilibrium position with lower inflection point of the S-shaped profile than before. Fig. 10 shows the solids volume fraction contours near the valve. After the valve is partially closed, the solids flow out is more than the solid flow in temporarily. So the solids accumulate in the downcomer, and a dense zone is formed behind the valve, thus increasing the imposed pressure over the valve, forcing more solids flowing into the riser until a new equilibrium between the input and entrainment is built again. The axial voidage profiles before and after the valve is partially closed are shown in Fig. 11. The voidage at the top and bottom sections remains almost unchanged, whereas the height of the inflection point of the S-shaped curve drops a little. All these results are in agreement with the phenomenon of closing the valve in Li and Kwauk (1994). The increase of valve resistance in simulation is realized only by the decrease of connecting area. However, as indicated by Nikolopoulos et al. (2012), the TFM may not effectively simulate the inter-particle friction forces in the recirculation 16
system, because the respective stress tensor does not incorporate compressibility of flow due to change of effective particle density. As a result, the induced normal and shear stresses may need more accurate models to account for valve resistance. And that may be the reason why the downward displacement of S-shaped curve is not very obvious in this simulation. More study on the treatment of mechanical valve in simulation is needed.
3.3 Valve closed Fig. 12 shows the variation of the solids flux and mean solids volume fraction in the riser against time before and after the valve is fully closed. The mean solids volume fraction drops instantly. In contrast, the solids flux remains almost at the same level for a period of time prior to decaying. The variation of axial voidage profiles in the riser after closing the valve is shown in Fig. 13. The time represents how long it takes since the valve is closed. The axial voidage profiles are obtained from the cross-sectionally averaged voidage at different heights. At 0s the initial curve of axial voidage is S-shaped. The closing of valve triggers the downward displacement of the inflection point of the S-shaped curve, with little change for the voidages at the two ends. At 12s the dense region almost disappears. Afterwards the whole riser becomes more and more dilute, while the solids flux drops gradually. During this period of time, the axial voidage profile
takes the shape of dilute phase conveying, with little difference between the top and bottom. And the voidage keeps increasing until the bed is emptied. It should be noted that the original experiment with closing valve was conducted at gas velocity of 1.05m/s (Li et al., 1998), in which the saturation carrying capacity was much smaller than in this work and accordingly the time to empty the riser (around 10 minutes) was much longer. Such a long process is expected to demand transient simulation of more than half a year, which is beyond our computing capability. That is why we choose the testing case with 1.52m/s. Anyway, such a process is qualitatively in agreement with our experimental findings as detailed in Li et al. (1998).
A CFB with a mechanical valve is simulated in this work. The sliding mesh model is successfully used to simulate the on-line closing of the mechanical valve. The simulation results reveal the dynamic response of CFB when the valve is partially closed or fully closed, which agree well with experimental observation. This work is a first step toward the virtual process engineering, on which one may realize simulation as vivid as a real experiment in both operation and measurement.
Standard drag coefficient for a particle
Diameter of particle, m
Gravitational acceleration, m/s2
radial distribution function
Solid flux, kg/m2s
Solids pressure, Pa
Superficial gas velocity, m/s
Real velocity, m/s
Real slip velocity between gas and solids, m/s
Initial solids inventory, kg
drag coefficient in a control volume, kg/(m3s)
Granular temperature, m2/s2
Minimum fluidization voidage
collisional dissipation of energy, J/(m3s)
diffusion coefficient for granular energy, Pa s
viscosity, Pa s
Stress tensor, Pa
Bulk viscosity, Pa s
This work is financially supported by the Ministry of Science and Technology under Grant No. 2012CB215003, by the National Natural Science Foundation of China under Grant Nos. 91334204 and 21176240, and by the Chinese Academy of Sciences under Grant No. XDA07080100.
The governing equations for gas-solids flow are shown in Table A1.
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! Figure 1. Geometry and mesh of the CFB unit. Figure 2. The schematic of sliding zone for the mechanical valve in simulation. Figure 3. The schematic of three positions of the sliding zone relative to the static loop seal. Figure 4. A snapshot of the solids distribution in the CFB as well as the cross-sectional image of distribution of solids volume fraction and axial solids velocity at different elevations of the riser (Ug=1.52m/s; I=28kg). Figure 5. Gas flow streamlines in the cyclone (Ug=1.52m/s; I=28kg). Figure 6. Simulated pressure balance across the whole loop of CFB (Ug=1.52m/s; 27
I=28kg). Figure 7. The variation of solids flux and mean solids volume fraction in the riser with solids inventory. Figure 8. The axial distribution of cross-sectionally averaged voidage in the CFB riser with different solids inventories. Figure 9. The time series of solids flux and the mean solid volume fraction in the riser before and after the partial closing of the valve (Ug=1.52m/s; I=28kg). Figure 10. The solids volume fraction contours before and after the valve is partially closed (Ug=1.52m/s; I=28kg; (a) before; (b) after). Figure 11. The axial voidage profile in the riser before and after the valve is partially closed (Ug=1.52m/s; I=28kg). Figure 12. The time series of solids flux and mean solids volume fraction in the riser before and after closing the valve (Ug=1.52m/s; I=28kg; the red dot line tags the time when the valve begins to close; the closing process takes 0.5s).
Figure 13. The variation of axial voidage profiles after the valve is fully closed (Ug=1.52m/s; I=28kg).
! Table 1. Fitting functions of the heterogeneity index, HD, for the CFB riser (HD=a·Reb ,
ρp=930kg/m3, ρg=1.1795 kg/m3, µg=1.8872×10-5 PaÂs, dp=54 µm, Ug=1.52 m/s, Gs=14.3 kg/(m2Âs), εmf=0.4, Re=dp|uslip|/vg).
Table 2. The detailed geometric parameters of the CFB
Table 3. Material properties and simulation settings in Ansys Fluent.
Table A1. Summary of governing equations for gas-solid flow.
Table 1. Fitting functions of the heterogeneity index, HD, for the CFB riser (HD=a·Reb ,
ρp=930kg/m3, ρg=1.1795 kg/m3, µg=1.8872×10-5 PaÂs, dp=54 µm, Ug=1.52 m/s, Gs=14.3 kg/(m2Âs), εmf=0.4, Re=dp|uslip|/vg).
Fitting formulae (HD=a·Reb, 0.001 Re 100)
Range (εmf εg 1)
°a = 235.48668 − 1666.97504ε g + 3921.62004ε g2 − 3058.91479ε g3 ® 2 3 °¯b = 4.63415 − 33.66493ε g + 81.33114ε g − 35.32964ε g
°a = 429.22082 − 2943.10294ε g + 6717.54689ε g2 − 5096.01902ε g3 ® 2 3 °¯b = −125.27204 + 842.13627ε g − 1887.23004ε g + 1409.9084ε g
° a = 582.77336 − 3259.42796ε g + 6079.47116ε g2 − 3781.28192ε g3 ® 2 3 °¯b = 81.6591 − 531.63534ε g + 1131.62258ε g − 788.26231ε g
° a = −0.27879 + 1.39427ε g − 2.11157ε g2 + 1.12754ε g3 ® 2 °¯b = 0.53789 − 0.07791ε g − 0.18593ε g
° a = −782.53911 + 2525.11521ε g − 2715.89268ε g2 + 973.77958ε g3 ® 2 3 °¯b = 234.68943 − 756.05136ε g + 813.40881ε g − 291.88676ε g
° a = −98.38128 + 100.06638ε g ® 2 °¯b = 978.72169 − 1959.15883ε g + 980.4769ε g
Table 2. The detailed geometric parameters of the CFB. Height of the riser
Diameter of the riser
Diameter of the downcomer
Diameter of the cyclone body, mm
Height of the cyclone body, mm
Height of the conical part of cyclone, mm
Diameter of the vortex finder, mm
Height of the vortex finder, mm
Table 3. Material properties and simulation settings in Ansys Fluent. Air density ρg
Air viscosity µg, kgm-1s-1
Particle density ρp,
Particle diameter dp, µm
Unsteady, Second Order Implict
Second Order Upwind
Volume fraction discretization
Primary inlet gas velocity (m/s)
Aeration inlet gas velocity (m/s)
Solids inventories (kg)
20, 23, 27, 28, 29, 30, 32, 34, 36
Boundary condition for gas
Boundary condition for solids
Time step (s)
Granular bulk viscosity
Lun et al.
Angle of internal friction ( ) Solids pressure
Lun et al.
Lun et al.
Packing limit of solids
Table A1. Summary of governing equations for gas-solid flow Continuity equations of gas and solid phases: ∂ (ε g ρg ) + ∇ ⋅ (ε g ρg ug ) = 0 ∂t ∂ ( ε s ρs ) + ∇ ⋅ ( ε s ρ s us ) = 0 ∂t
Momentum equations of gas and solid phases: ∂ (ε g ρg ug ) + ∇ ⋅ (ε g ρg ug ug ) = − ε g∇p + ∇ ⋅ Ĳ g + ε g ρg g − β ( ug − us ) ∂t ∂ ( ε s ρ s us ) + ∇ ⋅ ( ε s ρ s us us ) = − ε s∇p − ∇ps + ∇ ⋅ Ĳ s + ε s ρ s g + β ( ug − us ) ∂t
Granular temperature equation: 3 ∂ (ε s ρs Ĭs ) + ∇ ⋅ ( ε s ρ s usĬs ) = ( − ps I + Ĳ s ) : ∇us + ∇ ⋅ (κ s∇Ĭs ) − γ s − 3β Ĭs 2 ∂t
Stress-strain tensors for gas and solid phases: T 2 Ĳ g = ε g µg ª∇ug + ( ∇ug ) º − ε g µg ∇ ⋅ ug ǿ ¬« ¼» 3
2 · T § Ĳ s = ε s µs ª∇us + ( ∇us ) º + ε s ¨ λs − µs ¸ ∇ ⋅ us I ¬ ¼ 3 ¹ © Solid shear viscosity:
µs = µs,col + µs,kin + µs,fr where, 4 5
µs,col = ε s ρ s d p g 0 (1 + es ) Θs / π
2 10 ρs d p Ĭ sπ ª 4 º 1 + g 1 + e ε ( ) 0 s s » 96ε s (1 + es ) g 0 «¬ 5 ¼
ps sin ĭ 2 I 2D
Solids bulk viscosity 4 3
λs = ε s ρs d p g 0 (1 + es ) Ĭs / π Solids pressure:
ps = ε s ρsΘs ¬ª1 + 2 (1 + es ) ε s g 0 º¼ Radial distribution function
1/ 3 g 0 = ª«1 − (ε s ε s,max ) º» ¬ ¼
Diffusion coefficient for granular energy
2 150 ρs d p Ĭsπ ª 6 º 1 + ε s g 0 (1 + es ) » + 2 ρsε s2 d p (1 + es ) g 0 Ĭs π « 384 (1 + es ) g 0 ¬ 5 ¼
Collisional dissipation of energy
12 (1 − es2 ) ε s2 ρs g0Ĭ3/2 s dp π
β = CD
ε sε g ρg ug − us dp
ε g-2.65 H D
where, 24 1 + 0.15 Re0.687 , Res ≥ 1000 ° s CD = ® Res ° 0.44, Res < 1000 ¯
ε g ρg d p ug − us µg