0.0 0.2 0.4 0.6 0.8 1.0
2.5 3.0 3.5 4.0 4.5
Mean axial velocity (m/s)
Non dimensional radius (r/R) H = 1.4 m H = 1.5 m H = 1.6 m
0.0 0.2 0.4 0.6 0.8 1.0
-0.02 -0.01 0.00 0.01 0.02
Mean radial velocity (m/s)
Non dimensional radius (r/R) H = 1.4 m H = 1.5 m H = 1.6 m
Visual observation of flow field in riser reveals large scale and meso-scale meta stable structures in axial and radial directions. Stationary nature of these snapshots cannot characterize the flow fluctuations. In order to characterize the fluctuating behavior of solids, second order moment of solid velocity (spatial correlation of fluctuation flow field
v ) is studied (Tennekes and Lumley, 1972; Monin and yaglom, 1975). Spatial
correlations of the second order moments are related to the Reynolds stresses,
q pv
v , i.e.
transport of momentum in q direction due to the fluctuation in p direction. It is to be noted that such an averaged way of representation leads to loss of information on scales of turbulences. Obtaining the averaged turbulence quantities are challenging for the multiphase flow systems and only limited data is available in the literature on CFB.
In the current work, solids turbulence is quantified following similar approach to the fluids.
From the instantaneous velocities and ensemble average velocities in each grid, fluctuation velocities and root mean square velocities are calculated as,
i j k
v
i j k
v
i j k
vq , , q , , q , , (3.11)
2 q RMS
q v
v (3.12)
Further, Reynolds stresses are obtained as,
i j k
v i j k
vq s
p
qs , , , ,
(3.13)
From equation (3.11) and (3.13) six independent symmetric stress tensors are obtained.
Figure 3.19 shows the root mean square (RMS) fluctuations at three different heights for the operating conditions of Ug – 8.8 m/s and Gs – 110 kg/m2s. Radial variation of axial RMS velocities at different height show similar trend and almost remains same for all the heights.
Similarly, radial RMS velocities do not change with the height. Once again this proves that flow is fully developed.
Axial RMS velocities of solids are low at the center and gradually increase towards the wall. Similar kind of profile has been previously reported by Tsuji et al. (1984), Caloz (2000), Mathisen et al. (2008), He et al. (2009), Gopalan and Shaffer (2013), Pantzali et al.
(2013) in CFB. Similar profile has also been reported for both Geldart Group A (Pantzali et al., 2013) and Group B (Caloz, 2000; He et al., 2009; Gopalan and Shaffer, 2013) particles. However, Caloz (2000) and He et al. (2009) reported axial RMS velocities attains the maximum value around r/R = 0.8 and decreases towards the wall. In this work, such a maximum value is observed at the wall itself. Similarly, Mathiesen et al. (2000), Mathisen et al. (2008) and Pantzali et al. (2013) also reported maximum fluctuations at the wall only.
The reason behind this phenomenon is not clear at this point. This might be due to the low volume fraction observed in these works which have not attained the critical volume fraction near the wall. In other words, fluctuations due to mean free path are still dominating and decrease in the mean free path is not critical enough to decrease the fluctuations.
Further it is to be noted that radial RMS velocity and radial mean velocity is low near the wall. Thus, solid – wall interactions are not dominant. However, the wall shear due to the velocity gradient is not negligible. Further, as information on gas turbulence and effect of clusters on solid turbulence is not available, it is difficult to quantify the forces responsible for the solid fluctuations.
Axial RMS velocities are approximately five times higher than the radial RMS velocities.
This confirms that fluctuations in axial and radial directions are not same. Therefore, the flow is anisotropic and fluctuations are dominant in the primary flow direction. Further, the radial variation of mean axial velocity and axial RMS velocity of solids are not same. This indicates that different mechanism/forces govern the mean velocity and fluctuations. Mean
velocity of solids is primarily due to the drag and majorly governed by the gas motion.
However, the fluctuations are majorly governed by the solid interactions (both solids-solids and solids-wall) and metastable structures.
Figure 3.20 shows the normal and shear Reynolds stresses for the same operating conditions, Ug – 8.8 m/s and Gs – 110 kg/m2s. Results show that normal Reynolds stress per unit density in the axial direction is significant. However, in the radial and tangential directions (shear Reynolds stresses) the values are almost zero. This indicates that flow is anisotropic. Previously, anisotropic condition in CFB has been experimentally proved by lot of studies (Tartan and Gidaspow, 2004; Bhusarapu et al., 2006; Pantzali et al., 2013).
Interesting observation is that shear stress rz is very less, almost negligible compared with
zz. Similar kind of observation has been made by Tartan and Gidaspow (2004), with the particle size of 530 micron and density of 2460 kg/m3. However Bhusarapu et al. (2006), Pantzali et al. (2013) and Ibsen et al. (2002) reported significant values of rz, for the solids particles having lesser momentum. Insignificant values of rz might be due to the particle properties, as solids have high momentum, change in the direction of the motion is less likely to happen. From the present and previous studies, it can be concluded that particle properties play a major role in the radial motion of solids. Other kinetic and shear stresses are negligible.
Figure 3.19 Azimuthally averaged RMS velocities at different heights for operating condition of Ug – 8.8 m/s and Gs – 110 kg/m2s
0.0 0.2 0.4 0.6 0.8 1.0
0.3 0.4 0.5 0.6 0.7
Axial RMS velocity (m/s)
Non dimensional radius (r/R) H = 1.4 m H = 1.5 m H = 1.6 m
0.0 0.2 0.4 0.6 0.8 1.0
0.00 0.02 0.04 0.06 0.08 0.10
Radial RMS velocity (m/s)
Non dimensional radius (r/R) H = 1.4 m H = 1.5 m H = 1.6 m
Figure 3.20 Azimuthally averaged Reynolds stress at different heights for operating condition of Ug – 8.8 m/s and Gs – 110 kg/m2s