Particle volume fraction (αs)
y (m)
0.04 0.08 0.12
0.05 0.1 0.15
φ=0.0 φ=0.5 φ=0.8 For e=0.9
Figure 5.5: Variation of steady state particle volume fraction profiles with φ at the midplane of the channel for e=0.9.
near the wall decreases when the value of φ increases. The particles which move away from the wall tend to accumulate at some distance away from the wall where wall effect is not influential and hence, it can be seen that for non-zero values of φ, the particle volume fractions tend to increase as we move away from the wall and becomes maximum at a certain distance from the wall. The maximum value of particle volume fraction at the midplane of the channel is found to be obtained for φ=0.8; at a particular value ofe. At the central or core region of the channel which is far away from the wall, steady state volume fractions for all the three values of φ are almost same and equal to the inlet value of particle volume fraction.
5.4 Effects of variation of particle-particle resti-
5.4 Effects of variation of particle-particle restitution coefficient 95 0.1. The particle phase density is considered to be equal to 2500 kg/m3 whereas the gas phase is considered to be air with a density of 1.2 kg/m3. The effects of e on the flow physics have been studied for three different values (e=0.99, 0.9 and 0.8) at a particular value of φ=0.5. The particle diameter is taken to be 530µm.
5.4.1 Effect on particle velocity profiles
The variation in the steady state particle phase velocity profiles at the midplane of the channel (x=0.2 m) with e are shown in Fig. 5.6 for a value of φ=0.5. As the
Particle velocity (us) (m/s)
y (m)
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
0 0.05 0.1 0.15
e=0.99 e=0.9 e=0.8 For φ=0.5
Figure 5.6: Variation of steady state particle velocity profiles witheat the midplane of the channel for φ=0.5.
value of φ is fixed, so there is negligible variation in the value of particle velocity at the wall for all the values of e. But, there exists a difference in particle velocities in the core region of the channel for different values of e. For e=0.99, the loss of momentum due to particle-particle collisions is minimum. As a result, maximum value of steady state particle velocity can be obtained at any section of the channel for e=0.99. For e=0.8, the loss of momentum due to particle-particle collisions is maximum out of the three values of e and so, steady state particle velocity at any section of the channel is minimum for e=0.8. Hence, with increase of values of e, particle velocity inside the domain increases. Table 5.3 shows that the particle velocity obtained inside the channel is maximum when the value of e is maximum, at a particular value of φ.
5.4.2 Effect on gas velocity profiles
The corresponding steady state gas phase velocity profiles at the midplane of the channel (x=0.2 m) obtained for the three values of e at a particular value of φ=0.5 are shown in Fig. 5.7. Also, the values of maximum gas velocities obtained inside the domain are shown in Table 5.3. It can be observed that at a particular value
Gas velocity (ug) (m/s)
y (m)
0.5 0.6 0.7 0.8 0.9 1 1.1
0.05 0.1 0.15
e=0.99 e=0.9 e=0.8 For φ=0.5
Figure 5.7: Variation of steady state gas velocity profiles with e at the midplane of the channel for φ=0.5.
of φ, as there is no variation in the particle velocity at the wall, so the drag force acting on the gas phase near the wall for all the values of e is same and hence, the steady state gas velocity near the wall is almost same for all the values of e. As Table 5.3: Maximum gas and particle velocity inside the channel for different values of e at a particular value of φ
Case φ e Max gas velocity (m/s) Max particle velocity (m/s)
1 0.5 0.99 1.16 1.148
0.9 1.12 1.112
0.8 1.105 1.09761
5.4 Effects of variation of particle-particle restitution coefficient 97 discussed in Section 5.4.1, particle velocity us is maximum for e=0.99 and so the tendency to increase the gas phase velocity also becomes maximum when e=0.99.
As a result, maximum gas velocity inside the channel is obtained for e=0.99 and it decreases with decrease in the value of e. It can be seen from Table 5.3 that at a particular value ofφ, the gas velocity obtained inside the domain is maximum when the value of e is maximum and it is minimum when the value of e is minimum.
5.4.3 Effect on particle phase volume fraction profiles
The steady state particle phase volume fraction profiles for different values of e at the midplane of the channel (x=0.2 m) have been plotted in Fig. 5.8 for φ=0.5. As discussed in Section 5.3.3, at a particular non-zero value of φ, the particle volume fraction tends to increase as we move away from the wall to a local maximum value for all the values ofe and forms a peak as seen in Fig. 5.8. But this local maximum
Particle volume fraction (αs)
y (m)
0.02 0.04 0.06 0.08 0.1 0.12 0.14
0.05 0.1 0.15
e=0.99 e=0.9 e=0.8 For φ=0.5
Figure 5.8: Variation of steady state particle volume fraction profiles with e at the midplane of the channel for φ=0.5.
value of particle volume fraction is found to increase with increase in the value of e. The reason behind this increase is that as the value of e is increased, the particle velocity increases due to less loss of particle momentum due to particle-particle collisions. In other words, the particles can move more freely and so, more and more particles which get deflected by the wall tend to accumulate in that section as the value of eis increased. Hence, the maximum peak value of steady state particle volume fraction has been obtained for e=0.99. In the central or core region of the
channel where wall effect is minimum, steady state volume fraction for all the three values of e are almost same and equal to the inlet value of particle volume fraction.
A point worth noting is that the length of the region where steady state particle phase volume fraction is constant and equal to the inlet value is minimum when e=0.99 and it increases as the value of edecreases.