due to the increase in the value of interphase momentum exchange coefficient (Kgs) with increase in the value of particle phase volume fraction which in turn increases the interphase drag exerted by the gas phase on the particles.
4.4 Study of effect of inlet slip between the phases 81 loss due to the collisions of the particles with the wall and particle-particle collisions are perfectly elastic. The particle diameter is considered to be equal to 300 µm.
The slip is provided between the phases in two ways.
• Gas velocity (ug)=1.0 m/s and particle velocity (us)=0.5 m/s.
• Gas velocity (ug)=0.5 m/s and particle velocity (us)=1.0 m/s.
In both the cases, the flow Reynolds number based on the channel height and inlet gas velocity is kept to be equal to 200. A uniform grid of 2480 quadrilateral cells is used for the simulations. The changes in the steady state phase velocity and volume fraction profiles as well as the mass flow rates of both phases are studied and compared with the case when both the phases enter with the same magnitude of velocity.
4.4.1 When gas velocity is greater than particle velocity at inlet (u
g>u
s)
Figure 4.8 (a) shows the comparison of steady state gas velocity profiles when (i) both the phases enter the channel with same velocity (ug=us=1 m/s, no inlet slip condition) and when (ii) particle velocity at inlet is less than the gas velocity (ug>us, ug=1 m/s and us=0.5 m/s) at the midplane (H=0.2 m) of the channel. It can be seen that although the inlet gas velocity for cases (i) and (ii) mentioned here are the same, yet the steady state gas velocity obtained at the midplane of the channel is less in case (ii) than that of case (i). It is due to the reason that in case (ii), as the inlet particle velocity is less than the inlet gas velocity, there exists a tendency of the gas phase to increase the particle phase velocity and hence there is more momentum transfer from the gas phase to particle phase in this case than in the case of same inlet velocities. As a result, the steady state gas velocity at any section for this case is less than the case where there is same inlet velocity for both the phases. Table 4.6 shows the steady state mass flow rates at different sections of the channel and it can be seen that the steady state mass flow rate for the gas phase remains the same in both the cases (i) and (ii). But as the steady state gas phase velocity at any section in case (ii) is lower than that of case (i), so the gas phase volume fraction has to increase in order to keep the mass flow rate same in both the cases. Hence, it can be seen in Fig. 4.8 (b) that the steady state gas phase volume fraction at the
Gas velocity (us) (m/s)
y (m)
0.75 0.8 0.85 0.9 0.95 1
0.05 0.1 0.15
ug=1m/s, us=1 m/s ug=1m/s, us=0.5 m/s
Gas volume fraction (αg)
y (m)
0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92
0.05 0.1 0.15
ug=1m/s, us=1 m/s ug=1m/s, us=0.5 m/s
(a) (b)
Particle volume fraction (αs)
y (m)
0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22
0.05 0.1 0.15
ug=1m/s, us=1 m/s ug=1m/s, us=0.5 m/s
(c)
Figure 4.8: Comparison of plots of steady state (a) gas velocity,(b) gas volume fraction and (c) particle volume fraction profiles in the midplane of the channel for (i) ug=us=1 m/s; (ii) ug=1 m/s, us=0.5 m/s
midplane of the channel is greater in case (ii) than in case (i). Since at any section of the channel, αg+αs=1, so, the corresponding steady state particle phase volume fraction decreases for case (ii) than in case (i); as seen in Fig. 4.8 (c).
4.4 Study of effect of inlet slip between the phases 83 Table 4.6: Steady state mass flow rates at different channel sections for ug>us at inlet
Case inlet x=0.1 m x=0.2 m
˙ mg
(kg/s)
˙ ms
(kg/s)
˙ mg
(kg/s)
˙ ms
(kg/s)
˙ mg
(kg/s)
˙ ms
(kg/s) (i) ug =us=1 m/s 0.00124 0.155 0.001238 0.15491 0.001239 0.15484 (ii) ug=1 m/s,
us=0.5 m/s 0.00124 0.0775 0.001236 0.07701 0.001234 0.07699
4.4.2 When particle velocity is greater than gas velocity at inlet u
s>u
g)
In this case, the carrier phase, i.e., the gas phase inlet velocity is less than the dispersed phase inlet velocity. In this case also, a distinct difference between the steady state particle velocity profiles can be observed for the cases when (i) both the phases enter the channel with same velocity (ug=us=1 m/s, no inlet slip condition) and when (ii) particle velocity at inlet is greater than the gas velocity (ug=0.5 m/s andus=1 m/s), as seen in Fig. 4.9 (a). It can be seen that although the inlet particle velocity for cases (i) and (ii) mentioned above are same, yet the steady state particle velocity obtained at the midplane of the channel is less in case (ii) than that of case (i). It is due to the fact that in case (ii), as the inlet particle velocity is more than the inlet gas velocity, there exists a tendency of the gas phase to decrease the particle phase velocity through the interphase drag force term which contains (ug-us) term.
As a result, the steady state particle velocity at any section for this case is less than the case where there is same inlet velocity for both the phases. Table 4.7 shows the steady state mass flow rates at different sections of the channel and it can be seen that the steady state mass flow rate for the particle phase remains the same in both the cases (i) and (ii). But as the steady state particle phase velocity at any section in case (ii) is lower than in case (i), so the particle phase volume fraction has to increase in order to keep the mass flow rate constant in both the cases. Hence, it can be seen in Fig. 4.9 (b) that the steady state particle phase volume fraction at the midplane of the channel is greater in case (ii) than in case (i). Since at any section
Particle velocity (us) (m/s)
y (m)
0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 0.05
0.1 0.15
ug=1m/s, us=1 m/s ug=0.5 m/s, us=1 m/s
Particle volume fraction (αs)
y (m)
0.2 0.25 0.3 0.35
0.05 0.1 0.15
ug=1m/s, us=1 m/s ug=0.5 m/s, us=1 m/s
(a) (b)
Gas volume fraction (αg)
y (m)
0.65 0.7 0.75 0.8
0.05 0.1 0.15
ug=1m/s, us=1 m/s ug=0.5 m/s, us=1 m/s
(c)
Figure 4.9: Comparison of plots of steady state (a) particle velocity, (b) particle volume fraction and (c) gas volume fraction profiles in the midplane of the channel for (i) ug=us=1 m/s; (ii) ug=0.5 m/s, us=1 m/s
of the channel, αg +αs=1, so, the corresponding steady state gas phase volume fraction decreases for case (ii) than in case (i); as seen in Fig. 4.9 (c).
4.5 Closure 85