In this section, the vaporization times of the FCC feed (vacuum gas oil) droplets by the two existing heterogeneous models of Buchanan (1994), one model suggested by (Nayak et al., 2005) and the new DPC model developed in the present work were predicted. As pre- viously discussed in section 3.2 that in the heterogeneous vaporization mode, the droplet life- time was assumed to govern in two distinct stages – inert heating without vaporization and only vaporization at constant boiling temperature. Vaporization times reported here therefore are the sum of the time accounted for in these two stages.
Chapter 3 68 Figure 3.5 Transient change of FCC feed droplet diameter predicted by the four heterogene- ous models – Buc (2), Buc (3), Nayak (phi = 14) and DPC. Conditions are: dd0 = 50àm, dp =
65àm, Td0 = 561K, TB = 700K, TG = Tp = 922K (a). Conditions are: dd0 = 50àm, dp = 65àm, Td0 = 561K, TB = 700K, TG = Tp = 922K. All the model predictions could be seen
attaining the saturation temperature limit (b).
Figure 3.5a presents the vaporization time predicted by the four heterogeneous vapor- ization models for a 50 μm FCC feed droplet. The initial flat profile indicates the inert heat- ing period without vaporization. Vaporization time calculated by the model of Nayak (2005) [Nayak (phi = 14)] exceeds the minimum heat transfer limit suggested by Buchanan (1994) [Buc (3)] by ~ 30%. The Nayak model can be considered as a refinement to the minimum heat transfer model of Buchanan (1994), with improvement on vaporization per collision be- ing quantified through an adjustable parameter. The adjustable parameter ϕ = 14 was report- edly produced a good agreement with Buchanan’s analysis (Nayak et al., 2005) which was also confirmed here. Both models consider instantaneous contact during droplet-particle col- lision assuming that the Leidenfrost effect occurs which prevents the further contact and hence can be considered to set the limit for minimum heat transfer or maximum vaporization time. In that regard, it could be seen that the proposed droplet-particle collision (DPC) model prediction lies between the two heat transfer limits set in Buchanan’s (1994) analysis and jus-
Chapter 3 69 tifies a more reasonable heat transfer process during droplet-particle collision. Figure 3.5b presents the temporal increase in droplet temperature during the droplet-particle collision process. All models could be seen predicting the boiling temperature consistently during the vaporization stage.
Table 3.5 Vaporization times (ms) predicted by the heterogeneous models for FCC feed drop- let. (Operating conditions are: dp = 65àm, Td0 = 561K, TB = 700K, TG = Tp = 922K).
Droplet diameter (àm)
30 50 80 100 300 500
Buc (2) Heatup time 0.023 0.038 0.061 0.076 0.227 0.379 Vaporization time 0.051 0.085 0.135 0.169 0.508 0.847 Total time 0.074 0.123 0.196 0.244 0.735 1.226 Buc (3) Heatup time 0.06 0.15 0.33 0.55 2.87 6.17 Vaporization time 0.22 0.54 1.19 2.22 11.58 24.94 Total time 0.29 0.69 1.52 2.77 14.45 31.11 Nayak (φN = 14) Heatup time 0.02 0.07 0.17 0.26 1.45 2.81 Vaporization time 0.26 0.82 2.11 3.17 17.51 33.85 Total time 0.28 0.88 2.29 3.44 18.96 36.66
DPC Heatup time 0.01 0.02 0.03 - - -
Vaporization time 0.18 0.19 0.33 - - -
Total time 0.19 0.21 0.36 - - -
Table 3.5 summarizes the vaporization times of different size of gas oil droplets at carrier gas temperature TG = 922K for all heterogeneous models utilizing the operating condi- tions provided in Table 3.3. In Table 3.5, legends “Buc (2)” and “Buc (3)” indicate the infi- nite heat transfer (maximum heat transfer) and the hard-sphere collision (minimum heat transfer) models of Buchanan (1994) respectively. The other legend, Nayak (φΝ = 14) indi- cates the phenomenological model proposed by Nayak et al. (2005) with an adjustable pa-
Chapter 3 70 rameter φΝ = 14 defined in Eq. (3.28). Comparisons of the four models are provided in Table 3.5 for gas oil droplet size range from 30-80 μm. Table 3.5 reflects that vaporization times predicted by all heterogeneous models are at least an order of magnitude lower compared to the homogeneous models for the corresponding droplet diameters. This is because the homo- geneous vaporization mode is primarily mass transfer driven in nature and for obvious reason is a slow process. The heterogeneous vaporization mode on the other hand is purely heat transfer driven and comparatively much faster than the homogeneous vaporization mode. It was observed that heat transfer coefficient in heterogeneous mode was much larger than the convection heat transfer coefficient. For instance Nusselt number in heterogeneous mode is around 10 times greater than the corresponding Nusselt number for homogeneous mode for a 50 àm FCC feed droplet. It could also be noticed that prediction of the DPC model is restrict- ed to droplet size of 80 μm only. This is due to the fact that this model is only valid if the maximum spreading diameter of the droplet dmax is smaller than the perimeter of the interact- ing particle and this condition could only be satisfied with the maximum droplet size of 80 μm and not beyond.
During the collision, the amount of heat transfer depends on the contact time between the droplet and particle. In real scenarios, formulating this contact time is understandably very complex and would depend on number of factors such as fluid properties, particle sur- face conditions and the nature of heat transfer involved. In the proposed DPC model, a theo- retical contact time based on the contribution of inertia and surface tension force [Eq. (3.38)]
was used. Also, the model was tested for another different formulation of contact time, i.e.
the time required to reach the maximum spreading state denoted by dmax/vslip_dp in Eq. (3.39) assuming absence of recoiling state after impact. The droplet impact velocity, vslip_dp, used in the present analysis was the slip velocity (6.1 m/s) between the interacting droplet and parti- cle. Vaporization times obtained by these two different formulations of contact time are pre-
Chapter 3 71 sented in Figure 3.6. It is obvious that selection of contact time affects the model prediction directly as it governs the heat transfer duration. The contact time calculated from
max / ,
breakup d vslip dp
τ = yields a smaller value (6.86 às for a 30 àm droplet) compared with
3 / 16 2
breakup Ldd L
τ =π ρ σ (10.01 às for a 30 àm droplet) and this results in a larger va- porization time (0.28 ms). From a theoretical point of view, this scenario can be considered as the limiting case of contact time in the absence of a more realistic physical model.
Figure 3.6 FCC feed droplet vaporization time predicted by the proposed DPC model with two different formulation of the droplet-particle contact time. Conditions are: d0 = 50 àm, dp
= 65àm, Td0 =561K, TB =700K, TG =Tp =922K. Larger vaporization time is predicted when the contact time of droplet on particle surface decreases.
In formulation of the heterogeneous model in section 3.2.2, it was assumed that dur- ing collision catalyst particle temperature remains invariant. This assumption is explained as follows. In case of heterogeneous vaporization, contact time of droplet-particle pair was esti- mated to be ~ 10 μs for 30 μm droplet and ~ 680 μs for 500 μm diameter droplets using Eq.
(3.38). Considering temperature of particle changing entirely through conduction mode of
Chapter 3 72 heat transfer, a suitable time scale for heat loss during collision of a droplet particle pair could be constructed as theat loss_ =m Cp pP∆T (kAp∆T e/ film) where mp is mass of particle, CpP is heat capacity of particle, k is thermal conductivity, Ap is particle surface area available for heat transfer and efilm is vapour film thickness at the contact area. Two limiting cases were examined using k for vapour film (minimum heat transfer) and k for particle (maximum heat transfer). Using the given operating conditions, theat_lossfor droplet size range 30 and 500 μm, were obtained in the range of 5-15 μs and 220-600 μs, respectively. The above analysis shows that the heat loss time scale is comparable with the droplet-particle collision time scale within the same order of magnitude which ascertains the fact that temperature change in the particle may be considered negligible. However, due to uncertainty in operating conditions, it is quite possible that temperature change may occur but given the small time scale of heat transfer (μs), it can be considered being replenished instantaneously.
It is worth mentioning the vaporisation times predicted by the DPC model were based on the constant slip velocity of 6.1 m/s. However, the choice of the slip velocity has shown to significantly affect the calculated vaporization times. For example, for a 500 àm droplet in Buchanan’s work, the calculated vaporization time is ~ 31 s if the constant slip velocity of 6.1 m/s was used and is ~ 120 s if the droplet terminal velocity was used. From the present model predictions, the vaporisation time significantly reduces from 0.30 to 0.08 ms (for a 50 àm gas-oil droplet at the gas temperature of 911 K) when the slip velocity increases from 5 to 10 m/s.
The spreading mechanism of the droplet on particle surface during collision depends largely on the advancing contact angle [Eq. (3.35)]. Determining the maximum spread factor, dmax, is another challenging issue that is affected by a number of factors such as fluid proper- ties, surface characteristics and nature of heat transfer on the particle surface. Additionally,
Chapter 3 73 all the reported models to compute maximum spread diameter either analytical or empirical, based on droplet impact on flat surface and do not account for surface curvature are applica- ble for cases like droplet interaction with spherical particle. Based on the considerations that in the film boiling regime due to presence of the insulating vapour film at liquid-solid inter- face, particle surface behaves as a super-hydrophobic surface (contact angle > 150o), a con- tact angle value of 170o was used in the DPC model. A sensitivity analysis performed on the variation of contact angle suggested insignificant impact on the vaporization time. Figure 3.7 presents this observation where less than 1% variation in vaporization was observed when the contact angle was changed from 150o to 180o.
Figure 3.7 Effect of advancing contact angle variation on FCC feed droplet vaporization time in the proposed DPC model. Conditions are: dd0 = 50 àm, dp = 65 àm, Td0 = 561K, TB = 700K, TG = Tp = 922K. Vaporization time varies insignificantly when the advancing contact
angle of the droplet on particle surface changes from 150o to 180o.
It was mentioned earlier that generation of secondary droplets due to breakup during droplet-particle collision process was not explicitly taken into account in the present model-
Chapter 3 74 ling framework. The collision was accounted for based on the uniform volume fraction of catalyst particles and feed droplets in the riser. However, in principle, it is possible to inte- grate the proposed DPC model in a CFD framework to introduce the collision when an ap- propriate size ratio of collision partners is found in a control volume. A droplet size distribu- tion involving a population balance sub-model can be included to account for droplet size variation during breakup and hence the associated transient change in interfacial area and cor- responding vaporization rate in a more rigorous manner.
It is worth mentioning that whilst the present study does not consider developments in numerical modelling of droplet evaporation it is acknowledged that there is considerable work in this area, especially in direct numerical simulation (DNS), over a number of years.
Examples include: consideration of turbulence effects on the dispersed evaporating droplet using DNS by Mashayek (1998); DNS of a confined three-dimensional gas mixing layer with a stream of evaporating hydrocarbon droplets by R. S. Miller and Bellan (1999); DNS of evaporating droplet considering the deforming fac-tor and an incompressible fluid flow of Schlottke and Weigand (2008).