Recognizing the lack of mechanistic approach in modelling heterogeneous vaporiza- tion process, in this section, we attempt to develop an expression for heat transfer coefficient that includes the interaction mechanism of the droplet-particle collision process. Figure 3.1 presents a schematic of a droplet-particle system after impact. During impact, the kinetic en- ergy of droplet is converted into surface energy and utilized towards creating a larger wet- ting/spreading area. Below a critical impact Weber number limit for breakup, a recoiling phase is also observed just after the maximum spreading state is reached due to restoring sur- face tension force effect. Often a parameter called maximum spreading diameter, dmax, is used to quantify this wetted contact area (Chandra & Avedisian, 1991b; Mitra et al., 2013) which can be obtained by performing an energy balance on the droplet before and after impact (dis- cussed later in this section).
Chapter 3 49 Figure 3.1 Droplet-particle collision mechanism
For this mechanism to be valid, the maximum spread diameter of the droplet needs to be less than or equal to particle perimeter. Considering size of atomized feed droplets in FCC vaporizer which may vary from 30 to 500 μm and catalyst particle size ~ 65 μm, the mecha- nism is apparently plausible. Even the bigger droplets may undergo size reduction by a ho- mogeneous vaporization process to some extent, and reach a suitable size before interacting with the catalyst particles. Considering droplet-particle collision is the major pathway of heat transfer, total heat transfer during droplet-particle collision was expressed as:
.
total fred col
no of collisions energy transferred
Q col Q
volume time collision
= × = ×
×
(3.29)
The first term in Eq. (3.29) gives collision frequency (Nayak et al., 2005; Pougatch et al., 2012) which can be interpreted as a number of droplet-particle collision occurring per unit
Chapter 3 50 control volume. Considering the effective droplet-particle contact region diameter as (dd + dp), the collision frequency was given as:
( )2
_ 4
freq slip dp P d d p
col =v n n π d +d (3.30)
where np= εp/(π/6dp3) and nd = εd/(π/6dd3) are the number density of particle and droplets re- spectively in the interaction region.
As catalyst particles remain at much higher temperature (~1000K) than the saturation temperature of gasoil droplets (~ 700 K), droplet is heated up quickly. It was assumed that catalyst particle and surrounding gas always remain in thermal equilibrium and heat lost by the catalyst particle during collision is instantaneously replenished. This assumption was ex- plained later with quantitative analysis in section 3.3.3. During heat transfer at the droplet- particle contact area, occurrence of a thin vapour film was considered and this prevents direct contact between droplet and particle. All resistances to heat transfer were considered to be lumped into this vapour film. With this physical concept, the heat transfer per collision Qcol
was written as,
( - )
V wet P B cont
col
film
k A T T
Q e
= τ (3.31)
where Awet is the wetted particle surface area utilized for heat transfer, kV is vapour film ther- mal conductivity, Tp and TB are the particle surface temperature and droplet saturation tem- peratures respectively. efilm is the vapour film thickness and τcontis the collision duration of the droplet-particle pair.
Introducing collision frequency, the expression for total heat transferred to droplet, Qtotal therefore was written as:
Chapter 3 51
( )
2 _
( ) -
4
d p V wet P B cont
total slip dp P d
film
d d k A T T
Q v n n
e
π τ
+
= × , (3.32)
Wetted surface area, Awet in Eq. (3.32) was obtained considering a spherical cap formed by the droplet on the particle surface as follows:
wet p p
A =πd h (3.33)
where the spherical cap height, hp, underneath the droplet is calculated as
[1- cos ]
2
p p
h = d α (3.34)
where α is polar angle given as dmax/dp and dmax is the maximum spreading diameter of the liquid film after impact. A theoretical estimation of the maximum spread diameter dmax
(Chandra & Avedisian, 1991b) based on energy balance of the system as stated earlier was used in this study:
( max ) (4 )( max )2
3We We
1- cos - 4 0
2 ReL d dd + θa d dd 3 + = (3.35)
where θa is the advancing contact angle at the maximum spreading state of the three phase contact line. At very high surface temperature especially in film boiling regime, this advancing contact angle approaches a value close to 180o due to the presence of the vapour film which makes the solid surface to appear as superhydrophobic area (Chandra &
Avedisian, 1991b; Mitra et al., 2013). In this study, a constant contact angle, θa of 170o was considered. Final expression for Awet was obtained by substituting dmax calculated from Eq.
(3.35) into Eq. (3.34) and then substituting hp calculated from the Eq. (3.34) into Eq. (3.33).
Due to a large difference between the temperature of the riser (TG = 800-1000K) and the average boiling temperature of the feedstock (TB = 560-700K), film boiling regime was assumed to occur at the surface of the feed droplets as it collides with catalyst particle. The
Chapter 3 52 minimum temperature at which the film boiling occurs is defined as Leidenfrost temperature, TLeid which can be estimated using the expression TLeid =(27 32)TC
(Chandra & Avedisian, 1991b). Using the critical temperature TC for gas oil as ~ 842K assuming its properties can be represented by C26 hydrocarbon (Whitson, 1994), TLeid for gasoil then was obtained as
~758K. Although Leidenfrost temperature depends on many other factors such as particle surface roughness and theromophyscial properties of interacting droplet and particle, this simplest estimation justifies the validity of the film boiling assumption in the FCC feed va- porization zone. In this conduction mode of heat transfer, the entire resistance was assumed to be contained in the vapour film at the contact area as the droplet spreads over the particle surface. Understandably, this is a dynamic process and to the authors’ knowledge no theoreti- cal expression to estimate the film thickness under such condition is so far reported in the lit- erature. However, some studies are available on measurement of vapour film thickness for sessile water droplet on a flat transparent hot surface (370oC) which is reported to be in the range of 5 - 100 μm (Burton et al., 2012) depending primarily on the droplet diameter (1 – 20 mm) and temperature difference between the droplet and hot surface temperature. We are un- sure of any such experimental report on a spherical solid surface i.e. particle which could be attributed to the complexities involved in experimental measurements. In the absence of any experimental data and availability of a suitable model specifically addressing vapour film thickness on a spherical surface, the following expression for estimating vapour film thick- ness beneath a stationary droplet suggested by Biance et al. (2003) based on a scaling analy- sis was rather used in the present DPC model:
Chapter 3 53
( ) 1/3
4/3
2
- 2
V V L p B
d film
V V
g k T T
e d
L à ρ
σ ρ
=
(3.36)
It is worth mentioning that the vapour film thickness underneath a sessile droplet observed experimentally using optical interference method shows a non-uniform thickness distribution with a concave region at the centre surrounded by an annular neck (Burton et al., 2012), 2012). The thickness of the vapour film is reported in the concave region where it is maxi- mum and decreases towards the periphery (droplet interface). In the present modelling, such complex variation of geometry in estimating the vapour film thickness was not accounted and it was considered that expression given in Eq. (3.36) yields a uniform thickness.
Heat transfer between the droplet and particle occurs as long as the droplet remains in contact with the hot particle where contact time of the droplet was defined as the residence time on the hot particle surface. Based on the one dimensional vibration analysis of a droplet on a hot surface, Wachters and Westerling (1966) suggested the following expression to es- timate the oscillation time (residence time of the droplet on a hot surface) of droplet:
3
16
L d cont
L
ρ d
τ π
= σ (3.37)
where σ is surface tension of the liquid.
The expression of contact time in Eq. (3.37) was actually derived based on oscillatory behav- iour of droplet upon impact on solid surface and in principle would comprise the time for both droplet spreading and recoiling phase. However such collision interactions occur at very high Weber number (We > 5000) (Mirgain et al., 2000) in a typical FCC feed vaporization zone. It is therefore expected that droplets in this operating conditions would break upon im- pact and it is highly unlikely that a recoiling phase would exist. It was shown by Wachters and Westerling (1966) that droplets invariably break upon impact on a flat solid surface at We
Chapter 3 54
> 80. The similar droplet break up behaviour was also shown by (Mitra et al., 2013) on a spherical surface in film boiling regime where droplets were observed to invariably break in the spreading phase itself at We > 80. The droplet breakup time in their study was found to be approximately half of the theoretically predicted contact time using Eq. (3.37). With this physical picture in consideration, the following expression for collision induced droplet breakup time was used,
3
2 16
L d break
L
ρ d τ π
= σ . (3.38)
An alternate form of the droplet breakup time was also formulated as the time scale of droplet completely spreading over the particle surface which could be given as:
max _ break
slip dp
d
τ = v (3.39)
where dmax is the maximum droplet spread diameter obtained from Eq. (3.35) and vslip_dp is the slip velocity between droplet and particle.
Regarding the collision induced droplet breakup time, it could be seen from Eq. (3.38) that there is a direct dependency of the breakup time on droplet diameter which indicates larger droplet would take longer time for breakup compared to smaller droplet. In Eq. (3.38), the slip velocity between droplet and particle does not appear explicitly which also governs the impact/collision Weber number greatly. This dependency is reflected in Eq. (3.39) which in- dicates that droplet contact/breakup time is inversely proportional to the slip velocity, a find- ing which is more obvious. On the other hand, the maximum spread diameter dmax also in- creases with Weber number [Eq. (3.35)] and the overall effect of Weber number on droplet breakup time is somewhat cancelled out. This was observed in the present modelling study (not shown) as long as the slip velocity between droplet and particle was varied in the range
Chapter 3 55 of 5 – 10m/s. Droplet breakup time predicted by Eq. (3.39) was found to be always less than Eq. (3.38) for the operating conditions used in the present study. Experimentally, this behav- iour was also noted in the studies of Mitra et al. (2013) on droplet impact dynamics on a spherical particle in film boiling regime where droplet breakup time was found to be almost independent of collision Weber number once the breakup regime is reached.
After breakup, the primary droplet would produce a number of secondary droplets and in reality a complex size distribution would follow since droplet size reduction would occur simultaneously by breakup and vaporization. Each of the secondary droplets may also under- go collision with neighbouring particles without vaporizing if mean free path is long enough.
The analysis of length and time scales shows that this is indeed possible since time scale of traversing the mean free path is much faster (in the order of few microseconds) compared to time scale of droplet vaporization by homogeneous vaporization mode (in the order of milli- seconds).
Breakup of impacting droplet inevitably leads to an increase in interfacial area by producing a number of smaller size secondary droplets which enhance heat/mass transfer and hence the vaporization process. However, a proper treatment of this phenomenon would re- quire a population balance model to quantify number of smaller droplets produced due to breakup and number of droplets disappeared due to vaporization in a comprehensive CFD model framework. This was beyond the scope of the present study and was therefore not ac- counted.
Now, substituting Eq.(3.33), Eq. (3.36), Eq. (3.38) and Eq. (3.39) into Eq. (3.32) (where τcont is replaced by τbreak), a final expression for collision induced heat transfer can be obtained. A heat transfer coefficient, hcol, based on the available heat transfer area, Ap, and the
Chapter 3 56 temperature difference between particle, Tp, and droplet, Td, now can be introduced as fol- lows:
( )
col total p p d
h =Q A T −T . (3.40)
The final formulation of the total heat transfer coefficient due to collision hcol was obtained by substituting expression of Qtotalfrom Eq. (3.32) and substituting particle area for heat transfer,Ap =npπd2p and droplet number density, nd =εd (πdd3/ 6) into Eq. (3.32) as:
( ) ( )
( )
2
_ - 1 /
4
slip dp V wet P B break d d p
col
film s d
v k A T T d d
h e T T
τ ε +
= − . (3.41)
Considering heat received from successful collisions with hot particles increases the tempera- ture of the droplet to the boiling temperature (inert heating phase without vaporization) and then the vaporization process begins, the heat balance equations can be written first for the heating up stage as:
d -
G B
dT T T τ dt =
(3.42)
where the time constant for heat transfer and is given as:
( LC dpL d) / 6hcol τ = ρ
(3.43)
Then for the vaporization stage as:
( )
2 d
col d G B V
h d T T L dm
π − = − dt . (3.44)
From Eq. (3.44), the transient droplet size reduction can be calculated using the mass of the droplet, md = π ρ( dd3 d) 6, then:
Chapter 3 57
( )
2 col G B
d
V L
h T T
dd
dt L ρ
− −
= (3.45)