Binary collision model

3 1

2

3

( )

8 6 5

( ) ( ) 1

20 120

( ) ( )

( )

l

p p

p p

p

r t

K K

r t t r t y

n t t r t

r t t ρ

σ

 − −

+ ∆ =  + + 

 

 

+ ∆ =  

 + ∆ 

 

&

(23)

The model constant K in Eq. (23) represents the ratio of the total energy in the distorted particle to the particle oscillation energy in the fundamental mode and has the value of K=10/3 [31].

Figure 9 Taylor analogy breakup model

Collisions between drops can result in the following regimes: (i) bouncing, (ii) permanent coalescence, (iii) separation (or grazing), and (iv) shattering; see Figure 10. These regimes are listed in the order of increasing the relative speed between the colliding binary drops [88]. Bouncing results from the gas film trapped between two colliding drops at relatively low collision speed. Permanent coalescence occurs when two drops combine into one single drop. At relatively high collision speed, separation follows a temporary coalescence, resulting in the formation of a string of two or more drops [89].

Shattering may occur at the high speed collision, resulting in the disintegration of two drops and subsequent formation of a cloud of numerous satellite drops.

These aforementioned collision regimes significantly differ depending on the liquid properties; water drop binary collision is quite different from hydrocarbon binary collision. The models reviewed in this section are applicable only for the water drop collision. The model should be modified according to the collision regimes if it is to be used for the analysis of the hydrocarbon drop collision [90-91].

2.7 Binary collision model

Figure 10 Various collision regimes in the order of increasing the relative sped between the colliding binary drops.

Coalescence-Separation Model by Ashgriz & Poo

Three major collision phenomena (or regimes) are considered herein:

coalescence, reflexive separation, and stretching separation. Coalescence refers to the phenomenon of two drops combining into a single drop. Reflexive separation refers to two drops combining temporarily and then repelling each other, resulting in rebounding. Stretching separation occurs at a relatively high value of the impact parameter (defined as “χ” in Figure 11(a)), at which most of the mass of the colliding drops manages to flow toward their initial trajectory direction. Because of the nature of their off-centered collision (large value of χ), the collision triggers the spinning of both drops. The following model by Ashgriz and Poo [89] gives the criteria that distinguish one regime from another and, therefore, the status of the post-collision can be obtained analytically.

To illustrate the most general scenario for a binary collision, the necessary parameters are introduced in Figure 11(a). The subscript “s” and

“l” refer to small and large drops, respectively. The relative quantity is denoted with “r.” d and u are the diameters and velocities of the drops. b is the distance between the centers of the drops before the collision. β is the angle between the trajectory of the small drop and the center-to-center line. γ is the angle between the trajectory of the small drop and the relative velocity vector.

χ is the impact parameter, the projection of the separation distance between the droplet center in the direction normal to that of u, or χ can be thought of as the distance from the center of one drop to the relative velocity vector placed on the center of the other drop. B is the dimensionless impact parameter. The relative velocity of the two colliding drops is:

2 2 2 cos

r l s l s

u = u +u − u u α (24)

Where

2 sin 2

l s l s

B b

d d d d

χ β γ⁻

= =

+ + (25)

sin 1 ^l sin

r

u

γ = ^{− }u α^

 

As B approaches zero, head-on collision occurs, and as B approaches unity, grazing or off-centered collision occurs. A small value of B results in reflexive separation and a large value of B results in stretching separation. If B exceeds unity, two drops pass by and not collide. Thus the range of B is 0 ≤ B

≤ 1.

Figure 11 (a) Schematic of the colliding binary drops. (b) Schematic of bouncing binary drops.

Reflexive separation results when the effective reflexive kinetic energy (K_r) is larger than 75% of its nominal surface energy:

(

^{3 3}

)

^2/3

r 0.75 l s

K ≥ σπ d +d (26)

Where

( ) ( )

( ) ( )

2 2 3 2/3 6

1 2

3 2

1 1

12 1

r l

K =σπd ^ + ∆ − + ∆ + We ∆η η+ ^

 ∆ + ∆ 

 

(27)

s/ l

d d

∆ =

( )

²

(

²

)

^{1/ 2}

1 2 1 1 1

η = −ξ −ξ −

( )

²

(

^{2 2}

)

^{1/ 2 3}

2 2

η = ∆ −ξ ∆ −ξ − ∆

(

¹

)

^{B/ 2}

ξ = + ∆

(a) (b)

are those of coalescence and reflexive separation, respectively. On the other hand, when the total effective stretching kinetic energy (K_si) is larger than the surface energy of the region of interaction, stretching separation occurs:

si si

Here the subscript “i” stands for “interacting” region. V represents the volume of the interacting regions. h is the width of the overlapping region. Eq. (30) can be rewritten in terms of the Weber number by setting the equal sign in Eq.

(29).

This Eq. (31) distinguishes the regime of stretching separation from that of coalescence; the regimes below and above the line from Eq. (28) are those of coalescence and stretching separation, respectively.

Bouncing Model by Estrade et al.

Estrade et al. [92] gives a criterion for the bouncing regime. Their model criterion assumes that a drop’s initial kinetic energy of deformation does not exceed the energy required to produce the limit deformation. The model also assumes a short delay time during deformation, no energy exchange between gas and drop and little dissipative energy.

Bouncing severe and nearly a “pancake” or “disk” is formed subsequent to the head-on collision, φ→ . If the head-on collision is so insignificant that nearly no 0 deformation results, then φ→ because h approaches the particle diameter, 2 namely h→2r.

Coalescence-Separation Model by O’Rourke

The collision model by O’Rourke [43] accounts for grazing and coalescence of the colliding binary drops; it does not account for the shattering effect that may be important in high-speed collision scenarios [88].

The collision model is based on the stochastic parcel method, which assumes the collision of the two parcels residing in the same computational cell. It is also assumes that the particles inside each parcel are uniformly distributed over the computational cells where the particles reside. The particles with the larger diameters are named “collectors” while the particles with the smaller in

diameters are called “donors.” The expected frequency of collisions between one collector and all the donor particles is governed by a Poisson process with frequency parameter

λ

having the following function form.

( )

²

The subscripts “l” and “s” refer to the properties for the collectors (large) and donors (small) and, respectively. Vcell is the volume of the computational cell and n_s is the number of donors. For a Poisson process, the likelihood that a collector undergoes n collisions is defined with the following probability density function (PDF) distribution, undergoes with donors. ∆t represents the computational time step. When no collision occurs, the value of PDF approaches PDF_o →e⁻ⁿ. The value of PDF theoretical works of Brazier-Smith et al. [44] on the collision of water particles.

In the Brazier-Smith study, three regimes (i.e., bouncing, coalescence, and separation) of particle collision are identified and found to be dependent on the impact parameter, x= RN2

(

rl+rs

)

, defined as the perpendicular distance between the center of one particle and the undeflected trajectory of the other [44]; see Figure. 11(a). This impact parameter of each particle is represented stochastically because individual particles inside a parcel are not tracked during computation, but only parcels are tracked in Lagrangian simulation.

The O’Rourke collision model assumes the follow post-collision state (see Figure 12); without satellite particle formation. According to O’Rourke, the critical impact parameter, xcrit, is:

where ^f

( )

^{γ =γ4−2.4γ +2.7γ} represents a curve-fit based on the data by Brazier-Smith et al for water drop collision. The ratio of the collector to donor is γ =r r_l/ _s =d_l /d_s or γ =1 /∆ =d_l/d_s, and the Weber number is

2 /

s liq l s s

We =ρ ur −ur r σ .

Once the outcome of the collision is known (suppose it is coalescence), then the number of coalescences for each collector is determined by satisfying the relation below:

1

1

0 0

( ) ( )

n n

k k

PDF k RN PDF k

−

= =

≤ ≤

∑ ∑

(38)

In document Spray Drying Technology (Page 104-110)