(3) Constitutive model

1) Breakup model

; Breakup mechanism can be explained by a balance of external force and surface

tension. The external force results from the pressure change on the bubble surface

due to the eddy in turbulent flow, and from the shear stress in viscous flow.

o Maximum Particle Size

o number density and size distribution of generated particles after breakup

reaction

o reaction frequency

★ Research status

① Taylor(1934) ; consider viscous stress and surface tension

- critical We number for the breakup

[eqn: rm We_crit == mu_c (partial v / partial r)_max D_max /2sigma`]

(3-43)

② Hinze(1955) ; relate the flow characteristics in diffusion flow with a principle                       of particle collapse

- Governing force for the particle breakup ; viscous stress, dynamic pressure force,

surface tension

[eqn: rm We_crit =C LEFT [ 1+ varphi (Vi) RIGHT ]]

(3-44)

where  [eqn: rm Vi == mu_d over root{rho_d sigma D}]

- Critical We is affected by the properties of the discrete phase and the

continuous phase. The density difference between those phase affects the

occurrence of breakup

- The dynamic pressure force due to velocity is important parameter to

determine the maximum particle size in turbulent flow. In this case,

[eqn: rm We_crit = {rho_c {bar v}^2 D_max }over sigma]

(3-5)

where   [eqn: rm {bar v}^2 = c_1 ( epsilon D)^2/3] , [eqn: c_1] = 2.0 (Batchelor)

When we neglect the viscous effect and assume the critical We be constant,

the experiment shows

[eqn: rm D_max LEFT ( rho_c over sigma RIGHT ) ^3/5 epsilon ^2/5 =0.725]

(3-46)

③ Levich(1962) ; assume the liquid drag creates vapor flow in the bubble and

thus results in the dynamic pressure (ρgvg2/2).

When the dynamic pressure > vapor surface tension, breakup occurs.

[eqn: rm We_crit = tau over {sigma /d_max} LEFT ( rho_d over rho_c RIGHT ) ^1/3]

(3-47)

From  eddy velocity model by Batchelor(1959)

[eqn: rm d_max = LEFT ( We_crit over 2 RIGHT ) ^0.6 sigma^0.6 over {(rho_c^2 rho_d )^0.2} epsilon ^-0.4]

(3-48)

④ Prince & Blanch(1990) ; phenomenological model for bubble breakup rate                                         in the two-phase diffusive turbulent flow

- two steps of collision process between eddy/bubble and breakup

-  model for the rate of collision (Kennard(1938))

[eqn: rm theta_ie =n_i n_e S_ie ( {bar u} `_ti ^2 +{bar u} `_te ^2 )^1/2]

To use this equation, the number of eddy with characteristic length or

information about eddy size distribution is required.

Azbel과 Athanasios(1983) ; eddy size distribution model

[eqn: rm dN(k) over dk = 0.1k^2 over rho_f]

where, N(k) : wave number per unit mass, or the number of eddy with k

- Assume minimum eddy size = 20% of bubble size

- collision efficiency ; contribution rate to breakup among bubbles colliding with eddy

- Bubble critical velocity, uci , is expressed using critical We defiend by Hinz

for the vapor bubble in water. By applying critical We(=2.3),

[eqn: rm u_ci =1.52 (sigma /r_bi )^1/2]

Finally, the breakup rate & total breakup reaction rate of bubble with rbi are

given as,

[eqn: rm beta_i =Sum_e theta_ie exp LEFT { -(u_ci^2 /u_te^2 )RIGHT } ~~]

[eqn: rm beta_T = Sum_i Sum_e theta_ie exp LEFT { -(u_ci^2 /u_te^2 )RIGHT }]

Fig. 3-6 bubble breakup reaction rate(Prince et al., 1990)

⑤ Millies, Drew, Lahey(1996)

; Bubble breakup in bubble cluster is explained with

a. independent bubble breakup

b. detached single bubble from bubble cluster.

⑥ H. Stadtke, A. Blahak, B. Worth(1997)

- assume just the eddy with similar size of bubble is involved in breakup

reaction. Then, the collision rate between bubble and eddy will be,

[eqn: rm s_be =n_b n_e A_be root{v_b^2 +v_e^2}]

where, the collision cross section, Sbe, is :

[eqn: rm A_be = pi over 4 LEFT ( d_b + d_e RIGHT ) ^2]

- assume a number density of eddy is a function of eddy, i.e maximum be

bubble size and	minimum be 20% of bubble diameter

- breakup efficiency dependent on the ratio of kinetic energy of collision eddy to surface tension

[eqn: rm beta_break,b =exp LEFT ( - v_crit^2 over v_e^2 RIGHT )]

- breakup reaction rate of eddy (considered every size) and bubble is like this :

[eqn: rm Gamma_break,b = Sum_e s_be ``beta_break,b]

- Breakup of liquid droplet is caused by Kelvin-Helmholtz and Rayleigh-Taylor instability.

And they assume the only liquid droplet (>stable maximum size) will breakup.

2) Coalescence model

; If particle collision occurs, contact area would be flat and thin film is generated.      After sufficient time passes, liquid film thickness decrease to critical thickness        and film will break. Finally the coalescence is accomplished.

o study to estimate the time for coalescence in stagnant water

o a study on the collision rate of bubbles with various size

- main causes of bubble collision

o due to turbulent of continuous phase

o due to the velocity difference of bubbles with different size

o due to velocity distribution of continuous phase at the back of large particle

o radial velocity distribution of the flow

☞ The model for the collision of two bubbles

☞ The model for coalescence efficiency of the collision bubble

☞ the model for the stable minimum particle size

① Marrucci(1969) ; two steps for the coalescence dynamics of two contact bubbles

1) the time until liquid film thickness decrease to quasi-equilibrium thickness

The quasi-equilibrium thickness of liquid film is derived from the mass equilibrium

equation as,

[eqn: rm crk^2 over sigma =(kh_0 )^2 + 1 over kh_0]

where c ; a quantity with force unit

[eqn: h_o] ; quasi-equilibrium thickness of liquid film

[eqn: rm k== LEFT ( {12pi sigma} over Ar RIGHT ) ^1/3]

The time until the liquid film thickness decreases to h0 is obtained with assuming

uniform expansion as,

[eqn: rm t ( h=h_0 )= pi over 16 1 over D cr over sigma]

where D : diffusion constant

2) liquid film breakup time from the quasi-equilibrium thickness

- The liquid film with quasi-equilibrium thickness get thinner and finally

reaches to critical thickness to breakup.

[eqn: &rm t(h=h_0 ->h_f ) &= xR over 2D f~~ &for ~~R = constant# &rm &= {x^2 C^2} over 16D^2 f^2 ~~&for ~~R=C root t]         where C : variable dependent on the liquid film structure and flow condition

f : variable group, function of [eqn: rm c``r``k^2 /γ]    (if f<2, coalescence occurs rapidly)

② Chesters(1975)

; The behavior of liquid film between two bubbles is assumed to plug flow of flat liquid

film. They derived the change rate of liquid film thickness considering bubble

velocity as,

[eqn: rm dh over dt = - 8sigma over {rho V R_eq^2 }h]

where V : bubble approaching velocity

Req, : equivalent radius

[eqn: rm 1 over R_eq = 1 over 2 LEFT ( 1 over R_1 + 1 over R_2 RIGHT )]

By assuming initial thickness h0, the following equation is obtained.

[eqn: rm t= {rho V R_eq^2 }over 8sigma ln LEFT ( h_0 over h RIGHT )]

③ Thomas(1981) : develop the models of the time for the liquid film discharge and

the minimum bubble size to occur the coalescence

- the time which reaches to critical thickness when the liquid film is discharged.

[eqn: rm tau= {3pi `mu` s^4}over{2`F`h^2}]

where s : the radius of liquid film

F : the force pushing two bubbles

Considering pressure inside two bubbles, the force acting on liquid film is,

[eqn: rm F=4pi s^2 sigma /d]

The, the time which the liquid film thickness reaches to critical thickness (liquid film

discharge time) can be obtained as,

[eqn: rm tau= 3 over 32pi mu F(d/sigma h)^2]

- minimum stable bubble size

Assume buoyancy effect is more negligible in liquid phase than turbulence,

[eqn: rm F SIM rho epsilon ^2/3 d^8/3]

By applying  F into the previous equation about the liquid film discharge time

[eqn: rm tau= 3 over 32pi mu rho epsilon ^2/3 d^8/3 (d/sigma h)^2]

Assume the characteristic time for two bubbles to contact, T

[eqn: rm T sim (d^2 / epsilon )^1/3]

and, if τ < T, coalescence occurs.

Using that assumption, the minimum bubble size for the coalescence will be,

[eqn: rm d_2 sim 2.4 LEFT ( {sigma^2 h^2}over{mu rho epsilon } RIGHT )]

④ Models for the collision rate of bubbles and collision efficiency

- collision rate by turbulent flow

[eqn: rm s_turb,b &= {36 root 2} over pi {alpha_p^2 c_1 `^{ 3} SQRT { epsilon d_b } } over d_b^4 1 over {1-2alpha_p}# rm &= c alpha_p^7/3 over {1-2alpha_p } {|v_g -v_l |}over d_b^4 `^{ 3} SQRT {C_w rho_l over rho_m }]

where αp : volume fraction of the particle in p-phase

ρm : two phase mixture density

- collision rate

Assume critical thickness of liquid film between two bubbles ∼ initial thickness×104

[eqn: rm beta_coal,b APPROX exp LEFT ( -1.30 `^{ 6} SQRT { { d_b^5 rho_l^3 epsilon ^2} over {sigma^3 } } RIGHT )]

- coalescence reaction rate of two bubbles

[eqn: rm Gamma_coal,b =0.5 s_turb,b ``beta_coal,b]