2. Interfacial Area Concentration

(1) Local Interfacial Area Concentration

; probability to see the interface at a point

The time-averaged IAC is given as,

[eqn: a_i (r) = 1 over T SUM _j=1^N_i ~ 1 over {vert v_i CDOT n_i vert_ j}]

∴[eqn: a_i (r) = 1 over T SUM _j=1 ~ 1 over {vert v_ij vert cos phi_j}]

where [eqn: phi _j] the angle between the interface velocity and the surface normal vector      direction.

The mean value is,

[eqn: bar a_i (r) = 1 over tau bar {~~~~1~~~~} over {vert vec v_i vert cos phi}]

where [eqn: tau = 1/ (2N_i )]

∴ [eqn: rm N_i], [eqn: vert vec v_ij vert cos phi_i]  ⇨  IAC

Consider a double probe method where the z-direction is the mean flow direction.

Then [eqn: v_ij ~ and ~ vec n_j] can be given in terms of unit vectors [eqn: rm n_x , ~ n_y ~and ~ n_z] using angles    with z and y axes given by [eqn: rm (alpha_i ~, beta _j ) ~and ~( mu _j ~, nu _j )]

∴

[eqn: rm vec v_ij ~= ~ vert vec v_ij vert ~left {cos alpha_j ``n_z ~+~ sin alpha_j ~cos beta_j ``n_y ~+~ sin alpha_j ~sin beta_j `` n_x right }]

[eqn: rm vec n_j ~=~ cos mu_j``n_z ~+~ sin mu_j ``cos nu _j ``n_y ~ + ~sin nu_j ``n_x]                         (3-40)

From the derivation in Appendix 1, the local mean IAC is given as,

[eqn: rm bar a _i ~=~ {4N_i left{sum_j 1 over {vert v_ij vert} / (sum_j ) right}} over { 1-cot alpha_o over 2 ln (cos alpha_o over 2 ) - tan alpha_o over 2 ln (sin alpha_o over 2 )}]                (3-41)

where,

[eqn: {sin 2 alpha_o} over {2 alpha_o} ~=~ {1- (sigma_z^2 / vert bar vec v_iz vert ^2 )} over { 1+ 3 (sigma_z^2 / vert bar vec v_iz vert ^2 )}]

(2) Correlations

① experimental correlations

② analytical correlations which depend on the flow regime and

thermal-hydraulic parameters

1) Experimental correlations

[eqn: rm a_i =f LEFT ( LEFT [ { partial p} over {partial z } RIGHT ]_fr ,vec v RIGHT )]

- Table 1

2) Correlations dependent on flow regime

- depend on flow paramters like bubble size, shape and roughness, or hydraulic          parameters like number of bubble or void fraction

- different model for each flow regime

For bubbly and mist flow (diffusion flow)

[eqn: rm a_i = 3alpha_d over r_sm]

where αd ; bubble volume fraction

rsm ; Sauter diameteer

Ishii와 Mishima(1980) for elliptical bubble ;

[eqn: rm r_v over r_sm ={2+ E^2 over root{1-E^2} ln {1+ root{1-E^2}}over {1- root{1-E^2}}}over{4E^2/3}]

where          [eqn: rm E= 1 over 1+0.163N_EO^0.75]    and   [eqn: rm N_EO == {4g triangle rho r_v^2 }over sigma]

For Taylor bubble

[eqn: rm a_i = 4.5 over D_H LEFT ( { alpha-alpha_gs} over {1-alpha_gs } RIGHT ) + 3alpha_gs over r_sm LEFT ( { 1-alpha} over {1-alpha_gs } RIGHT )]

where  αgs  ; void fraction in slug and film

☞ linearly inclreases as [eqn: alpha] increases for constant bubble size in the bubbly flow

☞ Sauter diameter and IAC increase as the heat flux increases.

☞ As the mass flux increases, IAC decrease and Sauter diameter increases.

- In annular mist,

[eqn: rm a_i = 4C_an over D root{ { alpha} over {1-alpha_fd } }+{ alpha} over {1-alpha_fd } LEFT ( 3alpha_fd over r_sm RIGHT )]

where αfd, Can ; liquid volume fraction, roughness of film surface

그림 .  기포 유동과 slug 유동사이의 계면 면적 변화

(Ishii and Mishima, 1981)

그림 . 환형 액적 유동에서 계면 면적 변화

(Ishii and Mishima, 1981)

3)  Model in nuclear safety codes

가.  RELAP5/MOD3

① In bubbly and mist flow, the bubble is assumed to be spherical and has the          size distribution of

[eqn: rm p^* =4d^*2 e^-2d^*]

where d* ; nondimensional bubble size (=d/d1)

d1 ;  most probable bubble size

Then,[eqn: rm d_o], and IAC will be

[eqn: rm d_0 =1.5d_1]

[eqn: rm a_i = { 6 bar alpha} over {d' } { INT d^*2 p^* dd^*} over {INT d^*3 p^* dd^* } = { 2.4 bar alpha} over { d_1}]

Or           [eqn: rm a_i = { 3.6 bar alpha} over { d_0}]

In RELAP5/MOD3, do is assumed the half of most stable bubble size which           obtained from critical We

[eqn: rm We _{crit} = {d _{max} rho _{c} (v _{g} -v _{f} ) ^{2}} over {sigma }]

The critical We is 10.0 and 3.0 or 12.0 for vapor and liquid droplet,                   respectively.

② Slug flow ;

[eqn: rm alpha_b = { alpha_g -alpha_gs} over {1-alpha_gs }]

Thus, for slug flow

[eqn: rm a_i = alpha_b over L +{ 3.6alpha_gs over d_0} (1-alpha_b )]

and

[eqn: rm alpha_gs =alpha_BS exp[-8(alpha_g -alpha_BS )/(alpha_SA -alpha_BS )]]

③ annular mist flow;

[eqn: rm a_i = (4C_ann )(1-alpha_ff )^1/2 +{ 3.6alpha_fd over d_0} (1-alpha_ff )]

where Cann ; paramter related to liquid film surface roughness

αfd ; liquid volume fraction in mist flow inside channel

[eqn: rm alpha_fd = { alpha_f -alpha_ff} over {1-alpha_ff }]

나.  COBRA-TF

- COBRA-TF divides the flow regime as dispersed bubbly, slug, churn-                 turbulent, film mist regime according to the void fraction.

- apply transport equation for IAC of liquid droplet

① bubbly flow

[eqn: rm a_i =N_b 4` pi` r_b^2]         where [eqn: rm N_b = alpha_g over {4pir_b^3 /3}],

[eqn: r _{b}] ; larger value of

-[eqn: rm r_b =min(0.5 { We_b sigma} over {rho_l (v_g -v_l )^2} ,~0.5 D_H ,~ 0.02 )]

-[eqn: rm r_b =0.5 ( 3 over 4 alpha_min over {pi N_{b,min}} )^1/3]

② slug flow ;

[eqn: rm a_i = (a_i )_small,b +(a_i )_large,b]

For larger bubble

[eqn: rm r_{large,b} = min left [ LEFT { 3 over 4pi (alpha_g -0.16 alpha_f ) cdot(Volume ~of~ cell) RIGHT }^1/3 , ~D_h over 2 right ]]

③ film flow ;

[eqn: rm a_i =root alpha_g P_w /A]

where [eqn: rm P_w] , A ;  wetted perimeter and flow area

④ churn-turbulent flow ;

- IAC from interpolation of bubbly flow and film flow

- For entrained liquid droplet

[eqn: rm { PARTIAL a_i}over {partial t}+del cdot (a_i vec v `_e )={3 S^'''}over{rho_f r_p}+S_ GAMMA `]