III. Flow Regime Analysis

☞ As the shape or distribution of bubbles changes according to the break of

the thermodynamic and local hydrodynamic equilibrium, the mixing of liquid and

bubble has its unique form and thermal-hydraulics characteristics.

➡ Flow Patten or Flow Regime

☞ When a given flow regime exit or  where transitions occur,

=> different correlation for τw, <α> at each region

∴ need appropriate analysis for each flow regime and couple these

analysis at flow regime transition points.

☞ Observation of flow pattern

① flash and cine photography ; reflection and refraction

② X-radiography

③ hot-wire or electrical and optical probe

1. Flow pattern in cocurrent flow

◆ In vertical flow

① bubbly flow ; discrete small and spherical bubbles in a continuous liquid

② slug flow   ; bubble coalescence having approximate diameter of pipe,

periodic

③ churn flow  ; breakdown of large bubbles in slug flow,  chaotic

④ annular flow ; separate flow, entrainment of droplets

◆ In horizontal tube

① bubbly flow ; bubbles travel in the upper half of the pipe.

② plug flow   ; similar to slug flow in the vertical flow

③ stratified flow  ; at very low [eqn: v_l ~~and~~ v_g]

④ wavy flow  ; as the vapor velocity increases, the interface disturbed.

⑤ slug flow

⑥ annular flow

◆ In helical coiled tube

(1) Bubbly flow

; Actually, no developed bubble flow due to coalesce and break of bubbles

But, for <α> < 50%

① ideal bubbly regime  ;  no interaction between bubbles

② churn-turbulent bubbly regime

;  exist a lot of interaction due to distortion and wake effects

☞ the effect of virtual mass in momentum equation

- caused by the relative acceleration of vapor phase with respect to liquid

As bubbles are accelerated, they increase kinetic energy of surrounding liquid.

Thus this inviscid effect is modelled as an increase in the mass of the accelerating bubble by adding some liquid displaced.

∴ the virtual mass of bubble = true mass + additional hydrodynamic mass

and additional force on bubble = additonal mass × its relative acceleration

➡  virtual mass force ; [eqn: under+ { rho _l ```xi <alpha> TRIANGLE z A_x-s }over{g_c} LEFT [ { D_g (<u_g > -<u_l `>)} over { D_t} RIGHT ]]

․quantify the mom. transfer between phase

․important 2-phase cirtical flow

․improve numerical stability

- In the mixture momentum equation, both terms are cancelled.

-[eqn: xi] is dependent geometric configulation and interaction between bubbles.

[eqn: = 1/2]  for spherical, noninteracting bubble

★ Note that the virtual mass force is not a viscous drag term(interfacial shear          term) due to relative slip between phases, which is given as,

[eqn: {tau_i ~p_i} over A_x-s] = [eqn: {3}over{4}left({C_d}over{<D_b >}right)rho_l ~[<u_g >_g - <u_l >_l ] ~| <u_g >_g - <u_l >_l |]

where [eqn: left({C_d }over{<D_b >}right)] is constant for idela bubblely flow

[eqn: =1.4~(1-<alpha> )^3 ~"in"^-1]

Kinematics of bubbly flow by Zuber and Hench

[eqn: rm <u_r > = U_t (1-<alpha> )^m ~~~~~~~~~~~~~~~where~U_t :~bubble~rise~velocity]     (4.3.1)

Recall the define

[eqn: <u_r > = <u_g `> - <u_l` > = {<j_g >}over{<alpha>}-{<j_l`>}over{(1-<alpha>)}]                             (4.3.2)

Fig. 4.8 shows m vs bubble size.

그림 4.8 Terminal rise velocity of

air bubble vs bubble

size[Lahey]

- The wake interaction will be strong as the

bubble become larger.

For churn-turbulent flow, m=-1	(4.3.3)

Then [eqn: V _{gj} prime & =(1-< alpha >)u _{r}]

[eqn: =``U _{t} ``(1-< alpha >) ^{m+1}]       (4.3.4)

Thus from Eq. (4.3.3),

[eqn: V_gj ' ~ = ~U_t]

By multiplying Eqs. (4.3.1) and (4.3.2)

by <α>(1-<α>)

[eqn: U _{t} < alpha >(1-< alpha >) ^{m+1} ``=~<j _{g} >(1-< alpha >)-<j _{l} `>< alpha >]        (4.3.5)

[eqn: =~<alpha> V_gj ']  from Eq. (4.3.3)

-> general kinetic relationship

■ Batch Process ,  [eqn: <j_l `> = 0]

Eq. (4.3.5) will be

[eqn: {<j_g >}over{U_t}=<alpha>(1-<alpha>)^m]

Thus, if[eqn: {<j_g >}over{U_t}={m^m}over{(m+1)^m+1}], the flow  regime

changes

(Ex ; [eqn: {<j_g >}over{U_t}>{1}over{4}] ,  for  m=1 (small bubble),  )

■ Counter-current flow ,  [eqn: <`j_l` >` =` -<`j_g `>]

Eq.(4.3.5) will be

[eqn: <`j_g ` > /U_t `=` <` alpha`> (1-<` alpha`>)^m+1]    (4.3.6)

From Eqs. (4.3.1) and (4.3.2)

[eqn: &<`u_r ` > ( 1-<`alpha`>) = {<`j_g`> } over { <`alpha`> } `=`<`u_g`> ##&~~~~~therefore~~<`u_r`>~``>~``<`u_g`>]

Fractional analysis to obtain [eqn: U_t]

① [eqn: "Buoyancy force"= 1over 6 pi {D}`_{b} ^{3 }` (rho_l - rho_g ) g over g_c]

② [eqn: "Drag force"`=`3 pi U_t `mu_l D_b]

③ [eqn: "Inertia force on bubble"`=`rho_l over g_c { U}`_{ t} ^{2 } left( 1 over 4 pi { D}`_{2 } ^{b } right)]

④ [eqn: "Surface force"`=`pi D_b `sigma]

Assume the ratio between these forces is constant.

[eqn: "Inertia" over "Buoyance" `=` { rho_l ` { U}`_{ t} ^{2 } } over { D_b ` g ( rho_l - rho_g ) } ` == ` { k}`_{1 } ^{2 }]

[eqn: "Inertia" over "Surface"`=` { rho_l ` { U}`_{t } ^{2 } D_b } over { g_c ` sigma} `==` { k}`_{2 } ^{2 }]          [eqn: therefore~~~{ { rho}`_{ l} ^{2 } { U}`_{t } ^{4 } } over { g` g_c ` sigma`(rho_l - rho_g ) }`=` { k}`_{1 } ^{2 } { k}`_{2 } ^{2 } `==` { k}`_{3 } ^{4 }]

Then,      [eqn: U_t `=` k_3 `left[ {g`g_c ` sigma`(rho_l - rho_g )} over {{ rho}`_{l } ^{2 } } right]^{ 1 over 4}]

For churn-turbulent flow,  k3=1.53

[eqn: therefore ~~~ V_gj ' `=` 1.53 `left[ {g`g_c ` sigma`(rho_l - rho_g )} over {{ rho}`_{l } ^{2 } } right]^{ 1 over 4}]

(2) Slug flow

- easily observable in adiabatic system  not in diabatic system

[eqn: D_b ``/ D_ch ~>~ 0.5]

- 특성 ; ① bullet-shaped Taylor bubble occupied most of channel area.

② slug length ∝ liquid flow rate

③ liquid may downflow.

- Concerns :  [eqn: rm {cases{&"bubble"~"rise"~"vel"ocity#&#"effect"~"of"~"viscosity"&}}]

- Transition region between bubbly & slug flow

[eqn: ;~~<alpha> = 0.1 ~ SIM ~ 0.3]

In case of churn-turbulent flow, [eqn: U_t ~= ~V_gj ']

From fractional analysis, bubble rese velocity is

obtained from inertia ([eqn: SIM rho_l`` U_t ^2])  to bouyancy force (～[eqn: (rho_l ``-rho_g )gD_b])

[eqn: THEREFORE~~ U_t ~=~ k_1` LEFT [ { g`D_b (rho_l - rho_g )} over {rho_l } RIGHT ] ^1/2]	(4.3.7)

where   [eqn: k_1] = 0.345	for circular channel

= 0.23	for noncircular channel

☞  inertia dominant case

★ viscous dominant case  ;	viscous force (～[eqn: {mu_f ``U_t}over{D_b }]) ≃ [eqn: TRIANGLE rho ``g ``D_b]

[eqn: therefore U_t = {k`g` TRIANGLE rho ` D_b ^2}over{mu_f}]	where k=0.01

★ surface dominant: no bubble move

; depend on Bo (Bond # :[eqn: { TRIANGLE rho`g`D_b ^2}over{4`sigma}])

- <α> from drift flux model in slug flow

[eqn: u_g = j + (u_g - j)]

From Eq.(4.3.5), [eqn: j_g = alpha`j + alpha(u_g - j)]

By cross-sectional averaging,  [eqn: <j _{g} >= C _{o} < alpha ><j>+V`_gj < alpha >]

[eqn: therefore~ <alpha> = {<j_g >}over{C_o `<j> + V_gj}]

In slug flow [eqn: C_o ~SIM~ 1.2,~~~V_gj ~=~ U_t]

◆ Criterion of slug-annular transition by Wallis

Define normalized superficial velocity as,

[eqn: &j_g ^* ~==~ { j_g rho_g ^1/2}over{(g`D_H` TRIANGLE rho)^1/2}~~~:~~~{"mom. flux of each phase"}over{"buoyance force"}## &j_l ^* ~==~ { j_l rho_l ^1/2}over{(g`D_H` TRIANGLE rho)^1/2}]

[eqn: therefore ~~j_g ^* ~=~ 0.4 + 0.6 j_l ^*]

즉, if  [eqn: j _{g} ^{*} `````````>~~0.4+0.6j _{l} ^{*} `,````~"annular"~"flow"]

※ Churn flow

- similar characteristics to slug flow, but much more chaotic and frothy

- Taylor bubble is distorted, its length increases and becomes narrow

- liquid falls downward where it accumulates and is lifted up by the gas

again

[eqn: therefore]  oscillatory motion of liquid.

□ Characteristics of churn flow

(3) Annular Flow

☞ <α> = 0.65 ～ 0.8

☞ important in BWR (upper portion of fuel channel)

★ boiling heat transfer

★ interest in interface

What happens in interface ?

1) ideal annular flow

- no entrainment

roll

wave

- smooth and symmetric interface

2) wispy annular flow

- at high G (> [eqn: 1.0``x``10^6 ~~lb/h-ft^2])

- entrained liquid flowing in large agglomerate

wave

undercut

3) spray annular flow

- most frequent in BWR

- entrained of relatively small liquid droplet

★ droplet size

bubble

burst

[eqn: We_crit = left[ {rho_g (<u_g >-<u_l >)^2 D_d } over {g_c sigma} right]_crit = 6.5～22 ``]

- liquid entrainment ([eqn: m_e prime prime])

liquid

impingement

★ Fully-developed entrainment fraction

E ≜ [eqn: W_e,e /W_e,T ``]

which is related as,

[eqn: pi_3 ≜ rho_g over rho_l left(1+E G_l over G_g right) left( {G_g mu_g} over {rho_g sigma} right)^2 left( mu_g over mu_l right)^{1/5} ``]

liquid

bubble

disintegration

▶ mechanism of liquid deposition

[eqn: m_d prime prime = kC ``] ,

where  C: mean concentration of droplet (kg/[eqn: m^3 ``])

[eqn: k``]: empirical mass transfer coefficient (m/s)

In "hydrodynamic equillibrium"

[eqn: m_d prime prime ≈ ``m_e prime prime ``]

By Whally and Hewitt ([eqn: k ``∝ sigma ``])

[eqn: k over {sqrt {tau_i / rho_g }} = 87`` sqrt{mu_e^2 over{D sigma rho_l}}]

■ modification of momentum equation accounting for

[eqn: G_lF = w_lF over A_x-s ``]

the effect of the

entrained liquid

[eqn: <u_g > = G<x> over {<alpha> rho_g } ``]                                                 ①

[eqn: <u_l >_F = {G(1-<x>) (1-E``)} over {<LAMBDA>rho_l } ``]                                              ②

[eqn: <u_l >_e = {G(1-<x>) E} over {(1-<alpha> - <LAMBDA>)rho_l } ``],                                               ③

where [eqn: <LAMBDA> = t_lF over A_x-s ``]: volumetric flux of liquid flim

[eqn: - {partial P} over {partial z} - & g over g_c <bar rho > sin theta - {tau_w P_f } over A_x-s ``]

[eqn: = 1 over g_c left[ {partial G}over {partial t} &+ 1 over A_x-s partial over {partial z} left{ A_x-s G^2 left( {E^2 (1-<x> )^2 }over{(1-<alpha>-<LAMBDA> )rho_g } ## &+{(1-E)^2 (1-<x>)^2 }over{<LAMBDA> rho_l } + <x>^2 over {<alpha> rho_g } right) right} right]]                      ④

spatial acceleration term

To obtain [eqn: <LAMBDA> ``], assume "no slip" between vapor phase and entrained liquid.

From Eqs. ① & ③,

[eqn: <LAMBDA> = 1-<alpha> - {rho_g <alpha> (1-<x>)E } over {rho_l <x> }``]

■ Contitutive relation for [eqn: tau_i ``], [eqn: tau_w ``]

From a simple force balance for idealized annular flow

[eqn: (2pi r DELTA z)tau(r) = pi r^2 left[ p-left(p + {d P} over dz DELTA z right) right] ``]

[eqn: - {g DELTAz }over g_c sin theta int_0^r [rho_g alpha +rho_l (1-alpha)]2pi r prime dr prime ``]

[eqn: therefore tau``(r)= r over 2 left[ - {d P}over dz right] - g over g_c sin theta over r int_0^r [rho_g alpha+rho_l (1-alpha) ] 2pi r prime dr prime ``]

For special case of no liquid entrainment and smooth interface ([eqn: alpha=1 ``])

[eqn: tau``(R-delta)`` ≜`` tau_i ``= ``(R-delta)over2 left( - {d P}over dz - g over g_c rho_g sin theta ``right)``]                          ⑤

And no voids in liquid film [eqn: alpha=0``], [eqn: R-delta ``=`` R sqrt <alpha> ``]

[eqn: tau``(R) ``≜``tau_w ``=`` R over 2 left( - {d P}over dz right) - g over g_c left( {sin theta} over R right) left[ int_0^{R sqrt alpha } rho_g r prime dr prime + int_{R sqrt alpha }^R rho_l ``r prime dr prime right] ``]

[eqn: = R over 2 left( - {d P}over dz - g over g_c <bar rho > sin theta ``right) ``]                                    ⑥

If the pressure gradient has been assumed to be same in both phases, from                 Eqs. ⑤ and ⑥

[eqn: tau_w = R over 2 left[ 2tau_i over (R-delta) - g over g_x (rho_l - rho_g ) (1-<alpha>)sin theta right] ``]

1) Engineering approach

let [eqn: R-r_i = delta ``], [eqn: pi ( R^2 - r_i^2 ``) ∼ R ``delta ``2 pi ``] for small [eqn: delta``]

[eqn: tau_w = f_w G^2 over 2rho_f ``],    [eqn: tau_i `` = f_i {rho_f ``(v_g -v_f )^2 }over 2 ``]

[eqn: f_i ``≒ ``K``f_w ``] by Wallis,    where [eqn: K= CASES{ &∼1 ~ ~~~~~~~~~for ~smooth ~surface ## & 1+300 delta / D ~~~for ~wavy ~surface } ``]

2) Second Method

〫turbulence effect

〫account for entrainment

－ Calculation of [eqn: W_ef ``] when [eqn: delta ``], [eqn: {d P} over dz ``] are known

1) calculate [eqn: tau_i ``] from [eqn: r_i = R-delta ``] and [eqn: {d P} over dz ``]

2) calculate [eqn: tau``(r) ``] in liquid film from [eqn: delta``], [eqn: tau_i ``], [eqn: {d P}over dz ``]

3) calculate [eqn: u(r) ``] within liquid film

[eqn: tau = mu_E {d u} over {d y} ``]

4) calculate of liquid film flow rate

[eqn: W_ef = int_0^delta 2``pi`` u``(r) rho_l ``dr ``]

2. Flow pattern map and transients

Fig. 4.9 Flow pattern map for

vertical flow

- [eqn: w_l]  vs [eqn: w_g] : 가장 일반적임

[eqn: rm {CASES{ & I. ~w ~as ~the ~controlling ~parameters # & II. ~w ~and ~x ~as ~the ~controlling ~parameters # & III. ~di mensional ~analysis # & IV. ~other ~analytical ~and ~empirical ~approach }}]

Flow pattern map by Hewitt & Robert [Fig. 4.9]

－ Category II

[eqn: rho_f ``j_f^2 = {[G(1-x)]^2 } over rho_f ``],   [eqn: rho_g `` j_g^2 = [Gx]^2 over rho_g ``]

→ cannot show the effects of physical properties of

fluis or geometry effect

Thus,  Baker's chart (horizontal tube)[Fig.4.10] used

[eqn: G_g / lambda ``] vs. [eqn: G_l ``lambda ``psi ``/ G_g ``] ,

Fig. 4.10 Flow pattern map for

horizontal flow

where  [eqn: lambda = left[ left( rho_g over 0.075 right) left( rho_l over 62.3 right) right]^(1/2) ``]

[eqn: psi = left( 0.005 over sigma right) left[ mu over 2.42 left( 62.3 over rho_l right) right]^(1/3) ``]

(1) bubbly flow - slug flow transition

[eqn: alpha ``] < 0.3

By Taitel and Dukler

[eqn: j_f over j_g = 2.34 - 1.07 { [g(rho_f - rho_g ) sigma ]^(1/4) } over {j_g rho_f^(1/2) } ``]    (4.3.8)

(2) Slug flow - churn flow transition

slug flow: [eqn: beta``] < 0.8 ~ 0.85

upper limit of slug flow - flooding

[eqn: therefore ``]transition to churn flow by break-up of long vapor bubble

By Porteons

[eqn: U_t ``]: Eq. (4.3.7)

[eqn: {j} over {U _{t}} =0.3 LEFT ( {rho _{f}} over {rho _{g}} RIGHT ) ^{0.5} ``]  for slug flow

Then the upper limit of slug flow

[eqn: j``=``0.105 ``rho_g^-0.5 [g ``D ``(rho_f - rho_g )]^0.5 ``]

(3) Churn flow - annular flow transition

low limit of annular flow - flow reversal

[eqn: j_g^* ``=`` 0.9``]

By Zuber, [eqn: j_g^* ``= ``4 ``left( rho_g over rho_f right)^0.5 [j_f^* ``+ ``0.35] ``]

By Taitel and Dukler

[eqn: j_g ~=~ 3.09 ``rho_g^-0.5 [g(rho_f - rho_g ) sigma ]^0.25 ~~~~~~~for~~X_tt ~<<~ 1]

[eqn: j_g ~=~ 30.9 {[g(rho_f - rho_g ) sigma ]^0.25} over {rho_g^0.5 X_tt} ~~~~~~~~~~for~~X_tt ~>>~ 1]

★ [eqn: beta ``] vs. Fr  ( [eqn: beta ``≜`` Q_g over {Q_l +Q_g } ``],  [eqn: Fr = {[(Q_l +Q_g )/Ap ]^2 } over {g D_p } ``])

Bubble - slug	[eqn: beta = 0.05(Fr)^0.2 ``]

Slug - churn                        [eqn: = 0.12 (Fr)^0.15 ``]

Churn - annular                     [eqn: = 0.65 (Fr)^0.05 ``]