Chapter 2  Bubble Dynamics

o Bubble Dynamics

Boiling Heat Transfer           o Pool Boiling

o Forced Convection

latent                          micro-             Boiling

heat transfer                     convection

★ Description of Heat Transfer in the Isolated Bubble

- dynamic bubble growth coupled with heat transfer

- superheated layer

- heat transfer though the bubble          1) free convection of liquid

surrounding to bubble

2) heat transfer from super-

heated layer and vaporization

of microlayer

- micro-convection

① Step (1) ; rapid vaporization of

microlayer

② Step (2) ; temperature rise due                     to poor heat transfer       Fig. 2.1 Wall temperature change

through vapor                     around bubble

③ Step (3) ; surface quenching by cold liquid

during bubble departure  - microconvection or micropumping

④ Step (4) ; temperature rise due to re-establish of thermal-boundary

1. Basic Definitions

① Surface tension (σ)  [lbf/ft]

≡ Energy per unit area to maintain an

interface

② Gibbs free-energy function

g ≡ h - Ts

Fig. 2.2  σ vs Tsat

③ Contact angle (β)

≡ Angle between the solid surface

and gas-liquid interface

④ Equilibrium

1) thermal equilibrium    ; [eqn: T_v = T_l]       Fig. 2.3 Contact angle on

2) mechanical equilibrium ; [eqn: p_v = p_l]              nonwetting(left) and

3) phase equilibrium      ; [eqn: g_v = g_l]              wetting(right) surface

★ Bubble equilibrium

☞ the condition of mechanical

equilibrium for the curved surface

[eqn: p_v - p_l = 2 sigma over r]

☞ [eqn: T_v = T_l]

☞  [eqn: g_v = g_l]

Assume                                     Fig 2.4 Saturation curve with

[eqn: T_v ~=~ T_l ~approx~ T_sat (p_v )]                          Gibbs function

Since [eqn: T_l = T_v] and the vapor must

be saturated,

[eqn: T_l ~>~ T_sat (p_l )]

Thus, the liquid is superheated

2. Nucleation

□ Homogeneous Nucleation

□ Surface(Cavity) Nucleation

(1) Homogeneous(or bulk) Nucleation

; The supperheated liquid in meta-

stable suffers form an abrupt phase     Fig. 2.5 Curved and flat interfaces

change due to the fluctuation of

superheat.

● Liquid superheated

● radiation attenuation

● neutron thermalization

★ Metastability in the Two-Phase

[eqn: bar bc] (metastable superheated liquid)

[eqn: bar de] (metastable subcooled vapor)

; important in BWR, with respect to

to explaining bubble nucleation and

transient accident phenomea

[eqn: bar cd] (completely unstable and cannot exist)

; [eqn: dp over dv ~>~ 0]

Fig. 2.6  P-V Diagram

Energy required to form a bubble

= (surface tension) + (latent heat)

= [eqn: 1 over J int from A sigma dA + int from V rho g h_fg dV]

= [eqn: {4 pi r_2} over J sigma + 4 over 3 pi r_3 rho g h_fg]

Note Van-de-Waals equation overestimates         the amount of superheat for homo-      Fig. 2.7 Superheat required for

geneous nucleation                              homogeneous nucleation

	Tsuperheat	Tsat
air	320oC	100oC
benzene	225oC	80oC

(Ex)

But, in actual, [eqn: sim] the order of 10oC

※ Kinetic view by Cole

; exist a probability that sufficient number of molecules with greater than

average energy to form a cluster with equilibrium radius

[eqn: T-T_sat = {T_sat v_fg} over h_fg rho_l over {rho_l - rho_g} left[ 16pisigma^3 over {3kT ln(nkT/nuJ)} right]]

where n is the number density of molecules

(2) Surface(or cavity) Nucleation

- boiling at single spots(scratch or cavity) on the solid heating surface

- much lower wall temperature than the energy for homo. nucleation

◉ Clausius-Clapeyron equation

; relate the vapor-liquid pressure differential to liquid superheat

[eqn: left( dT over dp right)_sat = Tv_fg over Jh_fg]                                               (2-1)

Mechanical equilibrium of a spherical bubble containing a noncondensible

gas will be,

[eqn: p_v + p_g - p_l = 2sigma over r]                                          (2-2)

Integrating the Clausius-Clapeyron equation along the saturation curve,

[eqn: T_v `- `T_sat `= ` ( p_v - p_l` ) { T_sat`` v_fg } over {J `h_fg}]                                    (2-3)

By combining Eqs.(2-2) and (2-3), also with the thermal equilibrium

[eqn: T_l` -` T_sat` =` ( 2sigma over r - p_g` ) { T_sat`` v_fg } over {J` h_fg}]                                    (★)

- the superheat required for spherical vapor bubble of radius r to remain        in equilibrium

★ If pg is neglected the equilibrium bubble size will be,

[eqn: r^* = 2sigma over {Jh_fg rho_v} T_sat over {(T_l - T_sat)}]       for [eqn: v_v ~≫~ v_l]                    (2-4)

Fig. 2.8 기포 성장

For an ideal gas,

[eqn: v_fg ~APPROX ~ v_g ~=~ {RT_sat} over p_v]              (2-5)

Thus, Eq.(★) will be,

[eqn: T_l` - `T_sat `=` ( 2sigma over r - p_g ) { R_v {T}`_{sat } ^{2 } } over {J` p_v` h_fg}]   (2-6)          - The presence of dissolved gas

reduces the superheat required.

(3) Bubble growth process

If superheat > Eq.(★) → bubble grows

If superheat < Eq.(★) → bubble condense

3.  Nucleation criterion for heated commercial surface

(1) Nucleation Criteria

In laminar sublayer, Fourier's law gives,

[eqn: q^'' `=` - k_f ```partialT over partialy]                                          (2-7)

- should provide the superheat greater than that of Eq.(★)

By combining Eq. (2-7) and equation obtained by differentiating Eq.(★),

[eqn: q`^'' `=` - k_f ``partialT over partialy RIGHT |_{y={r_*}} = { 2 k_f`` sigma } over {{ r}`_{* } ^{2 } } LEFT ( {T_sat`` v_fg } over { J h_fg } RIGHT )]                              (2-8)

Thus, the critical cavity radius to nucleate will be,

[eqn: r_* = LEFT ( { 2 sigma T_sat`` v_fg` k_f`} over { J h_fg`` q`^'' } RIGHT )^1/2]                                      (2-9)

For a linear temperature gradient through liquid in the laminar sublayer, i.e.

[eqn: T_w - T_l `=`{ q`^'' r^* } over k_f]                                        (2-10)

Then,

[eqn: T_w - T_sat `=`(T_l - T_sat `)`+`{ q`^'' r^* } over k_f]                              (2-11)

From Eq.(★),

[eqn: T_w - T_sat `=` { 2sigma over r_* {T_sat`` v_fg } over {J` h_fg } }`+`{ q`^'' r^* } over k_f]                   (C4-23 )     (2-12)

If only one cavity size([eqn: r^* = r_o]) is present,

[eqn: q`^'' ~=~ k_f {(T_w - T_sat )} over r_o - { 2sigmak_f over r_o^2 {T_sat`` v_fg } over {J` h_fg } }]                             (2-13)

But, if all cavity sizes are present, r* will be eliminated as,

[eqn: q`^'' `=` { {J` h_fg `k_f} over {8sigmaT_sat`` v_fg } } (T_w - T_sat`)^2]  (2-14)

Fig 2.9 강제대류하 수직가열면에서           의 기포 성장

☞  Nucleation Criteria

=> low bound of nucleation

Note that the number of active nucleating

cavities increases as [eqn: dot q] increases.

[eqn: dot q ~～~ n^m]

where m=1 at low [eqn: dot q] and decreases to

around ½ at high [eqn: dot q]

(2) Boiling (or Ebullition) process

T∞

Fig 2.10 Boiling Process

◎ Prediction of tc                               T(y,t)

[eqn: partialT over partialt ~=~ left( k_f over {rho_f c_pf} right) {partial^2 T} over partialy^2]       (2-15)                     v       Tw

I.C  ; T(y,0) = [eqn: T_l (inf)]

B.C ; T(∞,t) = [eqn: T_l (inf)],    T(0,t) = Tw

=>  [eqn: {T(y,t) -T_w} over {T_l (inf) - T_w} ~=~ erf left( y over {2sqrt{alpha_f t}} right)]                              (2-16)

Thus, tc can be obtained until reaching T(y,tc) = T* of Eq.(★)

● Forces exerted on a bubble

① surface force

② buoyancy force

③ liquid inertia due to bubble growth

④ hydrodynamic drag force

⑤ force due to the liquid convection around the bubble

The size of bubble at departure from the surface can be obtained from the

dynamic force balance,

[eqn: pi over 6 D_B^3 g over g_c (rho_l - rho_v ) ~=~ pi D_B sigma C_o (beta_c )],                             (2-17)

as,

[eqn: D_B `=` root{ { 6 C_o ``sigma` g_c } over { g ( rho_l` -`rho_v`)} }]                                       (2-18)

※ For small cavity size, account only for buoyancy and liquid inertia force

By Fritz and Wark,

[eqn: D_B `=` C_d` beta_c ` root{ { 2 sigma` g_c } over { g ( rho_l` -`rho_v`)} }]      where Cd=0.0148

By Cole and Rohsenow

[eqn: { E}`_{0 } ^{1/2 } `=` CASES { ```````` 1.5 TIMES 10^-4 ```( { J}`_{a } ^{* } )^5/4~~~~~~~~"for water" # ````````4.65 TIMES 10^-4 ```( { J}`_{a } ^{* } )^5/4~~~~~~~"for other fluids" }]

where

[eqn: &E_0 `=` { g`( rho_l` -` rho_v`) { D}`_{B } ^{2 }`} over sigma #& J_a^*`=`{ rho_l` C_pf` T_sat } over { rho_v ` h_fg } ~~~left( ≡{sensible~ heat} over {latent~ heat} right)]

☞ DB decreases as the mass flow rate increases due to the increase of drag

force of bubble on the active nucleation site (Levy, Koumoutsos)

☞ DB increases as the wall heat flux (Unal)

Departure frequency

Fig. 2.11 Tw during bubble departue

[eqn: f_B ~=~ 1 over {t_c + t_d}]

Jacob and Linke let the average bubble

departure velocity be,

[eqn: U_B ~=~ D_B over t_d ~~~~~~~ therefore~~ D_B ~=~ U_B t_d]

Then,   [eqn: D_B f_B ~=~ t_d over {t_c +t_d} U_B ~~~ under～ ~f(p)]

Max Jacob assumed td = tc , and by Peebles & Garber,

[eqn: D_B f_B ~=~ 1.18 over 2 left[ {sigma g (rho_l -rho_v )} over rho_l^2 right]^¼]

[eqn: under～] 920 ft/hr

=> nearly constant, because, before leaving the cavity the bubble grows             faster in the beginning and then slow down its growth before ejecting

But, td ≠ tc ,  thus cannot be constant

(例 ; DBfB [eqn: under～] 1200 ft/hr   for Carbon tetrachloride   by Westwater )

4. Bubble Dynamics

Continuity Equation

[eqn: ~~ 1 over r^2 partial over partialr (r^2 rho_l ~v_r )~ +~ partialrho_l over partialt ~=~0]                                         (2-19)

Momentum Equation

[eqn: - partialp over partialr ~=~ 1overg sub c~ rho subl~ Dv subr over Dt]                                                 (2-20)

For incompressible and inviscid liquid,     [eqn: v(r) ~=~ v(R) left( R over r right)^2]            (2-21)

At the interface, if there is no mass transfer,

[eqn: v(r) ~=~ {dot R} left( R over r right)^2]                                        (2-22)

Inserting the above equation into the momentum equation, we can obtain Rayleigh equation which governs  the dynamic controlled bubble growth.

[eqn: {rho_l over g_c} left( R R″``+``3 over 2 R′^2 right)``=``{h_fg (T_v ``-``T_sat )rho_g} over T_sat ``-``2sigma over R]

=> Rayleigh Equation

Energy Equation

[eqn: partialT over partialt ~+~ u(r) partialT over partialr ~=~ alpha_l 1 over r^2 partial over partialr left[r^2 partialT over partialr right]]

Note that   [eqn: u(r) ~=~ dot R ( R over r )^2 (1- rho_g over rho_l )]

I.C  ;  T(r,0) = T∞

B.C ;  T(R,t) = Tv  ([eqn: T_l = T_v =T_sat] )

T(∞,t) = T∞

T

Bubble Growth Mechanism

(1) Isothermal Bubble Dynamiccs (dynamic controlled)

- initial stage , [eqn: t^+ ``≪ ``1]

① [eqn: r``≫``r_crit]

② [eqn: T_l ``=``T_v ``≫``T_sat]

③ [eqn: P_v ``≫``P_v,ONB ~(`=``P_l ``+``2sigma over r``)]

With assumption of [eqn: lim _{} {} {R} over {R _{0}} ``` -> �`` INF] ,  where [eqn: R_0]: initial radius

[eqn: R(t)``=``Kt``+``R_0]

or  [eqn: R^+ `=```t^+]

(2) Isobaric Bubble Dynamics (thermally controlled)

- [eqn: t^+ ``≫``1]

With assumption of [eqn: T_v ``=``T_sat]

[eqn: R(t)``` ∝``root t]

Introducing [eqn: J_a]

[eqn: R(t) ``=``2 root { 1 over pi} ``J_a root {alpha_f `t}]  where, [eqn: J_a ``=``{rho_f `C_pf `(T_w - T_sat )} over {rho_g h_fg}]

By Mikic, Rohsenow and Graffith

[eqn: R^+ ``=``2 over 3 left[ (t^+ +1)^{3 over 2} - (t^+ )^{3 over 2} - 1 right]]