[Appendix 2.2] Bubble growth velocity with evaporation

Consider evaporation

②

[eqn: Delta R ``=``[ dotR `-`u(R) `] Delta t] : evaporation 되는 두께

②'                            ★ [eqn: dot R] ; interfacial velocity

[eqn: {dot R} Deltat]     [eqn: u`(`R`)`Deltat]  R

jump condition ; [eqn: {dot m}_1 + {dot m}_2 = 0]

(If no mass transfer, [eqn: {dot m}_1 = {dot m}_2 = 0])

①,①'

Mass balance

[eqn: 4 pi``R^2 [`` dot R ``-``u(`R`)``] rho_l Delta t &~=~ rho_g d over dt LEFT [ 4 over 3 pi R^3 right] Delta t ## &~=~ 4 pi``R^2 rho_g dot R Delta t]

Thus,       [eqn: u(R) = dot R (1- rho_g over rho_l )]

If no mass transfer [eqn: u``(``R``)`` =`` dot R]                       (ⅱ)

[Appendix 2.3] Derivation of Rayleigh Eqaution

If no mass transfer, the momentum equation will be re-written with Eq.(ii) as,

[eqn: rho_l over g_c left[ {{ddot R} `` R^2 ``+``2R``dot R^2 } over r^2 ``+`` dot R`` R^2 over r^2 left(`` -`` {2 dot R `R^2 } over r^3 right) right] ``=``-`` {partial P} over {partial r}]

[eqn: rho_l over g_c left[ {ddot R `` R^2 ``+``2R``dot R^2 } over r^2 -`` {2 dot R^2 `R^4 } over r^5 right] ``=``-`` {partial P} over {partial r}]

Integrating [eqn: INT _{ R}^{inf } ( ~~~`~) dr]

[eqn: rho_l over g_c `` left [ ``(` 2R``dot R^2 ``+`` R^2 `ddot R``)` int_R ^inf {dr over r^2 } ``-`` 2 dot R ``R^4 int_R ^inf {dr over r^5 } right] ``=`` -``int_R ^inf {{partial P} over {partial r} ``dr}]

∞

Since [eqn: int from R to inf { dr over r^2 } ~=~ -``1 over r]    [eqn: ~=~ 1 over R]   and   [eqn: int from R to inf { dr over r^5 } ~=~ 1 over 4 ``1 over R^4] ,

R

[eqn: rho_l over g_c left[ {ddot R `` R^2 ``+``2R``dot R^2 } over R -`` {2 dot R^2 `R^4 } over 4R^4 right] ``=``-``P_l (inf) ~+~ P_l (R)]

[eqn: therefore~~~ rho_l over g_c left[ {ddot R `` R ``+``3 over 2``dot R^2 } right] ``=~ P_l (R)``-``P_l (inf)]

Note that  [eqn: P_l (R) ``-`` P_l ( inf) ~=~ P_v ``-`` P_l (inf) ``-`` [``P_v ``-`` P_l (R) ``]]

=[eqn: Delta P]             = [eqn: {2 sigma } over R]

Then, from the Clasius - Clayperyon Eq,.

[eqn: Delta P ``=`` P_v - P_l (inf ) ``=`` h_fg`` { (``T_v ``-``T_sat ``)```rho_g ``} over T_sat]    from  [eqn: v_fg SIMEQ 1 over rho_g]

Finally, Rayleigh Equation for the dynamic controlled bubble growth is given as,

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

where [eqn: T_v ``-T_sat] : superheat