[Appendix 2.4] Derivation of Isobaric Bubble Dynamics

Consider the bubble growth of a

single spherical isobaric bubble in a

uniformly superheated liquid of large

extent and assume the thermal

boundary layer as a slab

By neglecting the radial liquid motion,

[eqn: {partial T} over {partial t} = alpha_f {partial^2 T } over {partial y^2}]

B.C. : [eqn: cases {&T(inf,t) = T_inf #& T(0,t) = T_v}]

I.C. :    [eqn: T(y,0) = T_inf]     for superheated liq.

⇒ [eqn: {T(y,t)-T_v} over {T_inf - T_v } = erf ( y over {2 root{alpha_f t}}) = 2 over root pi integral to {y over {2 root{alpha_f t}}} from 0 e^ {-ξ^2 } dξ}]

Note  erf(0)=0

erf(∞)=1

By using Leibniz rule

[eqn: d over dy integral_{ F_1 (y)}^{ F_2 (y)} f(x,y) dx = integral_{ F_1 (y)}^{ F_2 (y)} partialf over partialy dx + f (F_2 , y) dF_2 over dy - f(F_1 ,y ) dF_1 over dy]

∴ [eqn: dT(y,t) over dy = (T_inf - T_v ) 2 over root pi e^{-y^2 over{ 4alpha_f t}} 1 over {2 root {alpha_f t}}]

∴ [eqn: dT over dy ｜_y=0 = (T_inf - T_v ) 1 over root {pi alpha_f t}]

Now

[eqn: q^'' = k_f partialT over partialy ｜_y=0 = {k_f (T_inf -T_v ) }over{root{pi alpha_f t }} = h (T_inf -T_v )]	①

From energy balance through interface

[eqn: ρ_g h_fg d over dt [ 4over 3 pi R^3 ] = ρ_g h_fg 4 pi R^2 dR over dt]

[eqn: = h 4pi R^2 (T_inf -T_v )]	←①

∴ [eqn: {k_f (T_inf - T_v )} over root {pi alpha_f t} = ρ_g h_fg dR over dt]

Integrating

[eqn: R(t) = {2k_f (T_inf -T_v )} over {ρ_g h_fg } root {t over {pi alpha_f }}]

Hsu assumed, [eqn: T_v = T_sat]

∴ [eqn: R(t) ∝ root t]

Introducing Jacob number, [eqn: J_a]

[eqn: J_a ≜ {ρ_f C_pf (T_inf - T_sat )} over {ρ_g h_fg } = rm {sensible~ heat} over {latent~ heat}]

Then,

[eqn: R(t) = 2 root {1 over pi } J_a root {alpha_f t }]

[eqn: = 2 C_1 root {alpha_f t}]      where, [eqn: C_1 = root {1 over pi } J_a]