# Homogeneous Bubble Growth

Bubble growth within a superheated liquid drop will be considered first. Once a growing vapor bubble’s radius reaches that unstable equilibrium, it grows spontaneously. During the early stage, when the bubble radius is small, the Laplace-Young equation indicates that the pressure differential across the interface is at its maximum value. The resulting high interfacial velocity leads to significant inertia terms in the momentum equation. Meanwhile, the temperature of the interface is close to the superheat temperature of the surrounding liquid, so heat transfer into the vapor bubble experiences the highest driving temperature differential that occurs at any time during the process. As a result, early bubble growth tends to be limited by inertia, or the exchange of momentum between the vapor and liquid phases. In the physical model shown in the figure on the right, the instantaneous radius and the liquid-vapor interface velocity at time t are R(t) and dR / dt, respectively. For an incompressible, radially symmetric inviscid flow, the continuity equation in the spherical coordinate system is $\frac{1}{{{r^2}}}\frac{{\partial ({r^2}u)}}{{\partial r}} = 0 \qquad \qquad(1)$

where u is the radial velocity in the liquid phase. Assuming ${\rho _v} \ll {\rho _\ell }$, this velocity at the liquid-vapor interface is equal to the growth rate, i.e., $u(R,t) = \frac{{dR}}{{dt}} \qquad \qquad(2)$

Integrating eq. (1) over the interval of (R,r), where r is an arbitrary location in the liquid phase, one obtains $u(r,t) = \frac{{dR}}{{dt}}{\left( {\frac{R}{r}} \right)^2} \qquad \qquad(3)$

The momentum equation for the surrounding liquid is ${\rho _\ell }\left( {\frac{{\partial u}}{{\partial t}} + u\frac{{\partial u}}{{\partial r}}} \right) = - \frac{{\partial p}}{{\partial r}} \qquad \qquad(4)$

Substituting eq. (3) into eq. (4) yields $\frac{{{\rho _\ell }}}{{{r^2}}}\left[ {2R{{\left( {\frac{{dR}}{{dt}}} \right)}^2} + {R^2}\frac{{{d^2}R}}{{d{t^2}}}} \right] - \frac{{2{\rho _\ell }{R^4}}}{{{r^5}}}{\left( {\frac{{dR}}{{dt}}} \right)^2} = - \frac{{\partial p}}{{\partial r}} \qquad \qquad(5)$

Integrating eq. (5) from the bubble surface (r = R) to infinity ( $r \to \infty$) yields $R\frac{{{d^2}R}}{{d{t^2}}} + \frac{3}{2}{\left( {\frac{{dR}}{{dt}}} \right)^2} = \frac{1}{{{\rho _\ell }}}\left[ {{p_\ell }(R) - {p_\ell }(\infty )} \right] \qquad \qquad(6)$

Substituting Nucleation and Inception Eq (1) ${p_g} + {p_v} - {p_\ell } = \frac{{2\sigma }}{{{R_b}}} \qquad \qquad(1)$

with pg = 0 into eq. (6) gives $R\frac{{{d^2}R}}{{d{t^2}}} + \frac{3}{2}{\left( {\frac{{dR}}{{dt}}} \right)^2} = \frac{1}{{{\rho _\ell }}}\left[ {{p_v} - {p_\ell }(\infty ) - \frac{{2\sigma }}{R}} \right] \qquad \qquad(7)$

which is referred to as Rayleigh’s equation. During the inertia-controlled stage of bubble growth, the surface tension term 2σ / R is negligible compared to ${p_v} - {p_\ell }(\infty ).$ The pressure difference ${p_v} - {p_\ell }(\infty )$ can be evaluated by using the Clausius-Clapeyron equation, i.e., ${p_v} - {p_\ell }(\infty ) = \frac{{{\rho _v}{h_{\ell v}}\left[ {{T_\infty } - {T_{sat}}[{p_\ell }(\infty )]} \right]}}{{{T_{sat}}[{p_\ell }(\infty )]}}\qquad\qquad(8)$

Substituting eq. (8) into eq. (7) and neglecting the surface tension term 2σ / R in eq. (7), one can obtain an ordinary differential equation for the bubble radius: $R\frac{{{d^2}R}}{{d{t^2}}} + \frac{3}{2}{\left( {\frac{{dR}}{{dt}}} \right)^2} = \frac{{{\rho _v}{h_{\ell v}}}}{{{\rho _\ell }}}\frac{{{T_\infty } - {T_{sat}}[{p_\ell }(\infty )]}}{{{T_{sat}}[{p_\ell }(\infty )]}} \qquad \qquad(9)$

which is subject to the following initial condition: $R = 0\begin{array}{*{20}{c}} , & {t = 0} \\ \end{array} \qquad \qquad(10)$

The instantaneous bubble radius can be obtained by solving eq. (9): $R(t) = {\left\{ {\frac{2}{3}\frac{{{\rho _v}{h_{\ell v}}}}{{{\rho _\ell }}}\frac{{{T_\infty } - {T_{sat}}[{p_\ell }(\infty )]}}{{{T_{sat}}[{p_\ell }(\infty )]}}} \right\}^{\frac{1}{2}}}t \qquad \qquad(11)$

which indicates that the radius of the bubble is a linear function of time during the early stage of the bubble’s growth. During the later stage of bubble growth, the inertial force becomes insignificant as the pressure difference ${p_v} - {p_\ell }$ (remember Laplace-Young’s equation) diminishes and the interfacial motion slows. At the same time, the temperature difference between the vapor inside the bubble and the interface diminishes. With a lower level of forcing fraction, heat transfer becomes the process that limits bubble growth. $\frac{{\partial {T_\ell }}}{{\partial t}} + u\frac{{\partial {T_\ell }}}{{\partial r}} = \frac{{{\alpha _\ell }}}{{{r^2}}}\frac{\partial }{{\partial r}}\left( {{r^2}\frac{{\partial {T_\ell }}}{{\partial r}}} \right) \qquad \qquad(12)$

where the radial velocity, u, is obtained by eq. (3). The initial and the boundary conditions for eq. (12) are $T(r,0) = {T_\ell }(\infty ),{\rm{ }}T(R,t) = {T_{sat}}({p_v}),{\rm{ }}T(\infty ,t) = {T_\ell }(\infty ) \qquad \qquad(13)$

The energy balance at the liquid-vapor interface is ${\left. {{k_\ell }\frac{{\partial T}}{{\partial r}}} \right|_{r = R}} = {\rho _v}{h_{\ell v}}\frac{{dR}}{{dt}} \qquad \qquad(14)$ $R(t) = 2{C_R}\sqrt {{\alpha _\ell }t} \qquad \qquad(15)$

where CR is a constant. For a large Jakob number (corresponding to low pressure), CR can be obtained by (Plesset and Zwick, 1954) ${C_R} = \sqrt {\frac{3}{\pi }} {\rm{Ja}} \qquad \qquad(16)$

and the Jakob number is defined as $Ja = \frac{{{\rho _\ell }{c_{p\ell }}\left[ {{T_\ell }(\infty ) - {T_{sat}}({p_v})} \right]}}{{{\rho _v}{h_{\ell v}}}} \qquad \qquad(17)$

For a small Jakob number (corresponding to high pressure) ${C_R} = \sqrt {\frac{{{\rm{Ja}}}}{2}} \qquad \qquad(18)$

Equation (15) indicates that the bubble radius is proportional to the square root of time. Bubble growth in the two limiting cases described above has been analyzed. The inertia-controlled bubble growth model ignored heat transfer, which is only valid in the early stage of bubble growth. In the heat transfer-controlled bubble growth model, on the other hand, the effect of inertia on bubble growth was neglected; this assumption is appropriate only during the late stage of bubble growth. A comprehensive bubble growth model must include both inertia and heat transfer and provide a smooth transition between the two regimes. For bubble growth in an initially uniformly superheated liquid, Mikic et al. (1970) obtained the following expression of the relationship between bubble radius and time: ${R^ + } = \frac{2}{3}\left[ {{{({t^ + } + 1)}^{\frac{3}{2}}} - {{({t^ + })}^{\frac{3}{2}}} - 1} \right] \qquad \qquad(19)$

where R + and t + are nondimensional radius and time, respectively, defined as ${R^ + } = \frac{{RA}}{{{B^2}}}\begin{array}{*{20}{c}} {} & {{t^ + } = \frac{{t{A^2}}}{{{B^2}}}} \\ \end{array} \qquad \qquad(20)$

And $A = {\left\{ {\frac{{b\left[ {{T_\ell }(\infty ) - {T_{sat}}({p_\ell }(\infty ))} \right]{h_{\ell v}}{\rho _v}}}{{{\rho _\ell }{T_{sat}}({p_\ell }(\infty ))}}} \right\}^{\frac{1}{2}}} \qquad \qquad(21)$ $B = {\left( {\frac{{12{\alpha _\ell }}}{\pi }} \right)^{\frac{1}{2}}}{\rm{Ja}} \qquad \qquad(22)$ ${\rm{Ja}} = \frac{{{\rho _\ell }{c_{p\ell }}[{T_\ell }(\infty ) - {T_{sat}}({p_\ell }(\infty ))]}}{{{\rho _v}{h_{\ell v}}}} \qquad \qquad(23)$

where b in eq. (21) is a geometric parameter with a value of 2/3 for a perfect spherical bubble. Equation (19) provides a generalized solution for the bubble radius because it can be reduced to eq. (11) for small t + and to eq. (15) for larger t + . Prosperetti and Plesset (1978) developed a more advanced model for vapor bubble growth in a superheated liquid.

## References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Mikic, B.B., Rohsenow, W.M., and Griffith, P., 1970, “On Bubble Growth Rate,” International Journal of Heat and Mass Transfer, Vol. 13, pp. 657-666.

Plesset, M.S., and Zwick, S.A., 1954, “The Growth of Vapor Bubble in Superheated Liquids,” Journal of Applied Physics, Vol. 25, pp. 493-500.

Prosperetti, A., and Plesset, M.S., 1978, “Vapor Bubble Growth in a Superheated Liquid,” Journal of Fluid Mechanics, Vol. 85, pp. 349-368.