# Bubble Growth Within Superheated Liquid Droplets

Two immiscible liquids of different volatility are mixed together such that one liquid is dispersed in the form of droplets within the other, so that the droplets may be heated by direct contact across the liquid/liquid interface. Heat can be supplied by conduction, convection, nucleate boiling, or film boiling. When nucleation sites are not present, single-phase conduction or convection can exist beyond the normal saturation state of liquid 1; this is termed superheating. This state can exist as long as the liquid does not come into contact with a vapor phase with which it is in equilibrium. The upper limit to the temperature of the liquid at a given pressure before a phase change occurs is called the superheat limit. At the superheat limit, homogeneous nucleation will begin to occur inside liquid 1 and form a vapor droplet. These vapor bubbles are in metastable equilibrium with the surrounding liquid, and subsequent growth of the initial bubbles will complete the phase transition. The two steps in phase change related to superheated liquid droplets are (Avedisian, 1986):

1. The initial stage, during which microscopic bubbles form in the droplet

2. A second or bubble growth stage, in which the initial bubble grows as the liquid droplet vaporizes

Bubble growth within superheated liquid droplets is important in applications such as preparation of emulsified liquids, fuel coolant interactions in postulated nuclear reactor accidents, and heat exchangers. In these cases, droplets of the volatile liquid are dispersed in another stagnant liquid. Heating the field liquid leads to superheating of the droplets, homogeneous nucleation, and bubble growth. Bubble growth in a volatile droplet suspended in an immiscible nonvolatile liquid was developed by Avedisian and Suresh (1985) by solving the coupled energy and momentum equations for the temperature fields in both liquids. A numerical solution for a two-phase droplet modeled as a vapor bubble growing from the center of the liquid is presented. When the properties of the two liquids are very different, the bubble growth rate can experience a significant change when the thermal boundary layer extends into nonvolatile liquid. Consider the following conditions for superheat bubble growth. At t = 0 a vapor bubble appears in liquid 1. The initial temperatures of liquids 1 and 2 are the same, and there is no motion between the droplet and the liquid. The bubble begins to grow due to a perturbation caused by a slight reduction in ambient pressure or an increase in ambient temperature. Continuous growth of the bubble completely consumes liquid 1 until the mass of the droplet is entirely vaporized and only a vapor bubble in liquid 2 is left. The radius of the final bubble is

${R_f} = {\left( {1 - \varepsilon } \right)^{ - 1/3}}{S_o} \qquad \qquad(1)$

where $\varepsilon$ is the density ratio, and S0 is the initial overall drop radius. To formulate an equation of motion for a spherical bubble growing from the center of a spherical droplet, assume a vapor bubble growing from the center of a volatile liquid suspended in a nonvolatile liquid. The velocity, pressure, and temperature in the two liquids are governed by the continuity, momentum, and energy equations for one-dimensional and unsteady conditions, given as

$\frac{1}{{{r^2}}}\frac{\partial }{{\partial r}}\left( {{r^2}{v_i}} \right) = 0 \qquad \qquad(2)$
$\frac{{\partial {v_i}}}{{\partial t}} + {\nu _i}\frac{{\partial {v_i}}}{{\partial r}} = - \frac{1}{\rho }\frac{{\partial {p_i}}}{{\partial r}} + {\nu _i}\left[ {\frac{1}{{{r^2}}}\frac{\partial }{{\partial r}}\left( {{r^2}\frac{{\partial {v_i}}}{{\partial r}}} \right) - \frac{{2{v_i}}}{{{r^2}}}} \right] \qquad \qquad(3)$
$\frac{{\partial {T_i}}}{{\partial t}} + {\nu _i}\frac{{\partial {T_i}}}{{\partial r}} = {\alpha _i}\frac{1}{{{r^2}}}\frac{\partial }{{\partial r}}\left( {{r^2}\frac{{\partial {T_i}}}{{\partial r}}} \right) \qquad \qquad(4)$

where i = 1 and 2, vi is the radial velocity, pi is the pressure within liquid i, Ti is the temperature within liquid i, and αi is the thermal diffusivity of liquid i.

The boundary and initial conditions are

${T_1}\left( {r,0} \right) = {T_2}\left( {r,0} \right) = {T_o} \qquad \qquad(5)$

${T_1}\left( {R,t} \right) = {T_v}\left( t \right) \qquad \qquad(6)$

${T_1}\left( {S,t} \right) = {T_2}\left( {S,t} \right) \qquad \qquad(7)$

${k_1}{\left. {\frac{{\partial {T_1}}}{{\partial r}}} \right|_{S,t}} = {k_2}{\left. {\frac{{\partial {T_2}}}{{\partial r}}} \right|_{S,t}} \qquad \qquad(8)$

${T_2}\left( {\infty ,t} \right) = {T_o} \qquad \qquad(9)$

$R\left( 0 \right) = {R_o} \qquad \qquad(10)$

$\dot R\left( 0 \right) = 0 \qquad \qquad(11)$

where k1 and k2 are the thermal conductivity of liquid 1 and 2, respectively, R is the bubble radius and S is the droplet radius. The interfacial energy balance around the bubble yields

${k_1}{\left. {\frac{{\partial {T_1}}}{{\partial r}}} \right|_{R,t}} = {\rho _v}{h_{\ell v}}\dot R \qquad \qquad(12)$

where $\dot R$ is the velocity of the bubble wall. T0 and R0 are the initial conditions of the limit of superheat and initial unstable bubble radius. These terms, defined by the critical nucleus state for homogeneous nucleation, are intrinsic properties of liquid 1 and are functions of ambient pressure for a given nucleation rate. Radial velocity is continuous in this problem, because there is no mass transfer across the liquid’s interface. The velocity can be found by integrating eq. (2) and applying a mass balance around the bubble.

${v_i} = \varepsilon \frac{{{R^2}}}{{{r^2}}}\dot R \qquad \qquad(13)$

where $\varepsilon$ is the density ratio $\left( {1 - {\rho _v}/{\rho _1}} \right)$. Equation (13) is the same as bubble growth in an infinite medium.

The Rayleigh equation for this problem is obtained by integrating eq. (3) over r twice: from R to S and again form S to $\infty$.

$\int_R^\beta {\left[ {\frac{{\partial {v_i}}}{{\partial t}} + {v_i}\frac{{\partial {v_i}}}{{\partial r}} = - \frac{1}{\rho }\frac{{\partial {p_i}}}{{\partial r}} + {v_i}\left( {\frac{1}{{{r^2}}}\frac{\partial }{{\partial r}}\left( {{r^2}\frac{{\partial {v_i}}}{{\partial r}}} \right) - \frac{{2{v_i}}}{{{r^2}}}} \right)} \right]} dr \qquad \qquad(14)$

where Β = S or $\infty$. When r = R,

${p_v} - {p_{R1}} = \frac{{2{\sigma _1}}}{R} - 2{\mu _1}{\left. {\frac{{\partial {v_1}}}{{\partial r}}} \right|_{r = R}} \qquad \qquad(15)$

and when r = S

${p_{S1}} - {p_{S2}} = \frac{{2{\sigma _{12}}}}{S} - 2\left( {{\mu _2} - {\mu _1}} \right){\left. {\frac{{\partial {v_1}}}{{\partial r}}} \right|_{r = S}} \qquad \qquad(16)$

The equation of motion for a bubble growing in the center of a droplet is found by combining eqs. (13) and (12), integrating twice, and substituting eqs. (15) and (16):

$\begin{array}{l} \left[ {R\ddot R + 2{{\dot R}^2}} \right]\left( {1 - \bar \varepsilon \frac{R}{S}} \right) - \varepsilon \frac{{{{\dot R}^2}}}{2}\left( {1 - \bar \varepsilon \frac{{{R^4}}}{{{S^4}}}} \right) + 4{v_1}\frac{{\dot R}}{R}\left( {1 - \bar \mu \frac{{{R^3}}}{{{S^3}}}} \right) \\ = \frac{{{p_v} - {p_o}}}{{{\rho _1}\varepsilon }} - \frac{{2{\sigma _1}}}{{\varepsilon {\rho _1}R}}\left[ {1 + \frac{{{\sigma _{12}}}}{{{\sigma _1}}}\frac{R}{S}} \right] \\ \end{array} \qquad \qquad(17)$

where $\bar \varepsilon$ is the density ratio $\left( {1 - {\rho _2}/{\rho _1}} \right)$, $\bar \mu$ is the viscosity ratio $\left( {1 - {\mu _2}/{\mu _1}} \right)$, σ12 is the liquid 1/ liquid 2 interfacial tension, and p0 is pressure at infinity in liquid 2.

Sideman and Isenberg (1967) developed an analytical, closed-form solution for boiling a droplet in another immiscible liquid with the following assumptions:

2. Constant and uniform temperature of liquid 1

3. Inviscid and uniform translatory motion of liquid 2 around liquid 1 droplet

This closed form solution is valid only after boundary layer penetration of liquid 2. During early bubble growth the assumptions are not valid, because the temperature field is transient and the boundary layer is entirely in liquid 1. The analytical solution for boiling of a droplet in another immiscible liquid is given as

$\bar S = {{(1 - \varepsilon)}^{-1/3}}{{ (1 - \varepsilon {{[ {{(\frac{9}{2\pi})}^{1/2}}(1 - \varepsilon)JaP{{e}^{1/2}}{\tau - 1}]}^2})}^{1/3}} \qquad \qquad(18)$

where $\bar S = S/{S_o}$, Pe is the Peclet number $\left( {{U_\infty }{S_o}/{\alpha _2}} \right),$ and Ja is the Jakob number.

## References

Avedisian, C.T., 1986, “Bubble Growth in Superheated Liquid Droplets,” Encyclopedia of Fluid Mechanics, Chapter 8, Gulf Publishing Company, Houston, TX.

Avedisian, C.T., and Suresh, K., 1985, “Analysis of Non-Explosive Bubble Growth within a Superheated Liquid Droplet Suspended in an Immiscible Liquid,” Chemical Engineering Science, Vol. 40, pp. 2249-2259.

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.

Sideman, S. and Isenberg, J., 1967, “Direct Contact Heat Transfer with Change of Phase: Bubble Growth in Three-Phase Systems,” Desalination, Vol. 2, pp. 207-214.