# Heterogeneous Bubble Growth

For a vapor bubble attached to a heating surface, growth occurs in a nonuniform temperature field. The criteria for initiation and growth of a bubble near a wall were analyzed by Han and Griffith (1965). They assumed that a surface cavity, which serves as a nucleation site, has a hemispherical vapor cap on it. As a result, the bubble can grow only if the thermal layer adjacent to the nucleation site is sufficiently thick. A relation between cavity size and surface temperature elevation was derived on the basis of an assumed relation between the thermal layer thickness and the nucleus size at incipience. For saturation conditions, they found the maximum and minimum cavity sizes that can be active under a constant temperature:

${({R_c})_{\max ,\min }} = \frac{\delta }{3}\left\{ {1 \pm {{\left[ {1 - \frac{{12\sigma {T_{sat}}}}{{\delta {\rho _v}{h_{\ell v}}({T_w} - {T_{sat}})}}} \right]}^{1/2}}} \right\} \qquad \qquad(1)$

where δ is the thermal layer thickness.

Figure 1 Criteria for growth of hemispherical bubble nucleus.

Howell and Siegel (1966) examined bubble characteristics on single nucleation sites of known diameter (0.1~1mm) on highly polished surfaces in pool boiling (see Fig. 1). Since the bubbles were large and slow-growing, only surface tension and buoyancy forces were important at departure. If the vapor bubble extends through the liquid thermal layer (Rc > δ) as shown in Fig 1 (a), evaporation occurs from the portion of the nucleus surface within the thermal layer while condensation occurs over the remaining portion of the nucleus. For the vapor bubble to grow, the amount of evaporative heat transfer must be greater than that of the condensation, which leads to the following criterion:

${T_w} - {T_{sat}} > \frac{{4\sigma {T_{sat}}}}{{{\rho _v}{h_{\ell v}}\delta }}\begin{array}{*{20}{c}} , & {{R_c} > \delta } \\ \end{array} \qquad \qquad(2)$

If the nucleus is wholly contained in the thermal layer, as shown in Fig. 1(b), the temperature required for the bubble to grow is

${T_w} - {T_{sat}} > \frac{{2\sigma {T_{sat}}}}{{{\rho _v}{h_{\ell v}}{R_c}}}\frac{1}{{1 - {R_c}/(2\delta )}}\begin{array}{*{20}{c}} , & {{R_c} < \delta } \\ \end{array} \qquad \qquad(3)$

For the growth rate of a vapor bubble attached to the wall, Mikic et al. (1970) suggested that Homogeneous Bubble Growth eqs. (19)-(23) are applicable, provided that the geometric parameters b = π / 7 and ${T_\ell }(\infty )$ are replaced by the heating surface temperature Tw.

As noted before, the microlayer plays a very important role during bubble growth near a heated surface (Figure 3 from Nucleation and Inception)

van Stralen et al. (1975a) proposed the following relation for bubble growth using microlayer theory, which is applicable to both inertia and heat transfer controlled region for pure liquids:

$R(t) = \frac{{{R_1}(t){R_2}(t)}}{{{R_1}(t) + {R_2}(t)}} \qquad \qquad(4)$

where

${R_1}(t) = 0.8165\sqrt {\frac{{{\rho _v}{h_{\ell v}}({T_w} - {T_{sat}})\exp [ - {{(t/{t_d})}^{1/2}}]}}{{{\rho _\ell }{T_{sat}}}}} t \qquad \qquad(5)$
$\begin{array}{l} {R_2}(t) = 1.954\left\{ {{R^*}\exp \left[ { - {{\left( {\frac{t}{{{t_d}}}} \right)}^{1/2}}} \right] + \frac{{{T_\infty } - {T_{sat}}}}{{{T_w} - {T_{sat}}}}} \right\}{\rm{Ja}}{({\alpha _\ell }t)^{1/2}} \\ {\rm{ + }}0.373\Pr _\ell ^{ - 1/6}{\left\{ {\exp \left[ { - {{\left( {\frac{t}{{{t_d}}}} \right)}^{1/2}}} \right]} \right\}^{1/2}}{\rm{Ja}}{({\alpha _\ell }t)^{1/2}} \\ \end{array} \qquad \qquad(6)$
${R^*} = 1.3908\frac{{{R_2}({t_d})}}{{{\rm{Ja}}{{({\alpha _\ell }{t_d})}^{1/2}}}} - 0.1908\Pr _\ell ^{ - 1/6} \qquad \qquad(7)$

R * can be determined if the departure time td is known. van Stralen et al. (1975b) compared the above equation with experimental bubble growth rate data for water over a range of temperature, and obtained good agreement.

## 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.

Han, Y.Y., and Griffith, P., 1965, “The Mechanism of Heat Transfer in Nucleate Pool Boiling. I – Bubble Initiation, Growth and Departure,” International Journal of Heat Mass Transfer, Vol. 8, pp. 887-904.

Howell, J.R., and Siegel, R., 1966, “Incipience, Growth, and Detachment of Boiling Bubbles in Saturated Water from Artificial Nucleation Site of Known Geometry and Size,” Proceedings of the 3rd International Heat Transfer Conference, Chicago, IL, Vol. 4, pp. 12-23.

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.

van Stralen, S.J.D., Sohal, M.S, Cole, R. and Sluyter, W.M., 1975a, “Bubble Growth Rates in Pure and Binary Systems: Combined Effects of Relaxation and Evaporation Microlayers,” International Journal of Heat and Mass Transfer, Vol. 18, pp. 453-467.

van Stralen, S.J.D., Cole, R., Sluyter, W.M., and Sohal, M.S, 1975b, “Bubble Growth Rates in Nucleate Boiling of Water at Subatmospheric Pressures,” International Journal of Heat and Mass Transfer, Vol. 18, pp. 655-669.