# Bubble Detachment

A vapor bubble attached to a heating surface is subjected to different forces. While surface tension and liquid inertia forces tend to prevent such a bubble from detaching, buoyancy and dynamic forces act directly on the bubble to lift it from the heating surface. When the bubble grows large enough, the detachment-oriented forces will dominate and the bubble will detach from the heating surface. The inertial force is a strong function of liquid superheat, which, as indicated by the following equation from [Nucleation and Inception]]: $\Delta T = T - {T_{sat}} = \frac{{4\sigma {T_{sat}}}}{{{p_v}{h_{\ell v}}{D_c}}}{K_{\max }}$

is inversely proportional to the size of the cavity. Therefore, a bubble formed from a small cavity grows faster than one formed from a large cavity. The bubble size at departure can be found by performing a force balance on the bubble. For a small cavity size (rc < 10μm), the bubble size at departure is dominated by a balance between buoyancy and liquid inertial forces. When the cavity size is large, the bubble grows more slowly, thus the inertial force becomes unimportant. Instead, a force balance between buoyancy and surface tension dominates the bubble size at departure. Fritz (1935) was the first to correlate the bubble departure diameter by balancing buoyancy and surface tension forces, and he proposed the following equation: ${D_b} = 0.0208\theta \sqrt {\frac{{2\sigma }}{{g({\rho _\ell } - {\rho _v})}}} \qquad \qquad(1)$

where θ is the contact angle measured in degrees. Further experimental investigations indicated that eq. (1) can provide a correct length scale for the bubble diameter at departure, but significant deviation of the bubble diameter at departure has been reported. Cole and Rohsenow (1969) proposed much better correlations for the bubble diameter at departure:

Forwater: ${D_b} = 1.5 \times {10^{ - 4}}\sqrt {\frac{{2\sigma }}{{g({\rho _\ell } - {\rho _v})}}} {\left( {{\rm{J}}{{\rm{a}}^*}} \right)^{\frac{5}{4}}} \qquad \qquad(2)$

For other fluid: ${D_b} = 4.65 \times {10^{ - 4}}\sqrt {\frac{{2\sigma }}{{g({\rho _\ell } - {\rho _v})}}} {\left( {{\rm{J}}{{\rm{a}}^*}} \right)^{\frac{5}{4}}} \qquad \qquad(3)$

Where ${\rm{J}}{{\rm{a}}^*} = \frac{{{\rho _\ell }{c_{p\ell }}{T_{sat}}}}{{{\rho _v}{h_{\ell v}}}} \qquad \qquad(4)$

At high-heat flux, the bubble departure diameter can be obtained from (Gorenflo et al., 1986) ${D_b} = {C_1}{\left( {\frac{{{\rm{J}}{{\rm{a}}^{\rm{4}}}h_{\ell v}^2}}{g}} \right)^{1/3}}{\left[ {1 + {{\left( {1 + \frac{{2\pi }}{{3{\rm{Ja}}}}} \right)}^{1/2}}} \right]^{4/3}} \qquad \qquad(5)$

where the constant C1 obtained by fitting experimental data for boiling of R-12, R-22, and propane are 14.7, 16.0, and 2.78, respectively. The bubble release frequency fb is related to the waiting time between detachment of the one bubble and initiation of the next bubble, tw, and the growth time, tg, of a bubble before its detachment. Attempts to obtain bubble release frequency by prediction of tw and tg were rarely successful because (a) these models generally did not consider evaporation from both the base and surface of bubbles; (b) cavity size variations significantly alters the growth time; (c) bubble activity, fluid flow, and heat transfer near a nucleation site can influence both bubble growth and waiting time; and (d) bubble shapes always change during growth (Kandlikar et al., 1999).

While the bubble release frequency depends distinctly upon the bubble diameter at departure, which differs from site to site, it is a constant for each individual nucleation site. It is recommended that the correlation have the form of ${f_b}D_b^n = {\rm{const}},$ where n is the exponent. Ivey (1967) suggested that the value of exponent n for inertia-controlled bubble growth is 2. When heat transfer controls bubble growth, the value of n is equal to 1/2. Malenkov (1971) proposed the most comprehensive correlation for the product of bubble release frequency and bubble diameter at departure: ${f_b}{D_b} = \frac{{{V_b}}}{{\pi \left( {1 - \frac{1}{{1 + {V_b}{\rho _v}{h_{\ell v}}/q''}}} \right)}} \qquad \qquad(6)$

where Vb is the bubble departure velocity and is calculated by ${V_b} = \sqrt {\frac{{{D_b}g({\rho _\ell } - {\rho _v})}}{{2({\rho _\ell } + {\rho _v})}} + \frac{{2\sigma }}{{{D_b}({\rho _\ell } + {\rho _v})}}} \qquad \qquad(7)$

Eastman (1984) developed an analysis related to dynamics of bubble departure that is presented here. During nucleate boiling, the forces that hold the bubble to the wall (negative) are larger than the forces that pull the bubble from the wall (positive) when the bubble is very small. As the bubble grows, the positive forces grow faster than the negative forces, until the total force becomes positive and pulls the bubble away from the surface. The negative forces are surface tension and drag force. The positive forces are buoyancy (for an upward facing surface), internal pressure, and inertia. These forces and directions are shown in the figure on the right. The five forces that act on a bubble during growth from a heated wall, are: internal pressure, surface tension, buoyancy, drag, and inertia. Analyses show that buoyancy is related to the gravitational acceleration and the inertial force is dependent on the deceleration of the bubble. The bubble departure can be determined by a force balance: ${F_d} + {F_s} = {F_i} + {F_p} + {F_B} \qquad \qquad(8)$

where Fi and Fp are liquid inertial force and pressure force, respectively. The surface tension force, Fs, is caused by the attraction of the liquid to the surface that acts around the perimeter of the bubble base. The surface tension force is proportional to the fluid surface tension, σ, and the contact angle, θ, i.e. ${F_s} = 2\pi {R_b}\sigma \sin \theta \qquad \qquad(9)$

where Rb is the base radius. The base contact angle goes to 90° as the bubble nears departure. When the bubble is growing in a viscous fluid, it will be subject to drag force. In realistic applications, this force is negligible for most fluids. A very rough estimate of this force was made by Keshock and Siegel (1962), and this approximation will be used here. In estimating drag force, it is also assumed that the bubble is spherical and growing away from the wall at a velocity equal to the change of its radius with time, i.e., dR / dt. The drag force can be calculated as: ${F_d} = {C_d}\frac{{{\rho _\ell }}}{2}{\left( {\frac{{dR}}{{dt}}} \right)^2}\pi {R^2} \qquad \qquad(10)$

where drag coefficient Cd can be calculated by using the following experimental correlation ${C_d} = \frac{{45}}{{{\mathop{\rm Re}} }} \qquad \qquad(11)$

where ${\mathop{\rm Re}} = \frac{{2{\rho _\ell }}}{{{\mu _\ell }}}R\frac{{dR}}{{dt}} \qquad \qquad(12)$

For a spherical bubble, which is submerged in a stagnant fluid, the buoyancy force is equal to the weight of the fluid displaced. So the buoyancy force is ${F_B} = \frac{{4\pi {R^3}}}{3}\left( {{\rho _\ell } - {\rho _v}} \right)g \qquad \qquad(13)$

Vapor inertia is negligible for most of the bubble growth; however, the inertia for the liquid surrounding the bubble needs to be accounted for. The liquid around the bubble is moved by the growth of the bubble. The affected mass of the fluid is that occupied by 11/16 of the bubble volume (Han and Griffith, 1965). The growth rate of the bubble decreases after the initial unbinding. The inertia of the liquid works against the decrease in velocity and tries to pull the vapor away from the surface. According to Newton’s second law, the inertial force can be approximated as follows, based on the above argument: ${F_i} = \frac{{11}}{6}\pi {R^3}{\rho _\ell }\frac{{{d^2}R}}{{d{t^2}}} \qquad \qquad(14)$

The pressure force results from the contribution of the dynamic excess vapor pressure and capillary pressure. It can be written as follows: ${F_p} = \left( {\frac{{2\sigma }}{R} + {p_v}} \right)\pi R_b^2 \qquad \qquad(15)$

where the vapor pressure is given below ${p_v} - {p_\ell } = \frac{{2\sigma }}{R} \qquad \qquad(16)$

The departure radius is assumed to be the radius at which the total force changed signs from negative to positive. Equations (9)-(16) were used to calculate the departure radius for saturated water for 1g and 0.229g, respectively. Analysis shows that buoyancy is directly related to the local gravitational acceleration. The inertial force is independent of gravity but dependent on the deceleration of the bubble.

## References

Cole, R., and Rohsenow, W.M, 1969, “Correlations for Bubble Departure Diameters for Boiling of Saturated Liquid,” Chemical Engineering Progress, Vol. 65, pp. 211-213.

Eastman, R.E., 1984, “Dynamics of Bubble Departure,” AIAA 1984 Thermophysics, AIAA-1984-1707, pp. 1-5.

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.

Fritz, W., 1935, “Maximum Volume of Vapor Bubbles,” Phys. Z. Vol. 36, pp. 379-384.

Gorenflo, D., Knabe, V., and Beiling, V., 1986, “Bubble Density on Surface with Nucleate Boiling – Its Influence on Heat Transfer and Burnout Heat Flux at Elevated Saturation Pressure,” Proceedings of the 8th International Heat Transfer Conference, Vol. 4, pp. 1995-2000, San Francisco, CA.

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.

Ivey, H.J., 1967, “Relationship Between Bubble Frequency, Departure Diameter and Rise Velocity in Nucleate Boiling,” International Journal of Heat and Mass Transfer, Vol. 10, pp. 1023-1040.

Kandilikar, S.G., Dhir, V.K., and Shoji, M., 1999, Handbook of Phase Change: Boiling and Condensation, Taylor and Francis, Philadelphia, PA.

Keshock, E.G., and Siegel, R., 1962, Forces Acting on Bubbles in Nucleate Boiling Under Normal and Reduced Gravity Conditions, NASA TN D-2299.

Malenkov, I.G., 1971, “The Frequency of Vapor Bubble Separation as Function of Bubble Size,” Fluid Mech. Sov. Res., Vol. 1, pp. 36-42.