# Heat Transfer in Nucleate Boiling

Heat transfer in nucleate boiling processes has been described through a variety of heat transport models, all of which have limitations of some sort because they were developed under ideal conditions. For nucleate boiling on an upward-facing horizontal surface in the isolated bubble region, heat transfer mechanisms include (1) transient heat conduction from the heating surface to the adjacent liquid, (2) natural convection on inactive areas of the heating surface, and (3) evaporation from the micro layer underneath the vapor bubble (Forster and Greif, 1959). The liquid adjacent to the heating surface becomes superheated, is pushed outward by the vapor bubble, and then mixes with the bulk liquid. An expression of the heat flux during partial nucleate boiling is (Kandlikar et al., 1999)

$q'' = \frac{{C_1^2}}{2}\sqrt {\pi {{(k\rho {c_p})}_\ell }{f_b}} D_b^2{N''_a}\Delta T + \left( {1 - \frac{{C_1^2}}{2}{N_a}\pi D_b^2} \right){\bar h_{nc}}\Delta T + {N''_a}\frac{\pi }{4}D_b^2{\bar h_{evp}}\Delta T \qquad \qquad(1)$

where the three terms on the right-hand side represent the contributions of transient conduction, natural convection, and micro layer evaporation. In order to use eq. (1) to calculate heat transfer, it is necessary to know the proportionality constant C1 for bubble area influence, the bubble diameter at departure Db, the bubble release frequency fb, the number density of the nucleate site N''a, and average heat transfer coefficient of natural convection and evaporation, ${\bar h_{nc}}$ and ${\bar h_{evp}}$.

Mikic and Rohsenow (1969) justified eq. (1) without the last term by using empirical correlations for several of these parameters. Mikic and Rohsenow’s (1969) correlation was recently revisited by Bergles (2005), who concluded that the Mikic-Rohsenow correlation is the ultimate pool-boiling model because all of the parameters can be obtained by various empirical correlations and only one experimental observation – the cavity size distribution. The problem is that it is impossible to get closer than one empirical constant since the nucleation site distribution is unknown a priori. The third term on the right-hand side of eq. (1) was introduced by Judd and Hwang (1976) to account for microlayer evaporation at the base of the bubble. The above nucleate boiling model considers only isolated bubbles. The analysis breaks down when bubbles coalesce and become slugs and columns. In addition, the above model includes expressions of bubble departure diameter Db and frequency fb, which are only correlations for most cases. There may be a high degree of uncertainty associated with the resulting correlations for heat transfer. Complete and reliable mathematical models for heat transfer in nucleate boiling have yet to be developed. Consequently, empirical correlations based on experimental data are widely used to predict heat transfer in nucleate boiling.

Jakob and Linke (1935) proposed one of the earliest and most popular models. Their model assumes that bubble growth and departure cause the convective heat transfer of liquid from the solid surface. This reasoning allowed Rohsenow (1952) to construct the correlation for nucleate boiling in the following single-phase forced convection type:

$Nu = \frac{1}{{{C_{s,\ell }}}}{{\mathop{\rm Re}} ^{1 - m}}\Pr _\ell ^{1 - n} \qquad \qquad(2)$

where ${C_{s,\ell }}$ is an empirical constant that accounts for the effect of different combinations of heating surface and liquid. The Nusselt number is defined as

${\rm{Nu}} = \frac{h}{{{k_\ell }}}{\left[ {\frac{\sigma }{{({\rho _\ell } - {\rho _v})g}}} \right]^{1/2}} = \frac{{q''}}{{{k_\ell }({T_w} - {T_{sat}})}}{\left[ {\frac{\sigma }{{({\rho _\ell } - {\rho _v})g}}} \right]^{1/2}} \qquad \qquad(3)$

where the bracket term at the right-hand side, together with its exponent (1/2) is the characteristic length.

The Reynolds number using the same characteristic length is

${\mathop{\rm Re}} = \left( {\frac{{q''}}{{{\rho _\ell }{h_{\ell v}}}}} \right){\left[ {\frac{\sigma }{{({\rho _\ell } - {\rho _v})g}}} \right]^{1/2}}\frac{1}{{{\nu _\ell }}} \qquad \qquad(4)$

where $q''/({\rho _\ell }{h_{\ell v}})$ is superficial velocity of the liquid. The heat flux in nucleate boiling, q'', is proportional to (TwTsat)3; therefore, the exponent m in eq. (2) is chosen to be 1/3. The exponent n is, however, dependent on the specific combination of the surface and the liquid. Substituting eqs. (3) and (4) into eq. (2), the following empirical correlation is obtained (Rohsenow, 1952):

$q'' = {\mu _\ell }{h_{\ell v}}{\left[ {\frac{{g({\rho _\ell } - {\rho _v})}}{\sigma }} \right]^{\frac{1}{2}}}{\left[ {\frac{{{c_{p\ell }}({T_w} - {T_{sat}})}}{{{C_{s,\ell }}{h_{\ell v}}\Pr _\ell ^n}}} \right]^3} \qquad \qquad(5)$

which applies to clean surfaces and is insensitive to the shape and orientation of the surface, which is obviously an approximation. It is still by far the most widely used empirical correlation for heat transfer in nucleate boiling. The coefficient ${C_{s,\ell }}$ and the exponent n depend on the liquid-surface combination, and their representative values are shown in the table below. It is necessary to point out that the error of heat flux estimated by eq. (5) can be as high as $\pm 100\% .$

If the heat flux is specified and it is necessary to estimate the temperature of the heated surface, one can rearrange eq. (5) to obtain

${T_w} - {T_{sat}} = \frac{{{h_{\ell v}}}}{{{c_{p\ell }}}}\Pr _\ell ^n{C_{s,\ell }}{\left[ {\frac{{q''}}{{{\mu _\ell }{h_{\ell v}}}}{{\left( {\frac{\sigma }{{g({\rho _\ell } - {\rho _v})}}} \right)}^{\frac{1}{2}}}} \right]^{\frac{1}{3}}} \qquad \qquad(6)$

under these circumstances, the error is within 25%.

Values of ${C_{s\ell }}$ and n for various combinations of surfaces and fluids for eqs. (5) and (6).
 Surface material Surface finish Fluid ${C_{s \ell}}$ n Brass Water 0.0060 1.0 Chromium Benzene 0.0101 1.7 Chromium Ethyl alcohol 0.0027 1.7 Copper Carbon tetrachloride 0.0130 1.7 Copper Isopropanol 0.0130 1.7 Copper Lapped n-Pentane 0.0049 1.7 Copper Polished n-Pentane 0.0154 1.7 Copper Polished Water 0.0130 1.0 Copper Scored Water 0.0068 1.0 Nickel Water 0.0060 1.0 Platinum Water 0.0130 1.0 Stainless steel Chemically etched Water 0.0130 1.0 Stainless steel Ground and polished Water 0.0060 1.0 Stainless steel Mechanically polished Water 0.0130 1.0 Stainless steel Teflon pitted Water 0.0058 1.0

## References

Bergles, A.E., 2005, “Bora Mikic and Pool Boiling,” Proceedings of the 2005 ASME Summer Heat Transfer Conference, San Francisco, CA.

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.

Forster, D.E., and Greif, R., 1959, “Heat Transfer to a Boiling Liquid – Mechanism and Correlation,” ASME Journal of Heat Transfer, Vol. 81, pp. 43-53.

Jakob, M., and Linke, W., 1935, “Heat Transmission in the Evaporation of Liquids at Vertical and Horizontal Surfaces,” Physik Z., Vol. 36, pp. 267-280.

Judd, R.L., and Hwang, K.S., 1976, “A Comprehensive Model for Nucleate Boiling Heat Transfer Including Microlayer Evaporation,” ASME Journal of Heat Transfer, Vol. 98, pp. 623-629.

Mikic, B.B., and Rohsenow, W.M., 1969, “A New Correlation of Pool-Boiling Data Including the Effect of Heating Surface Characteristics,” Journal of Heat Transfer, Vol. 91, pp. 245-250.

Rohsenow, W.M., 1952, “A Method for Correlating Heat-Transfer Data for Surface Boiling of Liquids,” Transactions of ASME, Vol. 74, pp. 969-976.