# Minimum Heat Flux

The minimum heat flux point in the boiling curve is the boundary between the transition boiling regime and the film boiling regime. The minimum heat flux, q''min, is reached when the heat flux is equal to the minimum vapor formation rate that can sustain a stable vapor film over the heating surface. In controlled-heat flux pool boiling, nucleate boiling is reestablished when the heat flux falls below the minimum heat flux. The minimum heat flux is referred to as the Leidenfrost point. The vapor in the vapor film can be released by regular generation of vapor bubbles at some point and time. The minimum heat flux is related to the vapor bubble release rate, and can be estimated by (Zuber, 1959) ${q''_{\min }} = {e_b}{n''_b}{f_{\min }}\qquad \qquad(1)$

where eb represents the energy per vapor bubble, ${n_b}^{\prime \prime }$ is the number of vapor bubbles released per unit area and per release cycle, and fmin is the minimum number of cycles per second needed to compensate for normal collapse rate. The energy per bubble is assumed to be equal to the latent heat carried away by the bubble with a radius equal to λD / 4 and released at the node of the Taylor wave: ${e_b} = \frac{{4\pi }}{3}{\left( {\frac{{{\lambda _D}}}{4}} \right)^3}{\rho _v}{h_{\ell v}}\qquad \qquad(2)$

Zuber (1959) postulated that the vapor generated at the liquid-vapor interface was released at the node and antinode of a two-dimensional Taylor wave pattern, or once per half cycle, i.e., ${n''_b} = \frac{2}{{\lambda _D^2}}\qquad \qquad(3)$

The bubble release frequency, fmin, is ${f_{\min }} = {C_1}{\beta _{\max }}\qquad \qquad(4)$

where βmax is the frequency of the most rapidly growing disturbance obtained by an interfacial stability analysis: ${\beta _{\max }} = \left[ {\frac{{4{{({\rho _\ell } - {\rho _v})}^3}{g^3}}}{{27{{({\rho _\ell } + {\rho _v})}^3}\sigma }}} \right]\qquad \qquad(5)$

The minimum heat flux can be obtained by substituting eqs. (2) – (4) into eq. (1) and using ${\lambda _D} = 2\pi \sqrt {\frac{{3\sigma }}{{({\rho _\ell } - {\rho _v})g}}}$

from Critical Heat Flux to obtain λD, i.e., ${q''_{\min }} = C{\rho _v}{h_{\ell v}}{\left[ {\frac{{g\sigma ({\rho _\ell } - {\rho _v})}}{{{{({\rho _\ell } + {\rho _v})}^2}}}} \right]^{\frac{1}{4}}}\qquad \qquad(6)$

where C = C12 / 12)(4 / 3)3 is a new constant. Berenson (1961) recommended C = 0.09 by fitting the experimental data for pool boiling so that eq. (6) becomes ${q''_{\min }} = 0.09{\rho _v}{h_{\ell v}}{\left[ {\frac{{g\sigma ({\rho _\ell } - {\rho _v})}}{{{{({\rho _\ell } + {\rho _v})}^2}}}} \right]^{\frac{1}{4}}}\qquad \qquad(7)$

Equation (7) provides minimum heat flux for a horizontal surface; it is accurate within approximately 50% for most fluids at moderate pressure and is less accurate for higher pressure. Lienhard and Witte (1985) provided a much more detailed analysis on this subject.

## References

Berenson, P.J., 1961, “Film Boiling Heat Transfer from a Horizontal Surface,” ASME Journal of Heat Transfer, Vol. 83, pp. 351-356.

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.

Lienhard, J.H., and Witte, L.C., 1985, Rev. Chemical Engineering, Vol. 3, pp. 187-280.

Zuber, N., 1959, “Hydrodynamic Aspects of Boiling Heat Transfer,” USAEC Report AECU-4439.