Critical Heat Flux

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In nucleate boiling, heat flux increases and reaches a maximum value with increasing surface temperature. Further increase in the surface temperature results in decreasing heat flux because the transition from nucleate boiling to film boiling takes place. The maximum heat flux that can be obtained by nucleate boiling is referred to as critical heat flux (CHF). In the case of controlled heat flux, a slight increase of heat flux beyond the CHF can cause the surface temperature to rise to a value exceeding the surface material’s maximum allowable temperature. This in turn can cause severe damage or meltdown of the surface. The CHF is also referred to as burnout heat flux for this reason. The value of the critical heat flux is affected by hydrodynamic instabilities, wetting criteria, heat capacity, heater geometry, heating method, and pressure.

The CHF phenomenon has received considerable attention in the past, and different mechanisms have been proposed to interpret its cause. Heat transfer in nucleate boiling involves (1) heat conduction through liquid from the heating surface to the liquid-vapor interface, (2) evaporation at the liquid-vapor interface, and (3) the escape of vapor from the heating surface. The critical heat flux phenomena occur when either generation or escape of vapor is restricted. Based on the assumption that CHF proceeds from the vapor escape limit, Zuber (1959) proposed one of the earliest successful models. This model was later refined by Lienhard and Dhir (1973). In the late stage of nucleate boiling, vapor bubbles join together to form vapor columns and slugs. The vapor columns and slugs are subject to the Kelvin-Helmholtz instability, which occurs when small disturbances in the interface between counter-flowing liquid and vapor are amplified. As a result, the columns and slugs become distorted and block the liquid from wetting the heating surface. A vapor blanket forms over part of or the entire heating surface, and separates the heating surface and the liquid.

Zuber (1959) postulated that the CHF is reached when the Helmholtz instability appears in the interface of the large vapor columns leaving the heating surface. These vapor columns are assumed to be in a square array. To determine the centerline spacing of the vapor columns, Lienhard and Dhir (1973) used the most dangerous wavelength in the two-dimensional wave pattern for the Taylor instability of the interface between a horizontal semi-infinite liquid region above a layer of vapor, $λ D$ . The diameter of each column is assumed to be $λ D / 2.$

For vertical liquid and vapor flow, the critical Helmholtz velocity is (see Kelvin-Helmholtz instability)

${u_c} = \left| {{{\bar u}_\ell } - {{\bar u}_v}} \right| = \sqrt {\frac{{\sigma \alpha ({\rho _\ell } + {\rho _v})}}{{{\rho _\ell }{\rho _v}}}} \qquad \qquad(1)$

where the wave number $α$ and the wavelength $λ D$ have the following relationship:

$\alpha = \frac{{2\pi }}{{{\lambda _D}}} \qquad \qquad(2)$

Substituting eq. (2) into eq. (1) and considering ${\rho _\ell } \gg {\rho _v}$ , the critical Helmholtz velocity becomes

${u_c} = \sqrt {\frac{{2\pi \sigma }}{{{\rho _v}{\lambda _D}}}} \qquad \qquad(3)$

Since the density of the vapor is significantly lower than that of the liquid, the upward vapor column velocity is much higher than the downward liquid velocity. Therefore, the critical Helmholtz velocity can be approximated as

${u_c} = {u_v} = \frac{{{{q''}_{\max }}}}{{{\rho _v}{h_{\ell v}}}}\frac{{{A_s}}}{{{A_c}}} \qquad \qquad(4)$

where $A s / A c$ is the ratio of the total surface area to the area of the surface occupied by the vapor column. Since the spacing of the centerline of the vapor column is $λ D$ and the diameter of the vapor column is $λ D / 2,$ the area ratio is

$\frac{{{A_s}}}{{{A_c}}} = \frac{{{\lambda _D}^2}}{{\pi {{({\lambda _D}/2)}^2}/4}} = \frac{{16}}{\pi } \qquad \qquad(5)$

Substituting eq. (5) into eq. (4), the critical Helmholtz velocity becomes

${u_c} = \frac{{16{{q''}_{\max }}}}{{\pi {\rho _v}{h_{\ell v}}}} \qquad \qquad(6)$

Combining eqs. (3) and (6), the critical heat flux is obtained:

${q''_{\max }} = \frac{{\pi {\rho _v}{h_{\ell v}}}}{{16}}\sqrt {\frac{{2\pi \sigma }}{{{\rho _v}{\lambda _D}}}} \qquad \qquad(7)$

The most dangerous wavelength, $λ D$ , as obtained by the Raleigh-Taylor instability analysis, was given by eq. (5.219), i.e.,

${\lambda _D} = 2\pi \sqrt {\frac{{3\sigma }}{{({\rho _\ell } - {\rho _v})g}}} \qquad \qquad(8)$

Substituting eq. (8) into eq. (7), the critical heat flux becomes

${q''_{\max }} = 0.149{\rho _v}{h_{\ell v}}{\left[ {\frac{{\sigma g({\rho _\ell } - {\rho _v})}}{{\rho _v^2}}} \right]^{\frac{1}{4}}} \qquad \qquad(9)$

Equation (9), which was obtained by Lienhard and Dhir (1973), is identical to the result of Zuber’s (1959) original model except that the constant on the right-hand side of the latter correlation is $π / 24 = 0.131.$ Kutateladze (1948) obtained the critical heat flux by using a dimensional analysis based on the similarity between critical heat flux and column flooding as described by the Helmholtz instability. He also suggested that the constant should be 0.131. However, Lienhard and Dhir (1973) showed that the constant of 0.149 could provide better agreement with the available experimental data. The critical heat fluxes predicted by eq. (9) reasonably agreed with the experimental data for water, hydrocarbons, cryogenic liquids, and halogenated refrigerants. The experimentally measured CHF for the boiling of liquid metal, on the other hand, is two to four times higher than that predicted by eq. (9). This discrepancy is due to a strong conduction-convection mechanism of liquid metal that the above model did not account for. Equation (9) indicates that the critical heat flux is proportional to $g 1 / 4$ , which means the critical heat flux will be reduced under microgravity conditions. Equation (9) is applicable to saturated boiling. The critical heat flux increases with subcooling of the liquid according to Zuber et al. (1961) by the following correlation

$\frac{{{{q''}_{\max }}}}{{{{q''}_{\max ,sat}}}} = 1 + \frac{{5.3({T_{sat}} - {T_\ell })}}{{{\rho _v}{h_{\ell v}}}}{\left( {{k_\ell }{\rho _\ell }{c_{p\ell }}} \right)^{1/2}}{\left[ {\frac{{\sigma ({\rho _\ell } - {\rho _v})g}}{{\rho _v^2}}} \right]^{ - 1/8}}{\left[ {\frac{{({\rho _\ell } - {\rho _v})g}}{\sigma }} \right]^{1/4}} \qquad \qquad(10)$

Equation (9) is applicable to an infinite horizontal heater surface, since there is no characteristic length in the expression. In order to use eq. (9), which was obtained by assuming equality in spacing between vapor columns and the most dangerous wavelength, $λ D$ , the length scale of the heater surface, $L$ , must be greater than $2λ D$ . Considering eq. (8), this criterion can be written in terms of confinement number:

${\rm{Co}} = \frac{{\sqrt {\sigma /[({\rho _\ell } - {\rho _v})g]} }}{L} \le \frac{1}{{4\pi \sqrt 3 }} \qquad \qquad(11)$

An extensive summary of the critical heat fluxes for various geometric configurations can be found in Lienhard and Dhir (1973) and Lienhard and Hasan (1979). There has been some debate regarding how surface conditions affect the value for critical heat flux. Some experimental results (in accordance with Zuber’s theory) show that there is no relationship between the two. To be more specific, surface-roughening techniques such as scoring and sandblasting have been found not to greatly influence the critical heat flux. In contrast, some experiments have shown that certain surface conditions (such as oxidation and deposition) do seem to increase the value of critical heat flux by increasing wettability of the fluid. The critical heat flux is significantly reduced on essentially nonwetting surfaces. In general, if the surface condition can directly increase the wettability of the fluid, then the value of the critical heat flux may be increased. In other circumstances, no general conclusion will apply in every instance.

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.

Kutateladze, S.S., 1948, “On the Transition to Film Boiling under Natural Convection,” Kotloturbostroenie, No. 3, pp. 10-12.

Lienhard, J.H., and Dhir, V.K., 1973, Extended Hydrodynamic Theory of the Peak and Minimum Heat Fluxes, NASA CR-2270.

Lienhard, J.H., and Hasan, M.Z., 1979, “On Predicting Boiling Burnout with the Mechanical Energy Stability Criterion,” ASME Journal of Heat Transfer, Vol. 101, pp. 276-279.

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

Zuber, N., Tribus, M., and Westwater, J.W., 1961, “The Hydrodynamic Crisis in Pool Boiling of Saturated and Subcooled Liquids,” International Development in Heat Transfer: Proceedings of 1961-62 International Heat Transfer Conference, Boulder, CO, pp. 230–236.

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Critical Heat Flux

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