The *peak heat flux, maximum heat flux*, or *the burnout heat flux* represent the upper limit of fully developed Nucleate Boiling. In an engineering system, this heat flux also defines the limit for the safe operation of a component. After the occurrence of the maximum heat flux condition, heat removal from the heater surface degrades substantially. This, in turn, leads to a large increase in wall superheat, or melting, or burnout of the heater. Fully developed nucleate boiling of saturated liquids is generally associated with the formation of vapor columns and mushroom type bubbles on the heater surface. Vapor columns form as a result of the merger of bubbles normal to the heater surface, whereas mushroom type bubbles form as a result of merger of bubbles at neighboring sites. The mushroom type bubbles are believed to be attached to the heat surface through several stems.

According to Zuber (1958), the maximum heat flux condition occurs when vapor velocity in the large jets leaving the heater surface reaches a critical value. At this value of the vapor velocity, the jets in a countercurrent flow situation become unstable and cannot support additional outflow of vapor. This leads to the accumulation of vapor at the heater surface and a drastic reduction in the rate of heat transfer from the surface. The instability of the vapor jets occurs away from the heater surface and surface conditions do not affect the maximum rate of heat removal. By assuming that 1) all of the energy dissipated at the heater surface is used in phase change; 2) the vapor jets are located on a square grid with a spacing equal to a Taylor wavelength (critical or “most susceptible,” see Taylor Instability); and 3) the jet diameter is equal to half of the Taylor wavelength, Zuber obtained an expression for the maximum heat flux on an infinite horizontal plate as:

where ρ_{G} is the vapor density, ρ_{L} the liquid density, s the surface tension and h_{LG} the latent heat of vaporizaiton.

Equation (1) has been found to be quite successful in predicting maximum heat flux on well-wetted, large horizontal surfaces. The numerical constant in Eq. (1) resulted from averaging of the results for critical and most susceptible wavelengths. The theory supporting Equation (1) is now known as the *hydrodynamic theory of boiling*. Lienhard and Dhir (1973) have shown that Eq. (1) predicts the flat plate data better if a lead constant of 0.15, instead of π/24, is used. Also, Eq. (1) is applicable as long as the heater is at least 3 Taylor wavelengths wide. For smaller heaters observed, maximum heat fluxes can vary significantly from those given by Eq. (1).

The hydrodynamic theory has been developed further by Lienhard and co-workers (1973) to account for the geometry and finite size of the heaters. According to this extended hydrodynamic theory, maximum heat flux on finite heaters (cylinders, spheres, ribbons, etc.), can be written as:

where L' is the characteristic dimensionless length of the heater (e.g., radius of a cylinder or sphere, height of a ribbon, etc.) and is defined as Figure 1 shows the predictions of maximum heat flux for several geometries.

Hydrodynamic theory does not account for the effect of surface conditions on maximum heat flux. There is substantial evidence in the literature [e.g., Costello and Frea (1965) and Hasegawa et al. (1973)] that maximum heat flux is influenced by the degree of wettability of the heater. The recent work of Liaw and Dhir (1986), in which a static contact angle was used as an indicator of surface wettability, shows that maximum heat flux on a vertical surface decreases as the surface becomes less-wetting. Similar observations on horizontal surfaces have been reported by Maracy and Winterton (1988), and on horizontal cylinders by Hahne and Diesselhorst (1978). Figure 2 shows the maximum heat flux data as a function of contact angle. It is seen that maximum heat flux approaches that given by the hydrodynamic theory when the contact angle is less than 20°. Dhir and Liaw (1989) have inferred that for partially wetted surfaces, the rate of evaporation near the heater surface sets the upper limit of nucleate boiling heat flux, whereas for well wetted surfaces, this limit is probably set by the rate of vapor overflow.

The soundness of the assumption of instability of large vapor jets in hydrodynamic theory has been questioned in recent years by Haramura and Katto (1983) and several others, on the grounds that the vapor jets are too blunt to allow the development of the critical wave on the vapor-liquid interface. Haramura and Katto have suggested that instability of the vapor stems underneath the mushroom type bubbles determining the maximum heat flux. No verification of the latter assertion exists yet.

Several other variables, aside from those described above, influence the burnout heat flux and are discussed below.

Data from Bui and Dhir (1985) and Berenson (1962) show that maximum heat flux is generally higher on dirty surfaces. During boiling on a vertical surface placed in a pool of saturated water exposed to a laboratory environment for a long period of time, Bui and Dhir have noted that maximum heat flux on a surface contaminated with dust particles in water was about 25% higher than that on a clean surface. The cause of such an enhancement in maximum heat flux is not yet understood.

Several investigators [e.g., Houchin and Lienhard (1966) and Tachibana et al. (1969)] have found that on thin heaters, maximum heat flux occurs prematurely. It is postulated that hot spots under vapor bubbles or jets are unstable at high heat fluxes, and advance to increase the heater fraction that is dry. Maximum heat fluxes obtained under steady-state conditions on well-wetted surfaces show little effect of the heater material. Howerver, Lin and Westwater (1982) have found that maximum heat flux obtained during quenching is influenced by the heater material properties. They correlated the quasi-steady maximum heat flux data with the product of density, specific heat and thermal conductivity of the material.

The maximum heat flux initially increases with system pressure, attains its highest value between reduced pressures of 0.3 and 0.4, and thereafter, decreases with further increase in pressure. For well-wetted surfaces, Eq. 1, based on the hydrodynamic theory, yields predictions for the maximum heat flux that are in good agreement with the data obtained over a wide range of pressures.

According to Eq. (1), the maximum heat flux should vary with gravity as The high gravity centrifuge data confirm this dependence. However, the drop tower data of Siegel and Usiskin (1959) shows that the value of the exponent of gravity decreases as the magnitude of gravitational acceleration is reduced; at very low gravity, the maximum heat flux is independent of gravity. Recent low gravity data of Straub et al. (1992) shows that observed heat fluxes are significantly higher than those obtained from Eq. (1). At present, no rational basis exists to describe the observed behavior. For short duration microgravity conditions, many internal and external factors (e.g., thermocapillary forces, vibrations, etc.) can influence the process. (See also Microgravity Conditions.)

The maximum heat fluxes obtained during rapid heating of a surface are generally higher than those obtained under steady-state conditions. Studies of Sakurai and Shiotsu (1977) on platinum wires submerged in a pool of water show that for exponential heating periods of less than 100 ms, the maximum heat fluxes increase with a decrease in the exponential time period.

The maximum heat flux is found to increase with liquid subcooling. Zuber et al. (1961) have extended Eq. (1) to subcooled liquids by accounting for heat lost to the liquid in a transient manner. Their expression for maximum heat flux in subcooled liquid is:

where κ_{L} is the liquid thermal diffusivity and,

and C_{PL} is the specific heat of the liquid and ΔT_{sub} the degree of subcooling. From analysis, the constant, C_{1}, was found to have a value of 5.33.

According to Eq. (3), maximum heat flux increases linearly with subcooling. Elkassabgi and Lienhard (1988), from their experiments on horizontal cylinders, have found that only at low subcoolings the maximum heat flux increased linearly with subcooling. At very high subcoolings, the maximum heat flux attains an asymptotic value. The asymptotic value is related to molecular effusion limit. Separate correlations for different ranges of subcoolings have been developed.

Liquid velocity over heaters in the direction of gravity enhances maximum heat fluxes. However, for low velocities in the direction opposite to the gravity, the maximum heat flux can be lower than that obtained under pool boiling conditions. Correlations for maximum heat flux on discs cooled by impinging jets have been developed by Katto and Yokoya (1988), and on cylinders subjected to cross flow by Lienhard and co-workers (1988) (see also Boiling).

Finally, it should be stated that Eqs. (1) and (2) have generally been accepted for engineering applications involving pool boiling. However, their basis is still under debate.

#### REFERENCES

Berenson, P. J. (1962) Experiments on pool boiling heat transfer, *Int. J. Heat and Mass Transfer*, 5:985–999. DOI: 10.1016/0017-9310(62)90079-0

Bui, T. D. and Dhir, V. K. (1985) Transition boiling heat transfer on a vertical surface,*ASME J. Heat Transfer*, 107:756–763.

Costello, C. P. and Frea, W. J. (1965) A salient non-hydrodynamic effect in pool boiling burnout of small semi-cylinder heater, *Chem. Eng. Prog. Symp. Series*, 61:258–268.

Dhir, V. K. and Liaw, S. P. (1989) Framework for a unified model for nucleate and transition pool boiling, *ASME J. Heat Transfer*, 111:739–746.

Elkassabgi, Y. and Lienhard, J. H. (1988) The peak pool boiling heat flux from horizontal cylinders in subcooled liquids, *ASME J. Heat Transfer*, 110:479–486.

Hahne, E. and Diesselhorst, T. (1978) Hydrodynamic and surface effects on the peak heat flux in pool boiling, *Proc. 6th Int. Heat Transfer Conf.*, Toronto, Ontario, Canada, 1:209–219.

Haramura, Y. and Katto, Y. (1983) “A new hydrodynamic model of critical heat flux, applicable widely to both pool and forced convection boiling on submerged bodies in saturated liquids,” *Int. J. Heat and Mass Transfer*, 26:389–399. DOI: 10.1016/0017-9310(83)90043-1

Hasegawa, S., Echigo, R., and Takegawa, T. (1973) Maximum heat fluxes for pool boiling on partly ill-wettable heating surfaces, *Bull. JSME*, 96:1076–1084.

Houchin, W. R. and Lienhard, J. H. (1966) Boiling burnout in low thermal capacity heaters, ASME Paper No. 66-WA/HT-40.

Katto, Y. and Yokoya, S. (1988) Critical heat flux on a disk heater cooled by a circular jet impinging at the center, *Int. J. Heat and Mass Transfer*, 31:219–227. DOI: 10.1016/0017-9310(88)90003-8

Liaw, S. P. and Dhir, V. K. (1969) Effect of surface wettability on transition boiling heat transfer from a vertical surface, *Proc. 8th Int. Heat Transfer Conf.*, San Francisco, CA, 4:2031–2036.

Lienhard, J. H. and Dhir, V. K. (1973) Extended hydrodynamic theory of the peak and minimum pool boiling heat fluxes, NASA CR-2270.

Lienhard, J. H. (1988) Burnout on Cylinders, *ASME J. Heat Transfer*, 110:1271–1286.

Lin, D. Y. T. and Westwater, J. W. (1982) Effect of metal thermal properties on boiling curves obtained by the quenching method, *Proc. 7th Int. Heat Transfer Conf.*, Munich, Germany, 4:155–160.

Maracy, M. and Winterton, R. H. S. (1988) Hysteresis and contact angle effects on the peak heat flux in pool boiling of water, *Int. J. Heat and Mass Transfer*, 31:1443–1449. DOI: 10.1016/0017-9310(88)90253-0

Sakurai, A. and Shiotsu, M. (1977) Transient pool boiling heat transfer, Part 2. Boiling Heat Transfer and Burnout, *ASME J. Heat Transfer*, 99:554–560.

Siegel, R. and Usiskin, C. (1959) A photographic study of boiling in the absence of gravity, *ASME J. Heat Transfer*, 81:230–238.

Straub, J., Zell, M., and Vogel, B. (1992) Boiling under microgravity conditions, *Proc. 1st European Symposium for Fluids in Space*, Ajaccio, France, ESA SP-353, pp. 269–297.

Tachibana, F., Akiyama, M., and Kawamura, H. (1967) Non-hydrodynamic aspects of pool boiling burnout, *J. Nucl. Sci. and Tech.*, 4:121–130.

Zuber , N., Hydrodynamic aspects of boiling heat transfer, Ph.D. diss., University of California, Los Angeles, (also published as USAEC Report No. AECU-4439).

Zuber, M., Tribus, M., and Westwater, J. W. (1961) The hydrodynamic crisis in pool boiling of saturated liquids, *Proc. 2nd Int. Heat Transfer Conf.*, Denver, CO, Vol. 27.

#### References

- Berenson, P. J. (1962) Experiments on pool boiling heat transfer,
*Int. J. Heat and Mass Transfer*, 5:985â€“999. DOI: 10.1016/0017-9310(62)90079-0 - Bui, T. D. and Dhir, V. K. (1985) Transition boiling heat transfer on a vertical surface,
*ASME J. Heat Transfer*, 107:756â€“763. - Costello, C. P. and Frea, W. J. (1965) A salient non-hydrodynamic effect in pool boiling burnout of small semi-cylinder heater,
*Chem. Eng. Prog. Symp. Series*, 61:258â€“268. - Dhir, V. K. and Liaw, S. P. (1989) Framework for a unified model for nucleate and transition pool boiling,
*ASME J. Heat Transfer*, 111:739â€“746. - Elkassabgi, Y. and Lienhard, J. H. (1988) The peak pool boiling heat flux from horizontal cylinders in subcooled liquids,
*ASME J. Heat Transfer*, 110:479â€“486. - Hahne, E. and Diesselhorst, T. (1978) Hydrodynamic and surface effects on the peak heat flux in pool boiling,
*Proc. 6th Int. Heat Transfer Conf.*, Toronto, Ontario, Canada, 1:209â€“219. - Haramura, Y. and Katto, Y. (1983) â€œA new hydrodynamic model of critical heat flux, applicable widely to both pool and forced convection boiling on submerged bodies in saturated liquids,â€
*Int. J. Heat and Mass Transfer*, 26:389â€“399. DOI: 10.1016/0017-9310(83)90043-1 - Hasegawa, S., Echigo, R., and Takegawa, T. (1973) Maximum heat fluxes for pool boiling on partly ill-wettable heating surfaces,
*Bull. JSME*, 96:1076â€“1084. - Houchin, W. R. and Lienhard, J. H. (1966) Boiling burnout in low thermal capacity heaters, ASME Paper No. 66-WA/HT-40.
- Katto, Y. and Yokoya, S. (1988) Critical heat flux on a disk heater cooled by a circular jet impinging at the center,
*Int. J. Heat and Mass Transfer*, 31:219â€“227. DOI: 10.1016/0017-9310(88)90003-8 - Liaw, S. P. and Dhir, V. K. (1969) Effect of surface wettability on transition boiling heat transfer from a vertical surface,
*Proc. 8th Int. Heat Transfer Conf.*, San Francisco, CA, 4:2031â€“2036. - Lienhard, J. H. and Dhir, V. K. (1973) Extended hydrodynamic theory of the peak and minimum pool boiling heat fluxes, NASA CR-2270.
- Lienhard, J. H. (1988) Burnout on Cylinders,
*ASME J. Heat Transfer*, 110:1271â€“1286. - Lin, D. Y. T. and Westwater, J. W. (1982) Effect of metal thermal properties on boiling curves obtained by the quenching method,
*Proc. 7th Int. Heat Transfer Conf.*, Munich, Germany, 4:155â€“160. - Maracy, M. and Winterton, R. H. S. (1988) Hysteresis and contact angle effects on the peak heat flux in pool boiling of water,
*Int. J. Heat and Mass Transfer*, 31:1443â€“1449. DOI: 10.1016/0017-9310(88)90253-0 - Sakurai, A. and Shiotsu, M. (1977) Transient pool boiling heat transfer, Part 2. Boiling Heat Transfer and Burnout,
*ASME J. Heat Transfer*, 99:554â€“560. - Siegel, R. and Usiskin, C. (1959) A photographic study of boiling in the absence of gravity,
*ASME J. Heat Transfer*, 81:230â€“238. - Straub, J., Zell, M., and Vogel, B. (1992) Boiling under microgravity conditions,
*Proc. 1st European Symposium for Fluids in Space*, Ajaccio, France, ESA SP-353, pp. 269â€“297. - Tachibana, F., Akiyama, M., and Kawamura, H. (1967) Non-hydrodynamic aspects of pool boiling burnout,
*J. Nucl. Sci. and Tech.*, 4:121â€“130. DOI: 10.3327/jnst.4.121 - Zuber , N., Hydrodynamic aspects of boiling heat transfer, Ph.D. diss., University of California, Los Angeles, (also published as USAEC Report No. AECU-4439).
- Zuber, M., Tribus, M., and Westwater, J. W. (1961) The hydrodynamic crisis in pool boiling of saturated liquids,
*Proc. 2nd Int. Heat Transfer Conf.*, Denver, CO, Vol. 27.