Critical heat flux maxima during boiling crisis on textured surfaces

Enhancing the critical heat flux (CHF) of industrial boilers by surface texturing can lead to substantial energy savings and global reduction in greenhouse gas emissions, but fundamentally this phenomenon is not well understood. Prior studies on boiling crisis indicate that CHF monotonically increases with increasing texture density. Here we report on the existence of maxima in CHF enhancement at intermediate texture density using measurements on parametrically designed plain and nano-textured micropillar surfaces. Using high-speed optical and infrared imaging, we study the dynamics of dry spot heating and rewetting phenomena and reveal that the dry spot heating timescale is of the same order as that of the gravity and liquid imbibition-induced dry spot rewetting timescale. Based on these insights, we develop a coupled thermal-hydraulic model that relates CHF enhancement to rewetting of a hot dry spot on the boiling surface, thereby revealing the mechanism governing the hitherto unknown CHF enhancement maxima.


Supplementary Notes Supplementary Note 1 -Hydraulic and thermal characteristics of the boiling substrate
The configuration and dimensions of the boiling substrate used in the pool boiling CHF experiments are shown in Supplementary Fig. 3. The 650 µm-thick silicon sample is 5 cm x 5 cm square and has a 100 nm electrically insulating SiO 2 layer deposited on its backside. The 1 cm long and 2 cm wide portion of the 150 nm-thick Ti thin film patterned on the backside that lies between the two 300 nm-thick silver contact pads is the thin-film heater, which heats the substrate. This precisely defines a 1 cm x 2 cm boiling area on the top surface of the substrate in contact with the liquid. This size of the boiling area was chosen to be large enough to ensure boiling behavior similar to that on an infinitely large boiling surface. It should be mentioned that for heaters that are too small, CHF increases above the normal value. However, Gogonin and Kutateladze 1 and Theofanous et al. 2 have experimentally shown that the CHF of a flat heater is equal to that of an infinitely large boiling surface as long as the minimum heater dimension is larger than 2 s / Drg (~5 mm for water). Note that the maximum diameter of the critical dry spot in our analysis ( 2l~2 s / Drg ) is also equal to 5 mm, ensuring that the dry spot that leads to a CHF event is completely inside the heated area (1 cm x 2 cm).
Since only a portion (1 cm x 2 cm) of the 5 cm x 5 cm silicon substrate in contact with the liquid is heated, the accuracy of the pool boiling CHF measurements could possibly be affected by heat losses due to conduction away from the heated area.
However, here we show that due to the small substrate thickness ( t s ~ 0.6 mm) and because the heat transfer coefficients associated with pool boiling are much larger than those associated with liquid convection, the conduction losses are within the margin of error of CHF measurements and can be neglected. Supplementary Fig. 3b illustrates the heat flow modes in the substrate using a 2-D representation, where q in is the incoming heat flow rate per unit time (supplied by the Ti heater), q b is the heat flow dissipated in boiling, and q l is the heat lost by conduction away from the heated area. Since the mechanical supports at the edge of the sample are thermally insulating, the heat lost by conduction has to be dissipated via convective heat transfer to the liquid above. This 2-D 8 conjugate heat transfer problem can be solved by discretizing the solid domain along the x-direction ( Dx = 1 mm), assuming 1-D conduction along x, and uniform liquid convective heat loss out of the discretized element. The problem can now be represented using the thermal resistance circuit shown in Supplementary Fig. 3c, where the unit cell corresponding to conduction/convection in the discretized domain unit is repeated to model the entire domain. The unit cell conduction resistance (per unit width out of plane) is given by h v is the heat transfer coefficient for liquid natural convection. The following correlation 3 can be used to evaluate h v : Using water properties at 100°C and Using the method of series and parallel resistances, the repeating array of conduction/convection thermal resistance unit cell can be combined to get the net thermal loss resistance R l ( Supplementary Fig. 3d): where M is the number of discretized domain units. Using an average unheated substrate length of L l = (20 +15) / 2 = 17.5 mm (due to asymmetric heater dimensions), we get M = L l / Dx ~ 17. This yields a thermal loss resistance of R The boiling thermal resistance is given by , a conservative value of h b ~ 5´10 4 W m −2 K −1 can be assumed, which yields a boiling thermal resistance of R b = 0.0027 K m W −1 . The fractional heat loss by conduction can now be calculated using q Note that this is a conservative estimate given that parts of the 9 substrate are covered by thermally insulating Teflon clamps, and because we calculated = 100 W cm −2 (which yields the lowest value of h b ). This means that the conductive heat loss should lead to a CHF measurement error of < 2.5 W cm −2 for all the samples regardless of the measured CHF value, and further the error should be systematic (of the same sign and approximately the same value). For these reasons, and because the error is well within the inherent CHF data acquisition error of 10 W cm −2 , we can ignore these losses in our analysis.

Supplementary Note 2 -Experimental uncertainty analysis
The objective of the CHF experiments is to measure the changes in CHF on a textured surface due to the change in spacing b between the plain and nanograss micropillars that give the surface its texture. Since the CHF is obtained by gradually increasing the applied heat flux in maximum 10 W cm −2 increments, the uncertainty in CHF measurement should be approximately 10 W cm −2 . However, it should also be kept in mind that CHF measurements are not exactly repeatable due to the inherent statistical nature of the boiling crisis phenomenon and its dependence on experimental conditions such as sample cleanliness, water purity etc. We therefore measured the CHF values for two nominally similar samples for each type of textured surface studied (see Table 1). The uncertainty in CHF (vertical error bars in Figs. 2, 7, and 8) is taken to be equal to the difference of these two measurements or 10 W cm −2 , whichever is greater.
Although the spacing between the micropillars b can be determined quite accurately using SEM imaging, it varied slightly between the two nominally similar Rahman et al. 4 have proposed a correlation to explain surface texture-induced CHF enhancement based purely on wicking of liquid into the surface textures. The CHF of a wicking surface is given by where q CHF, Zuber '' is the expression for CHF obtained using conventional hydrodynamic theory, first proposed by Zuber 5 , that does not account for surface effects: (4) The wicking number Wi is given by (5) where the wicked volume flux is measured by tracking the flow of liquid from a capillary tube onto the textured surface: Here A w is the initial wetting area of the liquid from the capillary tube that wets the textured surface, and where m is the dynamic viscosity and v the radial velocity of the liquid, and K v is the permeability of the texture (see equation (2)). Using conservation of mass, the velocity v at radius r can be expressed in terms of the velocity v i of the liquid imbibition front at Since the total viscous pressure drop DP v has to be equal to the capillary pressure P c (given by equation (1)) at the imbibition front and because the imbibition front velocity can be expressed as v i = dR / dt , we can write this as Integrating this equation, we can calculate the time Dt required for the liquid imbibition front to travel from The volume of the liquid imbibed into the texture is given by and neglecting wetting of the micropillar tops, we can calculate the net change in system energy by accounting for the loss of the existing solid-vapor interface ( To develop a scaling expression for permeability of surface with a square array of square micropillars, we consider two extreme scenarios.   ) ]. This yields a surface texture permeability of

Supplementary Note 5 -Extrapolation of silicone oil imbibition results to water
To verify the scaling imbibition model and provide experimental validation of the theoretically calculated dry spot rewetting timescales, imbibition experiments were conducted using silicon micro-textured surface samples. Although DI water was used to obtain the pool boiling CHF data, it was found to be unsuitable for performing the imbibition experiments due to its relatively high vapor pressure and low liquid viscosity ( m) at room temperature. Under the intense lighting conditions used for the high-speed