Is a crisis in pool boiling actually a hydrodynamic phenomenon?

International Journal of Heat and Mass Transfer 73 (2014) 265–273

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Is a crisis in pool boiling actually a hydrodynamic phenomenon? Victor V. Yagov ⇑ Department of Engineering Thermophysics, National Research University Moscow Power Engineering Institute, Krasnokazarmennaya Street, 14, 111250 Moscow, Russia

a r t i c l e

i n f o

Article history: Received 21 September 2013 Received in revised form 24 January 2014 Accepted 29 January 2014 Available online 4 March 2014 Keywords: Critical vapor velocity Vapor film instability Liquid macrofilm Dry spots Interline Reduced pressure

a b s t r a c t Now many experimental facts are revealed, which contradict to the hydrodynamic theory of pool boiling crisis. Nevertheless, the majority of the latest modifications of boiling crisis model are based on the hydrodynamic approach; this makes actual the question submitted in the title of the paper. The wellknown Kutateladze correlation for CHF was obtained on basis of dimensions analysis. This factually predetermines that any crisis model, which considers only hydrodynamic effects ignoring influence of liquid viscosity and heat transfer at the heated surface, inevitably leads to the Kutateladze equation with small corrections in the form of functions of liquid/vapor densities ratio. This is obviously seen from analysis of all theoretical models beginning from Zuber and till the newest ones. If in relation to the Kutateladze approach the main objection is the convincingly established fact that the limiting vapor velocity does not determine the crisis origin in the real boiling process and at low reduced pressures an actual vapor velocity can exceed the critical one many times, then in the theoretical studies always assumptions are found, which contradict to either experimental measurements or to some scientific fundamentals. However, at moderate and high reduced pressures the equation of the hydrodynamic model agrees satisfactorily with the data. Consequently, now it is necessary not only to develop a new approach to pool boiling crisis, but also to explain rather good predicting capability of the Kutateladze–Zuber equation. The model developed by the present author presents an attempt to exceed the limits of the hydrodynamic approach. The crisis is a result of enlarging area of dry spots which are an intrinsic feature of nucleate boiling. Great difference of vapor specific volume at high and low reduced pressures makes it reasonable to derive separately the equations for CHF for these two cases. A simple interpolating formula allows calculating CHF at arbitrary pressure. The equation for high reduced pressures gives the calculated CHF values close to those computed on the formulas of the hydrodynamic theory. As the most part of the experimental data are obtained at moderate and high reduced pressures, this coincidence gives a possible explanation of agreement of the Kutateladze’s formula with the data. A weak variation of liquid kinematic viscosity at saturation line allows concealing its actual influence on CHF. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Actuality of the question submitted in the title of the paper is kept up today because the majority of the latest modifications of boiling crisis model are based on the hydrodynamic approach. This seems to be surprising if one bears in mind a rather long list of experimental facts which clearly contradict to the hydrodynamic model of pool boiling crisis. The present paper gives an analysis of both the limits of the hydrodynamic theory, its contradictions and of probable reasons of its durability in spite of many evident mismatches. After publication in 1934 the pioneering work by Nukiayma (see [1]) it became clear that pool boiling crisis means a ⇑ Tel.:+7 495 362 7501, +7 903 132 1624. E-mail addresses: [email protected], [email protected] http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.01.076 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.

catastrophic change of heat transfer mechanism. In atmospheric water boiling at critical heat flux (CHF) the wall superheat increases from 24–30 K at nucleate boiling up to 1200–1400 K at film boiling. For ordinary metallic heaters this leads to their burnout; the term burnout has been using for boiling crisis designation sometimes till now. As equipment exploiting boiling of liquids fails its normal operating beyond crisis, regularities of this phenomenon are investigated carefully beginning from the 1940th both experimentally and theoretically. However, at present general understanding of the main mechanisms of nucleate boiling heat transfer and crisis is not achieved. Undoubtedly, among different models of pool boiling crisis a hydrodynamic model suggested by Kutateladze [2] and theoretically derived by Zuber [3] is mostly popular; practically all handbooks and textbooks expound this theory and give a simple predicting equation followed from it. Nevertheless, even during

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Nomenclature b C0,C1,. . .C4 cp Dd dj g H hLG k l p Pr q R Ri r T t W

capillary (Laplace) constant, m numerical factors specific heat, J/(kg K) vapor bubble departure diameter, m vapor jet diameter, m gravitational acceleration, m/s2 curvature, m1 latent heat of evaporation, J/kg dimensionless factor (stability criterion) characteristic distance, m pressure, Pa Prandtl number heat flux density, W/m2 vapor bubble radius, m gas constant (individual), J/(kg K) radial coordinate, m temperature, K time, s characteristic velocity, m/s

k thermal conductivity, W/(m K) k⁄, k⁄⁄, kH characteristic wavelengths, m l dynamic viscosity, Pa  s m kinematic viscosity, m2/s q density, kg/m3 r surface tension, J/m2 Superscript ⁄ reference value Subscripts 0 cr cr.h cr.l d ds G H L s w Z ⁄

Greek symbols D thickness of a conductive liquid layer, m d0 liquid macrofilm thickness, m DT ¼ T W  T S wall superheat, K e void fraction

the 1960–70th when acknowledgment of the hydrodynamic theory was almost universal some experimental facts and internal contradictions of the theory forced to doubt in its consistency. Now it is possible to estimate a ‘‘mature skepticism’’ [4] about the hydrodynamic theory of crisis as a predominant view. The present paper has an objective to explain (partly, at least) an obvious contradiction between this general opinion and vast practical using the equation based on the hydrodynamic theory. 2. A short history 2.1. Kutateladze’s model Hydrodynamic (‘‘hydromechanical’’) model of boiling crisis has been proposed first by Kutateladze in 1950 [2]. The model is based on four assumptions; the following two of them are the most principal ones: - change of boiling modes occurs due to variation of hydrodynamics of two-phase boundary layer and is characterized with a specific critical value of velocity of liquid evaporation (i); - due to high turbulization of the boundary layer viscosity both of liquid and vapor does not influence on the process (ii). Then dimensionless numbers have been obtained on basis of momentum equations for liquid and for vapor and compatibility condition for normal momentum components at the interface. An additional assumption on infinite extent of the heated surface, i.e. on absence of any definite linear scale in the problem allows determining a general expression for a characteristic velocity: 1=2

qG W G p ffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ const; 4 rg Dq

ð1Þ

where Dq = qL  qG. Dimensionless number defined with the above Eq. (1) is the Kutateladze number Ku. In boiling WG is an average velocity of evaporation, WG = q/ (qGhLG). Consequently, the well-known Kutateladze correlation for CHF follows from Eq. (1):

characteristic value critical critical at high pressures critical at low pressures departure dry spot gas (vapor) Helmholtz liquid saturation wall Zuber equilibrium

qcr ¼ khLG qG1=2 ðrg DqÞ1=4 :

ð2Þ

Constant k termed as ‘‘stability criterion’’ (or the Kutateladze number) was initially determined for the best fitting the available experimental data as k = 0.16. Later the value k = 0.14 is commonly used (for example, [5,6]). In paper [2] the author pointed out that his theory is only the first approximation and the further investigations are necessary, in particular, in relation of the heated surface properties influence on boiling crisis. But rather good agreement between calculation and the data including the measurements at high intensity of body forces (from terrestrial g = g0 up to g = 2500 g0), which confirmed the dependence qcr  g1/4, made the Kutateladze model the most recognized one. Some experimental results, however, caused the doubts on the hydrodynamic model consistency. First of all influence of surface wettability on CHF cannot be explained in the frames of the model. Dramatic decrease of CHF value in boiling at nonwettable surfaces does not disprove the model, because nonwettability can be considered as additional effect caused the change of the boiling modes before the hydrodynamic factors turned out them. The experiments on boiling at the surfaces with porous coating created for the hydrodynamic approach more difficulties as the measured critical heat fluxes were essentially higher than predicted according to Eq. (2). This means that liquid can penetrate to the heated surface even at vapor velocities higher than critical one. It should be noted that the majority of the experimental results on boiling crisis at the enhanced surfaces were published later, in the 1960–80th. As is clear, Eq. (2) has been obtained on basis of dimensions analysis with accuracy to a numerical factor k; any detailed quantitative model was not developed. This means that up to 1958 while Zuber has not presented his theory [3], discussions on the hydrodynamic model of boiling crisis can be only qualitative ones (‘‘to believe’’–‘‘not to believe’’). Rather surprising feature of the model is that it is not immediately connected with the preceding process of nucleate boiling. In [5] the author emphasized this purposely. Although this gives rise to the certain doubts, in absence of a thoroughly developed model

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of the process, both the opponents and proponents of the theory can only rely on the extent of the agreement between the theory and the experimental data. 2.2. Zuber’s theory Zuber [3] built up a detailed model and in frames of the model gave strict theoretical basis for his predicting equation. From historical view it is noteworthy a remark by Lienhard [4] on connection of the Zuber’s theory and the Kutateladze’s model. According to Lienhard Zuber was fluent in Russian as nobody else among American specialists in heat transfer; he could appreciate the Kutateladze model, which has already been recognized in Russia, but does not attract attention in the West before Zuber. Zuber considered already existing vapor film on an upward horizontal heated surface. The film is unstable in Taylor’s sense; characteristic wavelengths (critical and the most dangerous) of this instability are the following ones:

k ¼ 2pb

ð3Þ

and

pffiffiffi k ¼ 2pb 3;

ð3aÞ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ½b ¼ r=ðg DqÞ is capillary (Laplace) constant. The Taylor’s instability wavelength determines the size lZ of the square lattice at whose nodes vapor jets are formed (Fig. 1). Assuming the jet diameter to be equal dj = lz/2 and the ultimate (critical) vapor velocity to be determined by the Helmholtz instability condition

W G;cr ¼ W H ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pr=ðqG kH Þ;

ð4Þ

Zuber has obtained the following correlation:

qcr ¼ W H hLG qG ðp=16Þ:

ð5Þ

The last multiplier in (5) is the ratio of the area of the vapor jets to the total area of the heated surface that corresponds to the assumption on a jet diameter. The wavelength of the Helmholtz instability was assumed to be

kH ¼ pdj ¼ plz =2:

ð6Þ

Substitution of this value into (4) gives for the two different values of the Taylor’s instability wavelength according to (3) and (3a) correspondingly:

WH ¼

rffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 rg Dq

p

q2G

with lz ¼ k ;

sffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 r g Dq pffiffiffi with lz ¼ k : WH ¼ p 3 q2G

ð7Þ

Turning to the Kutateladze analysis one can see that these two equations theoretically specify a value of the constant in Eq. (1).

Fig. 1. Vapor column spacing in the Zuber’s (1958) hydrodynamic CHF model.

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Now, using Eqs. (7) in (5) one obtains finally:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=4 qcr ¼ k ðqL þ qG Þ=qL hLG q1=2 ; G ðrg DqÞ

ð8Þ

pffiffiffiffiffiffiffi pffiffiffi 2p=ð16 4 3Þ ffi 0; 119 and where the constant k lies between ffiffiffiffiffiffiffi p 2p=16 ffi 0; 157 that is in the range determined by the difference between the critical wavelength and the most dangerous wavelength of the Taylor instability. It is clear that Eq. (8) is no more than a theoretical substantiation of the Kutateladze’s correlation (2), because the multiplier pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðqG þ qL Þ=qL notably differs from unity only at the close vicinity of the critical point of a substance. As Eqs. (2) and (8) practically coincide in some publications one can meet terms ‘‘Kutateladze– Zuber theory’’ or ‘‘Kutateladze–Zuber equation’’. Certainly, publication [3] made the hydrodynamic theory proposed by Kutateladze well known among specialists abroad Russia, who commonly seldom cite Russian studies, probably, due to their lack of knowledge of the Russian language. At the same time Kutateladze regarded rather critically to the new theory. In [6] he noted that the Zuber’s scheme is far from really observed picture: according to the scheme the near-wall void fraction at the boiling crisis incipience is ecr ffi p/16, while the measurements in bubbling gave ecr ffi p/4. 2.3. Contradictions and limitations of the hydrodynamic theory Kutateladze was not a single scientist who has found the contradictions in the Zuber’s theory, but his criticism objectively is directed opposite his own approach. The detailed theoretical analysis by Zuber allows revealing many weak or doubtful points of the hydrodynamic theory of boiling crisis. This is indicative that one of the initial apologists of the theory, who, in particular, developed it for heated surfaces of the limited size, Lienhard, listed 7 questionable problems in the Zuber’s analysis [4]. The main objections of the American specialists concerned mainly the heated surface properties which were not accounted for by the Zuber’s theory; in particular, the paper [7] has presented the experimental data with CHF values 2–4 times higher than calculated ones according to Eq. (8). But the most significant and general objection was formulated by Labuntsov [8] immediately when the Zuber’s paper became available for a Russian reader. Labuntsov has stated that the theory considers factually a stability of vapor film already existed on the heated surface, but not a transition from nucleate to film boiling. This means that the analysis content does not correspond practically to its claimed aim. Labuntsov remarked that the Kutateladze formula, probably, also relates to existed film boiling, not to the nucleate/film boiling transition. At present there are enough convincing experimental evidences of inconsistency of the hydrodynamic theory of boiling crisis. First of all visual observations and direct measurements of void fraction above the heated surface at high heat fluxes confirmed the above Kutateladze notion on disagreement of the Zuber’s scheme and the real picture. Gaertner [9] has presented very important information on a structure of two-phase region in the vicinity of the heated wall; factually his paper gave the first detailed description of a liquid macrofilm on the wall which kept underneath large mushroom-shaped vapor bubbles up to the crisis. However, on an ironic notion of Bergles [10] in motion pictures and photo frames ‘‘one sees what one wants to see to support a particular mechanistic model’’, so that the Gaertner’s work became a classical one as it allows to support any of three models of CHF (based correspondingly on the hydrodynamic instability, on the liquid macrolayer evaporation and on vapor bubbles coalescence). In this respect the measurements of void fraction of two-phase structure above the wall [11–13] are more convincing. They showed that at a distance of 0.3–0.5 mm from the heated surface the void fractions

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are in the range of e ¼ 0.85–0.92. It is apparent that under these conditions it is impossible to speak about vapor jets, but more realistic interpretation is a vapor continuum with liquid drops. This is well known that Eq. (1) for critical vapor velocity determines also a terminal velocity of a falling liquid drop with a size of an order of magnitude of the capillary constant; in this case the const value in the discussed equation is approximately equal to 1.5. Probably, the hydrodynamic limit of CHF has to be determined by this velocity rather than by the Helmholtz instability velocity (7). As there is no limitation for vapor jet area, in this case the CHF value must be an order of magnitude higher than according to Eqs (8), (2). These arguments were presented first by the present author [14] and later in [15,16]. In [17] the hydrodynamics of near-wall layer was studied when blowing air into water through a fine screen. It was shown that substantial excess of air injection velocity above the critical value determined by the hydrodynamic model does not lead to any changes in structure of two-phase boundary layer on a flat surface in water either at horizontal or vertical its orientation. Visualization of the process revealed that large gas bubbles similar to those which were described by Gaertner [9] formed at the surface rather than jets. A conclusion of [17] is that the critical heat flux in pool boiling is not, apparently, determined by hydrodynamic instability or some other limiting condition concerning the vapor evacuation. In the experiments of [15] critical heat flux in water pool boiling at the same horizontal surface varied two times due to the surface aging. Basing on their experiments revealed many important features of vapor–liquid–solid interaction in boiling the authors of [15,16] made a conclusion on absence of hydrodynamic limit in nucleate boiling. This statement seems to be excessively categorical one, if to bear in mind the experimental results on boiling in channels with closed bottom [18,19] and the obvious hydrodynamic effect of body forces on CHF mentioned above. The Kutateladze’s hydrodynamic model in its general formulation, without specification of the const k value can actually determine the limiting heat flux density at the wall when the other factors do not provoke the crisis much earlier. Such conditions arise when (at a distance from the wall sufficient in order to neglect the viscosity influence) the critical velocity of the relative motion of the liquid and the vapor is reached. In this case according to [18,19] the critical velocity of vapor can be determined by Eq (1) with const 2.3–2.4; this means that the hydrodynamic limit gives CHF values 10–15 times higher than calculated ones according to Eqs (2), (8). Apparently, liquid cannot wet the heated surface if an average velocity of vapor exceeds the falling velocity of the liquid drops; the above estimation of the latter velocity is rather close to the critical vapor velocity measured in [18,19]. Thus, the experimental studies conducted after publication of the basic papers have shown that the hydrodynamic theory of boiling crisis is not confirmed not only in relation to the detailed scheme [3] of the two-phase region near the heated wall, but also in relation to the value of the critical velocity of vapor [2,5]. As early as in 1960th it was established that the theory failed in describing the experimental data on CHF at alkali metals boiling [20,21]. As the measured value of CHF are several times larger than the predicted ones by the theory, liquid penetrates to the heated surface overcoming the opposite vapor velocity, which is much higher than ‘‘critical’’. In [20] an empirical correlation was proposed to describe the experimental data on pool boiling crisis of sodium, potassium, rubidium, and cesium as follows

qcr =qcr ¼ ðp=pcr Þ0:12 ;

ð9Þ

where qcr is the experimental critical heat flux at the reference pressure p⁄ = 0.003 pcr. Later the similar expression with the exponent

value 0.185 was used in [22] for fitting the experimental data on pool boiling crisis of water, ethanol and some other organic liquids at subatmospheric pressures. It is clear that Eqs (2) and (8) at low reduced pressures predict a much stronger dependence of qcr on pressure, practically qcr =qcr ¼ ðp=pcr Þ0:5 . At water boiling under the very low pressures (till 1–2 kPa) the measured CHF values exceed the predicted ones 3–4 times [23]; it means that an actual velocity of vapor evacuation is higher at the same ratio than the critical one according to the theory. In this situation the question arises: why does the Kutateladze’s formula (2) which ignores any relation between the crisis and the preceding nucleate boiling regime, which contradicts to many experimental results concerned to the process mechanisms and to the integral characteristics at low reduced pressures manifest a satisfactory agreement with CHF data in a rather wide region of moderate and high reduced pressures? We will try to answer this question below. As for the fact of rather good predicting capability of the hydrodynamic theory of boiling crisis, it can partly explain why numerous new models of boiling crisis are commonly finished with the predicting equations based on the Kutateladze’s or Zuber’s formula. (Why look for a good thing when you have one?) This is true in relation to the first attempt to develop a boiling crisis model connecting this phenomenon with dry spots enlarging underneath the mushroom bubbles [24] and also in relation to the recent papers [25–27]. At the same time there exist definite objective reasons of such results of many theoretical models. If one builds up a crisis model in boiling at unlimited surface basing on momentum equations for liquid and for vapor accounting for only surface tension and body forces, one inevitably will obtain the equation similar to Eqs. (2), (8). This simply follows from dimensions analysis (the Buckingham’s p-theorem). The results of different researchers can differ from each other only with additional multipliers presenting some functions of liquid/vapor density ratio. This is clearly seen from a list of the correlations for CHF presented in [25]. As for a content of ‘‘a new mechanistic model’’ developed in [25], it repeated some doubtful approaches connected with using the Kelvin–Helmholtz instability in analysis of the liquid macrofilm thickness. This classic instability theory considers a relative motion of two inviscid infinitely spread liquids; for submillimetric liquid layer on the solid surface this approach is hardly applicable. In [14] this was discussed in relation of the model by Haramura and Katto [28], and this notion is probably true in respect to the model of [25], where the macrolayer thickness was taken as a half of the critical wavelength of the Helmholtz instability. It should be noted that none of the existing correlations for CHF including the new ones of [25–27] describes the weak dependence of qcr on pressure at the region of low reduced pressures. The above consideration shows that there are grounded reasons for developing a really new approach to pool boiling crisis, which involves in analysis not only hydrodynamic mechanisms of the process. The present author proposed in 1988 [14] a model connecting crisis with irreversible growth of dry spots area at the heated wall; later the model was corrected in some details [29,30]. In the following section of the paper a short description of this approach is given; the main ideas of the previous publications are retained, but a new method of derivation of an equation for CHF at high reduced pressures is used. In conclusion of this section it is useful to note that besides rather good predicting capability of Eqs (2) and (8) at relatively high reduced pressures, there is, probably, one more reason of popularity of the hydrodynamic theory. This was well formulated by Lienhard [4]: ‘‘We create defensive fortifications for our ideas, and close our ears to the rest of the story.’’

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3. Crisis as a result of dry spots area enlarging 3.1. The main features of the model At nucleate boiling nucleation sites at the heated surface are the small dry spots. Their existence is a generic feature of nucleate boiling, which distinguishes this process from any other mode of convective heat transfer and allows understanding the reasons of extremely high heat transfer intensity in nucleate boiling especially under high reduced pressures [31,32]. It seems to be the most natural to connect the boiling crisis with the dry spot area increase. As it is mentioned above, this idea was expressed first long ago [24], but was used only as new proving of the Kutateladze’s formula (2). We have turned to this idea initially in order to obtain an equation for CHF in flow boiling of saturated liquids [33] and later for pool boiling [14]. Far from CHF in nucleate boiling the dry spots are small; their probable size is about of an equilibrium vapor bubble radius R⁄ = 2r/Dp. As a distance between nucleation sites is usually equal to (103–104) R⁄, the area of dry spots presents very small fraction of a heated surface total area. However, as a local heat flux in the vicinity of the dry spot boundary is considerably greater than an average one at the heated wall, this mechanism of heat transfer is very important in boiling and its contribution in the total heat flux increases at higher nucleation sites density. At the same time the dry spots bear a possibility of terminating the nucleate boiling. On the one hand, an increase of the nucleation sites density with heat flux increase leads to higher heat transfer intensity; on the other hand, this means the increase of the total area of the dry spots. In the vicinity of the critical heat flux the density of the population of the nucleation sites becomes so great that the adjacent dry spots merge, and their size becomes considerably greater than R⁄. According to the recent publications [34,35] the surface dryout occurs locally at a high heat flux region close to the CHF and spreads rapidly over the surface with slight increases in the heat flux. The dryout void fraction averaged over the heating surface, however, remains small, about 5–15%, even at CHF. These observations obtained in the experiments with subcooled water boiling at atmospheric pressure fully confirmed the previous results of [36,37] for saturated water and methanol boiling, which were obtained by means of the other measuring technique. Evaporation of liquid at the boundary of the dry spot is considered as a basis for an analysis of critical heat flux. Schematic presented in Fig. 2 is quite similar to the model presented for nucleate boiling in [31,32]. The liquid inflow into the zone of intensive evaporation is provided by the capillary pressure gradient:

dpL d ¼ 2 ðrHÞ; dr dr

ð10Þ

where H is a mean curvature of a liquid film surface. In a pure liquid

r is const, the gradient of the capillary pressure is positive and provides the liquid inflow to the zone of intensive evaporation. At

269

q << qcr the dry spot radius is assumed to be equal to an equilibrium vapor bubble radius R⁄, which value is commensurable with the macrofilm thickness D in the region of the intensive evaporation, so the both linear scales affect the gradient of the capillary pressure. At the near-critical heat flux the dry spot size R0 >> R⁄ and R0 >> D0 ; this allows to assume that the capillary pressure gradient is governed by the only linear scale – the mean thickness of the macrofilm D0 . The other assumption is made that the curvature of the macrofilm surface decreases from 1/D0 in the zone of the high intensity of evaporation to zero at the flat surface of the film with the radial coordinate increasing Dr  D0 . This means that ðdH=drÞ  1=D20 . At the known pressure gradient in the macrofilm the liquid mass flow rate is easily calculated. Assuming further that the liquid impelled by the capillary pressure gradient to the boundary of the dry spot is fully evaporated in the vicinity of this line, one can obtain an expression for the linear density of heat flux at the dry spot boundary as follows [31,32]:

ql ¼ C 1

hLG rd0

m

;

where C1 is an unknown const. The heat flux density per an area of the dry spot with radius R0 = lds can be expressed, consequently, as

qds ¼ C 01

hLG r d0 : m lds

For nucleate boiling heat transfer at heat fluxes essentially lower than the critical ones the similar equation determines the heat flux density per one dry spot, but an equilibrium bubble radius R⁄ is used instead of lds as a characteristic size of a dry spot. Under such conditions qds >> q, where q is an average heat flux at the heated wall in boiling. At high heat fluxes near boiling crisis the value of qds decreases with enlarging a dry spot size. The macrofilm thickness is controlled by liquid viscosity and the characteristic values of length l0 and velocity W0:

d0 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ml0 =W 0 :

ð11Þ

These characteristic parameters are determined differently for the limiting cases of low and high reduced pressures; but in the both cases the macrofilm thickness diminishes with increase of heat flux density. Near the crisis a dry spot size is assumed to be proportional to a large vapor bubble departure diameter Dd. This assumption seems to be absolutely natural as a dry spot area continues to grow while a large vapor volume does not detach from the wall. In the recent paper [35] it is stated that surface dryout occurs in the latter half of the vapor mass hovering period; this factually confirmed the above assumption. As the average macrofilm thickness d0 diminishes with the average heat flux q increase, so the equality q = qds = qcr determines the boiling crisis condition at the given combination of the controlling parameters and liquid properties. At q > qds

Fig. 2. Schematic of liquid meniscus at the interline.

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evaporation rate becomes higher than the liquid inflow rate. As a result the dry patch area has to grow indefinitely and boiling crisis occurs. So, for one-component liquid the general expression for CHF is as follows

qcr ¼ C 0

rhLG d0 : m Dd

ð12Þ

This is a basic equation determining the critical heat flux in pool boiling. Eq. (12) is comparable with the equations of the hydrodynamic theory (2) and (8) in its simplicity and a clear physical content. But there are arguments to estimate it as more corresponding to the real mechanisms of nucleate boiling. Parameters d0 and Dd in [14] have been derived separately for high (p/ pcr > 0.03) and low (p/pcr < 0.001) reduced pressures, and consequently, two different equations for qcr.h and qcr.l have been obtained. This approach seems to be reasonable, if to bear in mind that, for example, in water the liquid to vapor density ratio at p/ pcr = 0.1 and 0.0001 differs approximately two orders of magnitude, and in similar manner are related a characteristic internal velocities of nucleate boiling (bubble growth rate and an average velocity of evaporation). Under such difference the mechanism of boiling crisis is hardly the same at the considered limiting cases. For arbitrary pressure CHF is calculated by a simple interpolating equation:

 1=3 qcr ¼ q3cr:h þ q3cr:l :

ð13Þ

3.2. Low reduced pressures limit Nucleate boiling at subatmospheric pressures is featured with an irregularity of nucleation and with very high bubble growth rates [29]. Rather big values of an equilibrium vapor radius explain difficulties in nucleation at the heated wall and absence of fixed nucleation sites at relatively small heat fluxes. In some experiments the boiling crisis occurred at the very incipience of nucleate boiling, i.e. transition from natural convection to film boiling was observed when thin wires or small diameter tubes were used as heaters. But using the heaters of rather large size provides the conditions for nucleate boiling mode existence as a process preceding to the boiling crisis onset. However, low density of nucleation sites and high Jakob numbers, which correspond to the considered conditions, allow to assume that the characteristic velocity in Eq. (11) for mean thickness of the liquid macrofilm at low reduced pressure is an ensemble-average bubble growth rate. Our visual observations of the water boiling at pressures 1 and 2 kPa confirmed that even at near-critical heat fluxes vapor bubbles maintain to some extent their individuality. The characteristic linear scale under these conditions is the average distance between the neighboring nucleation sites. Assuming a uniform distribution of the sites at the wall one obtains

l0 ¼ C 2 R ; where for low pressures R⁄ is inversely proportional to DT2 [14,29]. The scale l0 determines a growing bubble size at the moment of the bubbles merging, so it is appeared also in the derivation of the ensemble-averaged bubble growth rate. Thus, in the low reduced pressure region the mean thickness of the liquid macrofilm is a function of the wall superheat. According to [31] at low reduced pressures a dominant mechanism of heat removal from the wall in nucleate boiling is convection caused by the vapor generation near the heated surface. Heat flux density is determined as q ¼ kDT=D, where k is thermal conductivity, D is a thickness of heat conducting layer. In nonmetallic liquids D is thinner than the macrofilm thickness, D = d0f(Pr). In [29] an expression connecting d0 with heat flux density has been derived.

Bubble departure diameter in boiling at low reduced pressures (Jakob numbers, Ja > 200) according to [38] is determined as follows

 2=5 6=5 2 Ri T s kcp Dd ¼ 0:44 ; 8=5 qG hLG  g 3=5

ð14Þ

where cp is specific heat, Ri is a gas constant. Using (14) and the equation determining the mean thickness of the macrofilm through heat flux and liquid properties in Eq. (12) the following equation for CHF at low reduced pressures has been obtained in [30]: 81=55

qcr:l ¼ 0:5

hLG

r9=11 ðqG Þ13=110 k7=110 g 21=55 f ðPrÞ : m1=2 c3=10 R79=110 T s21=22 p i

ð15Þ

A numerical factor 0.5 was fitted to the experimental data. A function of Prandtl number for nonmetallic liquids is determined as follows:

f ðPrÞ ¼

Pr9=8 1 þ 2Pr1=4 þ 0:6Pr19=24

!4=11 ;

ð16Þ

for liquid metals

f ðPrÞ ¼ 0:5: The latter corresponds to the condition that in liquid metals the heat conducting layer coincides with the liquid macrofilm, i.e. D = d0. In [14,29] for f(Pr) a simple power correlation was used; but analyzing the data of [39,40] on critical heat flux in cyclohexanol (C6H12O) boiling forced to use more sophisticated Eq. (16), which is similar to equations used in calculation of heat transfer at natural convection. At the saturation line the viscosity of cyclohexanol is approximately an order of magnitude higher than that of water at the same pressures. The values of the Prandtl number of cyclohexanol in the pressure range ps = 1.6–100 kPa lie between 123 and 13.8. In [29] the comparison of calculations according to (15) with the experimental data on CHF in liquid metal boiling is presented. Although liquid metal boiling heat transfer and crisis is featured with rather large scatter of the experimental points a reasonable agreement of the data and the prediction has been demonstrated. Comparison with the data for nonmetallic liquids is given and discussed below. 3.3. High reduced pressures limit With high pressures liquid evaporation at the boundaries of the dry spots is the predominant mechanism of heat removal from the wall in nucleate boiling [31,32]. High density of the nucleation sites at these conditions allows assuming that the mushroomshaped vapor bubbles grow due to average heat flux q from the wall. For a growing semispherical bubble a simple energy balance has to be fulfilled:

2 dR phLG qG R2 ¼ C 3 pR2 q; 3 dt where unknown const C3 is a measure of proximity of estimation of the area, from which the heat flux removed. From this equation follows that

dR q ¼ C 03 : dt qG hLG Large bubbles departure conditions can be estimated according to inertia scheme [38]. In this textbook a problem on rising in quiescent liquid a spherical gas bubble growing according to the

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271

known power law is solved. Using the obtained solution to the case of the constant bubble growth rate gives the bubble radius at the departure moment:

Rd ¼

1 g Dq 2 t 4 qL 0

where t0 is full growth time. From the above two equations it follows

 Dd ¼ C 4

q qG hLG

2

qL g Dq

:

ð17Þ

In the recent work [41] growth of the coalesced bubble near crisis was also estimated on the basis of average evaporation velocity at the wall. The experimental data on the coalesced bubble departure diameter at CHF under high pressures (up to 5 MPa for water) are rather unique and present the great interest for understanding the boiling crisis mechanism. Eq. (17) with C4 = 2–4 is found to be in rather good agreement with the measurements results. Probably, at relatively low pressures the width of the heater (4 mm) could limit the bubble size at detachment; independence of the measured diameter of coalesced bubbles on pressure in [41] could be caused with this effect. Unfortunately, the authors of [41] used rather doubtful empirical correlation for departure frequency in order to determine the bubble departure diameter. At high reduced pressures the characteristic velocity in Eq. (11) is the average velocity of evaporation, i.e. W0 = WG = q/(qGhLG); the characteristic linear scale is the distance between the neighboring dry spots. As the total area of dry spots even at rather low reduced pressures (atmospheric pressure for water) is a small part of the heated wall area, it is reasonable to assume that in the considered case of high reduced pressures l0 is proportional to the dry spot size. This means that the bubble departure diameter factually remains the only linear scale in Eq. (12). After simple transformations the latter changes to the form: 0:4 qcr:h ¼ 0:06hLG q0:6 ðg Dq=lÞ0:2 ; G r

ð18Þ

where l is dynamic viscosity; numerical factor was fitted to the experimental data. In the first publication of the model considered [14] the other way was used in order to obtain equations for large vapor bubbles growth rate and for their departure diameter at near-critical heat fluxes. At that time there was no experimental information on these processes. After publication [41] this is reasonable to correct the model in accordance with new experimental observations. But a final equation for CHF at high reduced pressures limit in [14] was the same as Eq. (18). The majority of the available experimental data on CHF have been obtained at moderate and high reduced pressures when the crisis conditions are similar to those which were considered in this section. The interpolating Eq. (13) is built so that the less constituent of the CHF loses its meaning quickly in comparison with the dominant one. Thus, in the most practical cases qcr.h presents the main part of the total critical heat flux, and qcr  qcr.h. 3.4. Comparison with the experimental data Fig. 3 depicts comparison of the calculations according to Eqs (13), (15), and (18) with the experimental data on boiling crisis at low pressures. The curves calculated in accordance with the interpolating Eq. (13) at their most part coincide with the calculation on Eq. (15) for the limiting case of low reduced pressures. In this region of reduced pressure CHF very weakly depends on pressure, and this regularity is well described by Eq. (15). The Kutateladze’s formula (2) cannot describe this experimental tendency. It is easy to see that new theoretical equations (for example, [25,26])

Fig. 3. Comparison of the calculated and experimental CHF data: 1, 3, 5 – calculations according to (13) with qcr.l on (15) and qcr.h on (18) for water, ethanol and isopropanol correspondingly; 2, 4, 6 – calculations according to (2) for the same liquids; experimental data: 7 – water [22,23,42], 8 – ethanol [22,23], 9 – isopropanol [22], and 10 – isopropanol [40].

based on the hydrodynamic approach also fail to describe these data. In [30] rather detailed analysis is given of applicability of Eqs. (15) and (16) for description of the unique data on CHF in cyclohexanol boiling. To the best of the present author’s knowledge the experimental data of Dhir [39,40] were not analyzed by specialists before. These data are actually unique not only due to very high viscosity of the liquid, but also due to that a part of the experiments were conducted using a centrifuge. The efficient acceleration g in some experimental runs was essentially higher than the standard gravitational one g0 (up to 25 g0). Two tables presented in [30] show that the calculated values of CHF are in good agreement with the measured ones both at normal and increased accelerations. As for moderate and high reduced pressures, a wide comparison of predicted and measured data was presented in the first publication of the model [14]. It is easy to understand that Eq. (18) is very similar to Eqs. (2) and (8) of the hydrodynamic theory and gives practically coinciding result in calculations in the majority situations at the reduced pressures region p/pcr > 0.03. As an example new experimental data of [25] on CHF in pool boiling of pentane, hexane and fluorocarbon FC-72 and data of [41] on CHF of water boiling were compared with the calculated ones according to Eq. (18). All experiments were conducted at the reduced pressures higher than 0.04, so that contribution of low pressure constituent qcr.l into the total CHF in accordance with Eq. (13) does not exceed 3%. In Fig. 4 comparison with the data of [25] for pentane and hexane is presented; besides the calculated curve on Eq. (18), the calculations according to Eq. (8) with k = 0.131 and according to the new equation of the authors of [25] are also shown. As is seen, agreement between the experimental points and the three predicting curves is quite satisfactory. Fig. 5 depicts the experimental data on CHF of water boiling in the range ps = 1–7 MPa and calculated curves in accordance of the Zuber’s Eq. (8) and of Eq. (18). The experimental values of CHF in [41] are higher than the majority of previously obtained data at similar conditions. The predicting curve on Eq. (18) lies at the lower boundary of the experimental points of [41], while the Zuber’s equation gives even lower values. Nevertheless, practically all the points in Fig. 5 deviate less than 30% from the predicting curve of the present author. The interpolating Eq. (13) satisfactorily agrees with the experimental data at any reduced pressures, but it is practically actual at the region p/pcr = 0.001–0.03. The approach considered allows also building up a model of crisis in binary mixtures boiling [43]. The model does not consider an influence of the heated surface properties on CHF; it relates to commercial surfaces, to some

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Fig. 4. Comparison of the calculated curve according Eq. (18) (line 3) with the experimental data of [25] on CHF for pentane and hexane; 1 and 2 – calculation in accordance to new model of [25] and to Eq. (8) correspondingly.

(18) gives for helium at atmospheric pressure qcr.h ffi qcr = 8.5 kW/ m2, which differs from the above averaged experimental value by 17.5%. This analysis, probably, explains the apparent absence of the viscosity effect on CHF at the moderate and high reduced pressures. The actual effect of viscosity appears to be concealed in the numerical k in the Kutateladze’s formula which on this reason is found to be in the satisfactory agreement with the majority of available experimental data on CHF. 4. Concluding remarks

Fig. 5. The experimental data of [41] on CHF of water boiling at ps = 1–7 MPa and calculated curves: 2 – Zuber’s Eq. (8), 3 – Eq. (18).

‘‘average’’ conditions. At the same time as distinct from the hydrodynamic theory the proposed approach allows in principle to introduce in analysis such properties of the heated wall as roughness and wettability. In particular, the latter effect was considered in [14]. Certainly, revealing quantitative regularities of the surface properties influence on boiling crisis seems to be a problem too difficult to be solved now. However, positive effect of the wall roughness and high wettability on CHF value, which is found in the experimental studies, appears to be quite natural in the light of the content of the model suggested. The both effects, probably, prevent the dry patches area enlarging. Analysis of Eq. (18) reveals that at the pressure region corresponding to its applicability CHF very weakly depends on a complex ðDq=lÞ0:2 . This is explained with very small variation of kinematic viscosity of liquids at saturation line. In [30] a table is presented where the numerical values of the discussed complex are given at the reduced pressures p/pcr = 0.03 and 0.9 for 20 different liquids, which represent the main types of substances with known values of qcr. For all liquids, except helium, the values of ðDq=lÞ0:2 are in the range 18–25 s0.2 m0.4 with maximal deviation of 16.5% from the average value 21.5. This deviation is considerably smaller than the usual scatter of the experimental data on qcr. For helium boiling at atmospheric pressure the excess near 49% of ðDq=lÞ0:2 above its mean value surprisingly coincides with the difference between the averaged experimental value of qcr ffi 10 kW/ m2 recommended in [44] and the CHF calculated on Eq. (2). Eq.

1. Answering the question in the title of the paper one can say: yes, crisis in pool boiling is a hydrodynamic phenomenon, but to some extent. Actually, the pool boiling crisis is more complex phenomenon, which hydrodynamic aspect is a dependence of the dry spot size on the departure diameter of the coalesced bubbles. 2. The hydrodynamic model of boiling crisis, apparently, determines the upper limit of the average velocity of evaporation, which is not achieved under usual conditions of boiling on a surface submerged in a liquid. This limiting value corresponds, in particular, to the maximum amount of vapor generated in vertical heated channels with a closed bottom. 3. Obvious contradictions of the Zuber’s hydrodynamic theory of pool boiling crisis stimulated a number of new models in this field. However, if a model considers only hydrodynamic effects ignoring influence of liquid viscosity and heat transfer at the heated surface, this inevitably leads to rather small corrections to the equations by Kutateladze and Zuber. Besides, some of these models used in analysis rather doubtful statements. None of the available predicting correlations can describe the experimental data on boiling crisis at low reduced pressures with very weak influence of pressure on CHF. 4. The model of pool boiling crises developed by the present author presents an attempt to exceed the limits of the Kutateladze and Zuber approach. The crisis is considered as a result of enlarging area of dry spots which are on their part an intrinsic feature of nucleate boiling heat transfer. Great difference of specific volume of vapor at high and low reduced pressures makes it reasonable to analyze separately the detailed regularities of the crisis incipience at these two cases. The predicting equations for CHF at the regions of high and low reduced pressures are found to be in good agreement with the experimental data at the corresponding conditions. A simple interpolating formula allows calculating qcr at arbitrary pressure.

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5. Eq. (18) gives the values of CHF close to those calculated in accordance with the formulas of the hydrodynamic theory; as the most part of the experimental data on boiling crisis relates to the moderate and high reduced pressures, this coincidence can be considered as a reason of satisfactory agreement of the Kutateladze’s formula (2) with the data. A weak variation of liquid kinematic viscosity at saturation line with pressure allows concealing an actual influence of this effect on CHF. Conflict of Interest I confirm that the proposed paper does not conflict with my University authority or with any persons. Acknowledgment The work was supported by the Federal Program of Science Schools Support, Grant Nos. 4977.2012.8 and 3783.2014.8. References [1] S. Nukiyama, The maximum and minimum values of the heat q transmitted from metal to boiling water under atmospheric pressure, Int. J. Heat Mass Transfer 27 (1984) 959–970. [2] S.S. Kutateladze, Hydromechanical model of heat transfer crisis in pool liquid boiling, J. Tech. Phys. 20 (11) (1950) 1389–1392 (In Russian). [3] N. Zuber, On the stability of boiling heat transfer, Trans. ASME 80 (3) (1958) 711–720. [4] J.H. Lienhard, Snares of pool boiling research: putting our history to use, in: Proceedings of the 10th International Heat Transfer Conference, vol. 1, Brighton, UK, 1994, pp. 333–348. [5] S.S. Kutateladze, Boiling heat transfer, Int. J. Heat Mass Transfer 4 (1961) 31– 45. [6] S.S. Kutateladze, Heat Transfer Fundamentals, fifth ed., Atomizdat, Moscow, 1979. pp. 304–324. (In Russian). [7] C.P. Costello, W.J Frea, A salient non-hydrodynamic effect on pool boiling burnout of small semicylindrical heaters, AIChE Preprint 15, Sixth Nat. ASMEAIChE Heat Transfer Conf. Boston, Aug. 11–14, 1963. [8] D.A. Labuntsov, About a new approach in boiling crisis theory, Therm. Eng. 8 (1961) 81–85 (In Russian). [9] R.F. Gaertner, Photographic study of nucleate pool boiling on a horizontal surface, Trans ASME J. Heat Transfer 87 (1965) 17–29. [10] A.E. Bergles, What is a real mechanism of CHF in pool boiling?, in: V.K. Dhir, A.E. Bergles (Eds.), Pool and External Flow Boiling, ASME, N.-Y., 1992, pp. 165–170. [11] Y. Iida, K. Kobayasi, An experimental investigation of the mechanism of pool boiling phenomena by a probe method, in: Proceedings of the Fourth International Heat Transfer Conference, vol. 5, Paris, 1970, B.1.3. [12] Yu.P. Dzhusov, A.A. Tsyganok, A.A. Gribov, N.N. Mityaev, The void fraction above the heated surface under water pool boiling, in: Therm. Eng. 6 (1986) 55–59 (In Russian). [13] A.M. Bhat, J.S. Saini, R. Prakash, Role of macrolayer evaporation in pool boiling at high heat flux, Int. J. Heat Mass Transfer 29 (1986) 1953–1961. [14] V.V. Yagov, A physical model and calculation formula for critical heat fluxes with nucleate pool boiling of liquids, Therm. Eng. 35 (6) (1988) 333–339. [15] T.G. Theofanous, T.N. Dinh, J.P. Tu, A.T. Dinh, The boiling crisis phenomenon part I: Nucleation and nucleate boiling heat transfer, Exp. Therm. Fluid Sci. 26 (2002) 775–792. [16] T.G. Theofanous, T.N. Dinh, J.P. Tu, A.T. Dinh, The boiling crisis phenomenon part II: Dryout dynamics and burnout, Exp. Therm. Fluid Sci. 26 (2002) 793– 810.

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