International Journal of Heat and Mass Transfer 70 (2014) 526–535

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International Journal of Heat and Mass Transfer

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An analysis of incipient boiling superheat in alkali liquid metals

0017-9310/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.11.047

⇑ Corresponding author. Tel./fax: +86 29 82663401.E-mail address: szqiu@mail.xjtu.edu.cn (S. Qiu).

Zaiyong Ma, Zicheng Qiu, Yingwei Wu, Suizheng Qiu ⇑, Guanghui SuSchool of Nuclear Science and Technology, State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China

a r t i c l e i n f o

Article history:Received 25 January 2013Received in revised form 16 November 2013Accepted 16 November 2013

Keywords:Incipient boiling superheatInert gas diffusionOxide layerDynamic effect

a b s t r a c t

Studies on the incipient boiling superheat (IBS) in alkali liquid metals have revealed that many factors canhave effects on IBS. However, existing results about the effects of most variables are inconsistent andeven conflicting. It is possible that effects of many variables are only apparent and the complex resultscan be explained with several simple factors. Three basic variables, i.e. the effects of inert gas diffusion,oxide layer and dynamic process were involved in our model to show their effects on IBS. Analysisshowed that the inert gas diffusion should be carefully considered because it may result in large variationof IBS in a short time. Existence of oxide layer could help reducing IBS by keeping large cavities un-wettedand enhancing nucleation contact angle. The effect of dynamic process on IBS which may cause apparentvelocity and heat flux effects may be explained with the variation of apparent nucleation contact anglecaused by interface speed, and a correlation with some empirical coefficients was proposed to describeits effect. To verify the model several sets of experimental data were carefully selected. Predictions con-sidering these effects were compared with these experimental data and good agreements were gained.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Sodium and other liquid metals have been chosen as the cool-ants of several kinds of Generation-IV reactors. In the design andsafety analysis of these reactors, IBS is an important parameter thatinfluences its following boiling process like flow pattern evolution,boiling heat transfer and heat transfer crisis. Because of high ther-mal conductivity, wetting and other characteristics, IBS in alkali li-quid metals is quite different from that in conventional fluids, andexisting models for conventional fluids are not applicable [1]. From1960s to 1980s, a great number of studies were done by manyresearchers. However, the principle has not been fully revealed yet.

The early experimental data were in large spread, and thesuperheats were found to vary in a range of 5–500 K, which indi-cates effects of many variables were not well understood [2]. Thevariables including heat flux, temperature ramp, velocity, inertgas, O2-impurity, operation history, surface condition, heatingsurface material have been investigated by many researchers [3],but influences of most of these variables on superheat are indefi-nite, with experimental results differing from one another.

The most inconsistent results are the effects of heat flux, tem-perature ramp and velocity. Heat flux sometimes seems to beunimportant to IBS [4], but clear effects were also found in manyexperiments and analyses, and superheat may increase or decreaseas heat flux increases [1,5,6]. The temperature ramp effect was

experimentally studied by Dwyer et al. [7], and it was found thatincrease of temperature ramp clearly increased IBS. But in anotherexperiment [8] where the temperature ramp effect was investi-gated under several typical conditions in nuclear reactor like valveclosure and power increase, no clear effect was found. However, itshould be mentioned that the temperature ramp in [8] (�10 K/s) ismuch larger than that in [7] (�0.1 K/s), and it is possible that thediscrepancy comes from the magnitude difference of temperatureramp. Many studies were also done on the influence of velocityor Reynolds number [9–13], and generally the superheat wasfound to decrease or unaffected with an increase in velocity. Boththe bulk superheat and wall superheat were employed to show thevelocity effect in the literature, and the nucleation was usuallysupposed to occur at the test section exit (hottest location). How-ever, as the studies of some researchers showed, the wall super-heat instead of the bulk superheat and also the local superheatrather than the exit superheat should be used to properly showthe effect of velocity [3,7,9]. It has been pointed out by someresearchers that the effects of heat flux and velocity observed bymany experiments is due to the failure of determining the locationof nucleation, and this was partly demonstrated by Kottowski andWarnsing [9], in whose experiment the velocity effect disappearedafter measuring the local wall superheat. However, Dywer et al.[1,7] conducted many experiments that were carefully executed,and apparent heat flux, velocity and temperature ramp effectswere observed, although the locations of nucleation weremeasured accurately enough.

The most consistent agreement in the trend concerns theinfluences of inert gas and O2-impurity, and superheat was found

Nomenclature

C empirical coefficient in Eq. (10)c dissolved inert gas concentration (mol/m3)cp specific heat capacity (J/kg K)D mass diffusivity coefficient (m2/s)Dh thermal equivalent diameter (m)h heat transfer coefficient (W/m2 K)hlv latent heat of vaporization (kJ/kg)K constant of Henry’s Law (Pa�1)k temperature ramp (K/s)L heated length (m)M molar mass (g/mol)n number of moles (mol)P pressure (Pa)q heat flux (W/m2)R interface curvature radius (m)Rc gas constant (J/mol K)Rmin nucleation radius (m)r interface location (m)rc oxide layer location (m)rmax cavity mouth radius (m)rmin deepest interface location (m)T temperature (K)V volume (m3)v velocity (m/s)x coordinate (m)

Greek symbolsa empirical coefficient in Eq. (10)c ratio of nucleation radius to deactivation radiusd parameter in Eq. (6)h half cone angle (rad)hr nucleation contact angle (rad)hrk nucleation contact angle for k temperature ramp (rad)hr0 nucleation contact angle for zero temperature ramp

(rad)q density (kg/m3)r surface tension coefficient (N/m)s time (s)

Superscript0 deactivation

Subscriptsd at x = dv vaporl liquidg inert gasin inletsat saturatedsub subcoolingw wall

Z. Ma et al. / International Journal of Heat and Mass Transfer 70 (2014) 526–535 527

to decrease due to increase in gas entrainment and oxide level inmany experiments [3]. Apart from experiments, the inert gas effectwas also analytically studied by several researchers. Singer andHoltz [6] studied the incipient pool boiling superheat, and the ef-fects of initial temperature, heat flux, initial inert gas partial pres-sure and heating technique were studied, but the results were notcompared with experimental data directly. In another analyticalstudy conducted by Holland and Winterton [14], aging effectcaused by diffusion of inert gas was gained both in theory andexperiment. The analytical studies given by Singer and Holtz [6]and Holland and Winterton [14] reflects the important role of inertgas, and because of this, an idea about the effects of some variableson superheat was raised that the observed influences of other vari-ables may be due to the diffusion of inert gas [3,15].

The operation history has also been found to be an importantparameter. The most famous model about this is the so-calledpressure–temperature history model developed by Holtz [16],and the model is partially successful in explaining experimental re-sults [17]. Because many experimental superheats are underesti-mated with deactivation radii used as nucleation radii in themodel, in the literature a factor c (ratio of nucleation radius todeactivation radius) is often introduced to compensate the differ-ence, and in these cases it is believed at deactivation state theinterface is stopped at the non-wetting non-metallic inclusions.Holland and Winterton [18] summarized experimental data (about30 data points) of some researchers and proposed a weighted meanvalue c = 0.35. In 1968, Chen [19] conducted many studies to studythe effects of operation history, but results showed that manysuperheat data were overestimated by Holtz’s model, meaning thatc can be larger than 1. He explained this by taking the quantity ofinert gas to be constant and considering partial pressure changecaused by volume variation. Later Dwyer [15] introduced thechanges of liquid–gas interface curvature radii and partial pressureof inert gas in the cavity and proposed c = 0.75 to explain Chen’sdata. They both gained relatively good agreement with

experimental data, but the development of Chen [19] and Dwyer[15] involved some constants that may vary much in practicalsystems.

It should be mentioned that though little literature can be foundabout the IBS in alkali metals in recent years, many related studieson other fluids have been done, involving the effects of dissolvedinert gas, surface conditions, wetting, contact angle and so on[20–24]. These studies provide some new ideas for researchingthe boiling incipience in alkali liquid metals, for example, theimportant roles of wetting and (dynamic) contact angle in boilingincipience.

As some researchers pointed out [3,25], though many factorshave been observed to have influences on nucleation superheatin liquid metals, some of these factors may be only apparent. Inthis paper, three basic factors which can influence nucleation ofliquid metals were studied, i.e. diffusion of inert gas, non-wettingoxide layer and dynamic effect. With these factors, some experi-mental data were explained. This paper may provide a betterunderstanding of the IBS in liquid metals.

2. Model development

In this paper, only surface nucleation is considered, and the sur-face cavities are assumed to be conical (see Fig. 1a). At equilibrium,by analyzing the forces acting on the gas–liquid interface, thefollowing familiar equation can be gained [15,16]:

Pv � Pl ¼2rR� Pg ð1Þ

where Pv, Pg and Pl are the vapor pressure, inert gas partial pressureand liquid pressure, respectively; R is the interface curvature radius,which may be positive or negative, and r is the surface tensioncoefficient.

When condition varies, the value of R will change to balance theforces and it is assumed that nucleation takes place when R equals

Fig. 1. A schematic of (a) conical cavity and (b) interface variation.

528 Z. Ma et al. / International Journal of Heat and Mass Transfer 70 (2014) 526–535

the nucleation radius Rmin, where 1/R reaches its maximum,namely the flip-over position [26]:

1Rmin

¼ cosðhr � hÞr

ð2Þ

here hr and h are the apparent nucleation contact angle and the cav-ity half cone angle, respectively, and r is the interface location (seeFig. 1a).

Usually Pl is known in a system, and Pv and r can be determinedwith wall temperature. Thus from Eqs. (1) and (2), it can be seenthat to obtain the nucleation wall superheat, the values of the inertgas partial pressure Pg, the nucleation contact angle hr, the inter-face location r and the parameters of the cavity geometry whennucleation happens need to be known. In our model, the effectsof inert gas diffusion, oxide layer and dynamic process on thesefactors are analyzed.

2.1. Inert gas diffusion

As previously mentioned, IBS is dependent on many factors.However, influences of many parameters might be explained byconsidering the effects of inert gas [3], so the effects of inert gasdiffusion should be carefully considered. The volume variation ofinert gas is involved in our model in view of the study of Chen[19], though in the literature it is usually neglected [6,14].

The inert gas concentration distribution is assumed to be one-dimensional (Fig. 1a), and the diffusion equation then can be writ-ten as follows [6,14]:

D@2cðx; sÞ@x2 ¼ @cðx; sÞ

@sð3Þ

where D is the mass diffusivity coefficient of inert gas, and c is inertgas concentration.

The initial condition for Eq. (3) is the distribution of initial inertgas concentration c0 [6]:

cðx;0Þ ¼ c0ðxÞ ð4Þ

At the gas–liquid interface, the inert gas concentration is equal tothe solubility and can be obtained according to Henry’s Law [6,14]:

cð0; sÞ ¼ qKPg=M ð5Þ

where q and M are the density and molar mass of liquid metal,respectively, K is Henry’s Law constant, Pg is the inert gas partialpressure in the cavity.

Another boundary condition is the boundary inert gas concen-tration cd at x = d:

cðd; sÞ ¼ cdðsÞ ð6Þ

The value of d should be determined considering both convenienceand accuracy, like a location where the value of cd doesn’t change orits variation over time can be gained easily.

The amount of gas n in the cavity can be calculated with theconcentration gradient at the interface [6,14,27]:

dnds¼ pr2D

@cð0; sÞ@x

ð7Þ

Assuming the inert gas to be ideal, we have:

PgV ¼ nRcTw ð8Þ

where Rc is the universal gas constant, Tw is wall temperature, and Vis the cavity gas volume. The value of Pg in this equation should sat-isfy Eq. (1).

2.2. Oxide layer effect

Non-metallic inclusions which are mostly oxides are not easy tobe wetted by liquid metals, and cavities with such oxide layermight be nucleation points [18]. In our model, three independentparameters are used to describe a conical cavity, namely the halfcone angle h, the cavity mouth radius rmax, and the oxide layerlocation rc. Because without oxide layer, the surface cavity is veryeasy to be wetted, so the part between rmax and rc in fact is oftencovered with liquid, and thus rc is a more important parameterthan rmax. Supposing a long enough deactivation time and a per-fectly non-wetted condition (contact angle p) [19], at the initialtime of heating the value of the gas–liquid interface location canbe calculated with Eq. (1) and Henry’s Law:

r0 ¼ R0 cosðp� hÞ ¼ � 2r0 cosðhÞP0v þ P0g � P0l

¼ � 2r0 cosðhÞP0v þ c0ð0ÞM=q0K 0 � P0l

ð9Þ

The interface location calculated with the above equation may belarger than rc. In this case, the interface location is set to be theoxide layer location, which means a contact angle smaller than p.

According to Cornwell [26], there is a contact angle hysteresisdue to the micro-roughness of the cavity wall. Without consideringthe dynamic effects, interface deformation with no variation of rcan occur in the range determined by the advancing and retardingpositions (Fig. 1b), and interface movement with variation of r can

Z. Ma et al. / International Journal of Heat and Mass Transfer 70 (2014) 526–535 529

happen only if the change of conditions is very large. Both theinterface deformation and movement can result in the volume var-iation of the gas in the cavity.

The interface movement towards the mouth is unstable andmay lead to boiling incipience, and retarding position is the criticalstable location and the retarding angle can strongly affect the IBS[26]. For cavities with oxide layer, because of the non-wettingproperties, the retarding angle can be very large, and thus thenucleation radius is large (Eq. (2)), resulting in small superheat.However, according to research about other fluids [20,21], becauseof the motion of the liquid–solid–gas boundary (dynamic effect),the nucleation contact angle can be much smaller than retardingangle. So the range of actual nucleation contact angle and incipientsuperheat may be very wide because of oxide layer when theintensity of dynamic effect varies.

2.3. Dynamic effect

Many experimental results have shown that when a quick in-crease in heat flux is imposed to the heater, nucleation will nothappen until the steady state is essentially reached [7,28]. This factreflects the dynamic process has an important influence on liquidmetal nucleation. In our opinion, this may be explained if the dy-namic process can influence the nucleation contact angle, as men-tioned in Section 2.2.

The mechanism of the dynamic process effect on incipientsuperheat may be described as follows. Let us look at Fig. 1b. Sup-pose the interface stays at position 1 initially. When heated, thecavity wall temperature increases and the interface deforms (posi-tion 2). It can be seen that the center of the interface moves morethan the edge, that is, the speed of the interface center is largerthan the edge. It can be expected this speed difference increaseswith an increase in the intensity of dynamic process. Because ofthe speed difference, the interface may depart from sphere, andalso, the interface movement is restrained because the centerspeed is much larger than the edge (position 3; note that for sim-plification here we suppose interface movement and interface var-iation cannot occur at the same time, and the actual situations aremore complex, see Ref. [21]). This leads to the apparent contact an-gle smaller than the retarding angle. The interface will go furtherbeyond retarding position, calling for higher superheat. Boilingincipience will not happen until the heating process is slow enoughor the wall superheat is high enough to allow the interfacemovement.

Suppose the possible apparent nucleation contact angle is notsmaller than h, meaning that the dynamic effect has a limited rangeand the minimum nucleation radius is the interface location. Forboiling incipience achieved by heating, the dimensionless parame-ter (hr0 � hrk)/(hr0 � h) may be used to show the intensity ofdynamic effect, here hrk and hr0 are the apparent nucleation contactangle for k temperature ramp and zero temperature ramp, respec-tively. According to our explanation of the dynamic effect, it can beanticipated that with the increase of the temperature ramp, theincrease of (hr0 � hrk)/(hr0 � h) will become slower and slower.The temperature ramp needs to be large enough to make thedimensionless parameter to be 1. Based on this, the tangentfunction may be used to estimate the effect of dynamic processfor temperature ramp k:

tanhr0 � hrk

hr0 � hp2

� �¼ Cka ð10Þ

where C and a are empirical coefficients. The nucleation radius Rmin

then can be expressed as:

1Rmin

¼ cosðhrk � hÞr

ð11Þ

When the interface reaches the oxide layer location, it is possiblethat rc becomes nucleation radius because of high interface speedor sudden increase of wetting properties, and in this case, if Rmin

calculated with Eq. (11) is larger than rc, rc should be used as thenucleation radius. However, it is also possible that the interfacespeed is not high and the increase of wetting properties is insignif-icant (rc = rmax) when the interface reaches the oxide layer location,and in this case, rc should not be used as the nucleation radius evenif Rmin calculated with Eq. (11) is larger than rc.

Based on the data represented by Dwyer et al. [7], the asymp-totic value (IBS with nucleation contact angle h) will be reachedat a relatively small temperature ramp (about 0.2 K/s). This meansthe range of the effect of dynamic process may be very narrow andnucleation superheat might be very close to the asymptotic valuein many practical operations.

3. Results and discussion

Researchers have done many experiments on IBS in alkali met-als, almost all of which were sodium and potassium. However, be-cause the effects of some variables were not considered or wellcontrolled by early experimental conductors [17], many importantparameters were not provided in many experiments, and thusmany experimental data are almost impossible to be analyzed.Also, many parameters like geometry of the nucleation cavitiesand inert gas concentration are very difficult to evaluate and canonly be estimated.

Several sets of experimental data were carefully chosen andanalyzed to verify our model. Inert gas effects were shown dis-tinctly in the experiment of Holland and Winterton [14], and theirdata were used first to show our basic consideration of inert gas.With the consideration of inert gas, data of Chen [19] were ex-plained with both the inert gas and oxide layer effects. The temper-ature ramp effects (one type of dynamic effect) were carefullystudied by Dywer et al. [1,7], and their data were employed tocompare with our theory on dynamic effect. In addition, theparameter trend found in some experiments and analysis were alsoused for better verification.

The values of diffusion coefficients, properties of alkali metals,and solubility of inert gas in alkali metals in our calculation werebased on Refs. [27,29–31].

3.1. Ageing effect caused by gas diffusion

The existence of inert gas in cavities helps reducing the IBS, andthis has been demonstrated by both experiment and theory [2].Ageing effect caused by gas diffusion was experimentally studiedby Holland and Winterton [14]. The main parameters of the exper-iment is summarized in Table 1. In the experiment, it is reportedthat the system was heated rapidly until nucleation occurred,and as we discussed in Section 2.3, the range of the effect of dy-namic process may be very narrow, so it may be safe to assumethat the temperature ramp was large enough when nucleation,making the nucleation contact angle very close to h (Eq. (10)), i.e.the nucleation radius equals the interface location.

The initial inert gas concentration is negligible in this experi-ment [14]. According to [32], the value of d can be estimated withffiffiffiffiffiffi

Dsp

, namelyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi10�8 � 102

pm ¼ 10�3 m, and thus the value of d was

taken to be 3 mm. In our calculation, the wall temperature was in-creased with time very fast to simulate the rapid heating process.Using the cavity parameters shown in Fig. 2, superheat can be cal-culated by solving Eqs. (1)–(8) simultaneously with numericalmethod.

The main difference of our inert gas diffusion model from thatgiven by Holland and Winterton [14] lies in that the inert gas

Table 1Experimental conditions of inert gas effect [14].

Deactivation temperature (K) Ageing temperature (K) Ageing time (s) Pressure (MPa) Fluid Cover gas

663.15 1073.1 ± 25 0–150 0.101 Sodium Argon

Fig. 2. Inert gas aging effect on IBS.

530 Z. Ma et al. / International Journal of Heat and Mass Transfer 70 (2014) 526–535

volume variation is taken into consideration. Also, the thermody-namic and transport properties are functions of temperature ratherthan constants during the transient process.

Fig. 2 shows a comparison of the experimental data [14] withthe calculated curves. It can be seen generally the trend of theaging effect is well predicted by the curves. However, the curvegiven by us presents a slower increase than that of Holland and

Fig. 3. Predictions of the deactivation pressure effect on IBS (a) constant deactivation t

Winterton [14] when the aging time is not large, which may be abetter prediction.

The effect of volume variation in Fig. 2 is not very obvious, butin a rapid pressurization process, the interface may be pusheddeeper. In this case the consideration of volume variation can beimportant to predict IBS better, and this will be shown in Fig. 3.

3.2. Oxide layer effect

As discussed in Section 2.2, the oxide layer may reduce thesuperheat by keeping large cavities unwetted and enhancingnucleation contact angle. Since inert gas also reduces the super-heat, the oxide layer effect can be easily covered by the inert gaseffect. So the inert gas effect should be carefully dealt with whenstudying the oxide layer effect.

As we previously mentioned Section 2.2, interface movementand deformation can happen when conditions change, and as aresult, the interface curvature radius also changes. Because themain barrier of nucleation is determined by the maximum inter-face curvature (1/R) when the interface moves in a cavity, and atthe interface location r the maximum curvature is 1/r (Section2.3), the nucleation radius should be a value between the deepestinterface location rmin and the oxide layer location rc, on conditionthat rmin is small enough compared with rc, making the interfacespeed is large enough when it reaches the oxide layer location.

emperature, (b) low deactivation subcooling and (c) high deactivation subcooling.

Z. Ma et al. / International Journal of Heat and Mass Transfer 70 (2014) 526–535 531

The value of rmin can be easily obtained from Eq. (9) if the com-pletely non-wetted condition is used, while the value of rc shouldbe related with surface conditions.

Chen [19] conducted many experiments to study the effects ofdeactivation pressure, deactivation subcooling and incipient boil-ing pressure, and many of the experimental data showed smallersuperheat than that predicted by Holtz’s model [16]. The oxidelayer effects are used by us to explain his data. The main parame-ters of the experiment are shown in Table 2.

The nucleation contact angle is unknown, but considering thenot very large heat flux and heating method (incremental increasein power), the temperature ramp when nucleation may be not verylarge. So according to the dynamic effect together with otheruncertain factors, a medium contact angle may be a proper estima-tion. The nucleation radii Rmin were then obtained based on anaverage method, namely, the geometric mean of rc and rmin. The va-lue of rc was taken to be 3.5 lm [18], and the values of rmin werecalculated with Eq. (9) directly.

The heating time was set to be very short in our calculation, inview of the existence of low nucleation superheat when deactiva-tion pressure is low, meaning that the inert gas diffusion aging ef-fect may not be strong. The value of d was taken to be 3 mm, whichis large enough for rapid heating to exclude the effect of boundaryinert gas concentration (see Appendix).

The boundary inert gas concentration was assumed to be negli-gible and was taken to be zero, and the initial inert gas concentra-tion was estimated with the following equation:

C 0 ¼ P0lKðTÞqðTÞ=M ð12Þ

where T is taken to be 630.7 K, and this temperature was gained byfitting the data of Chen [19] based on the solubility data presentedby Shpil’rain et al. [31]. For a detailed analysis of the estimation ofinert gas concentration, see Appendix.

With the parameters of cavities, the nucleation radii and defi-nite conditions determined, the IBS under different conditionscan be gained by solving Eqs. (1)–(9) and (12).

Comparisons of our predictions with Chen’s experimental data[19] about deactivation pressure effects are given in Fig. 3. It canbe seen that nearly all the experimental data falls in the possibleregion predicted by our model, and generally the average curve fitswell with the experimental data. Calculation shows that inert gaseffect is apparent in the region where deactivation pressure is low-er than nucleation pressure, while for large deactivation pressure,inert gas effect is negligible. When deactivation pressure is low,the gas–liquid interface will be pushed deep due to pressurization(interface movement), and this will enhance the superheat; but forlow deactivation pressure, the inert gas partial pressure can bevery large after pressurization because of decrease of cavityvolume, and this decreases the superheat. The competition of thistwo factors results in a relatively flat region ranging 0.06–0.11 MPaof the deactivation pressure, as shown in these figures.

Table 2Experimental conditions of Chen’s experiment [19].

Parameters Deactivation Incipient boiling

Temperature (K) 755.37–1093.15 974.82–1144.26Pressure (MPa) 0.034–0.276 0.055–0.166Liquid velocity (m/s) 0.061 0.061Heat flux (kW/m2) 0 6.296–157.4Liquid subcooling (K) 5.55–211.11Liquid superheat (K) 9.44–65Test section I.D. (mm) 15.8 15.8Test section length (mm) 304.8 304.8Fluid Potassium PotassiumCover gas Argon Argon

In Fig. 3(b), when deactivation pressure is high (0.26–0.28 MPa),a sudden increase of superheat can be observed in our predictioncurve. That is because the temperature at deactivation pressureis higher than the saturation temperature at nucleation pressure,and thus a superheat already exists after depressurization, so thissuperheat becomes the predicted value when the IBS needed islower than it. Besides, in Fig. 3(c) the predicted values tend tounderestimate the experimental data especially for low deactiva-tion pressure, and this may be explained if the cavities don’t con-tain much inert gas due to some reason, like a long heatingprocess resulting in great loss of inert gas.

3.3. Dynamic effects

3.3.1. Temperature ramp effectThe dynamic effects in fact results from the interface speed, and

thus both variation of pressure and temperature may cause thiseffect. Boiling can be gained by depressurizing or heating, anddynamic effects may be observed in both ways to boiling. In thispaper, only the dynamic effects due to heating were studied, andtemperature ramp was used to show the dynamic effects. Theeffect of temperature ramp on superheat was carefully studiedby Dwyer et al. [7]. They separated the temperature ramp effectby fixing velocity and heat flux and gradually increasing inlet tem-perature, and their results showed a clear temperature ramp effect.In Section 2.3, an explanation has been made and a possible corre-lation to describe it has been proposed. Their main experimentalconditions are shown in Table 3.

The deactivation parameters were not clearly given in their pa-per, making it impossible to consider the deactivation radius andinert gas. Our analysis showed that because of the low pressure(0.027 MPa) and the small solubility of argon in sodium, the partialpressure of inert gas in cavities is less than 30 Pa around saturatedtemperature, and thus the effect of inert gas can be neglected. Sothe superheat can be estimated with the following equation [6]:

Tw � Tsat ¼2r

Rmin

Tsatðq�1v � q�1

l Þhlv

ð13Þ

where qv and ql are the densities of vapor and liquid, separately,and hlv is the latent heat of vaporization.

The values of cavity parameters and empirical coefficients usedin our calculation are shown in Fig. 4, and the values of Rmin can begained based on Eqs. (10) and (11) with temperature ramp known.It should be mentioned that the study of Dywer et al. [1] showeddistinct effect of heat flux on superheat, and they also found thatthe effect of temperature ramp increased with the increase of heatflux. To agree with their study, the value of a was changed withheat flux.

Fig. 4 shows the comparison of calculated curves with experi-mental data of Dwyer et al. [7]. The dispersion of the experimentaldata is large, and accurate prediction seems impossible. It can beseen that the trend of the effect of temperature ramp on IBS has

Table 3Experimental conditions of the temperature ramp effect [7].

Parameters Values

Test section I.D. (mm) 13.208Test section O.D. (mm) 23.622Heated length (mm) 292.100Heat flux (kW/m2) 157.393–629.571Velocity (m/s) 0.244Wall superheat (K) 18.333–133.333Boiling pressure (MPa) 0.027Fluid SodiumCover gas Argon

Fig. 4. Effect of temperature ramp on IBS with different heat flux (a) 157.4 kw/m2, (b) 314.7 kw/m2 and (c) 629.5 kw/m2.

532 Z. Ma et al. / International Journal of Heat and Mass Transfer 70 (2014) 526–535

been predicted well. The wall superheats at zero temperature rampare the same for different heat fluxes, which is consistent withexperiment.

In these figures, only a is adjusted according to heat flux. How-ever, it is difficult to understand that heat flux could influence a,and a possible reason will be given in the next section.

3.3.2. Effects of heat flux and velocityMany researchers have studied the effects of heat flux and

velocity [4,9–11,13,33], but their results are inconsistent. Consid-ering the dynamic effects, some of the experimental results maybe explained.

Supposing the increase of heat flux follows the exponentialfunction, it can be expected the wall superheat and temperatureramp will also vary over time in a way like the exponential func-tion. Considering the effect of temperature ramp, the requirednucleation superheat can be gained based on Eqs. (10), (11), and(13). Fig. 5(a) shows a schematic of the relationship between re-quired superheat and actual wall superheat.

It can be seen in Fig. 5(a) that when nucleation happens steadystate has already been reached, and this agrees with the research ofDwyer et al. [7]. Thus the wall superheat can be gained with thefollowing heat balance equation [7]:

Tw � Tsat ¼ Tin � Tsat þ4qL

Dhqvcpþ q

hð14Þ

where Dh is the thermal equivalent diameter, and h is the convec-tion heat transfer coefficient which can be obtained with existingcorrelations [34–37].

From the above equation, an increase of superheat as heat fluxincreases and velocity decreases can be expected. However,

because of the limited range of the required wall superheat, forlarge heat flux and small velocity, nucleation will happen beforesteady state is reached, and then the IBS will not change if heat fluxis larger than some value or velocity is smaller than some value,and experimental evidences about it may be found in [9]. Besides,if the range of the required superheat is narrow which means thedynamic effect is not apparent, the effects of heat flux and velocitymay be covered by scatter of experimental data.

The study of Singer and Holtz [6] shows that IBS decreases asheat flux increases due to diffusion of inert gas. Thus when consid-ering the effects of inert gas, the trend of the effect of heat flux maybe very complex [33].

Another kind of velocity effect can be related with slow heatingprocess, and the temperature ramp is small enough to allow theoccurrence of temperature ramp effect. If the heat flux variationis the same for different velocities, it can be expected that for smallvelocity the wall temperature ramp is larger (this can be estimatedwith Eq. (14)). Higher temperature ramp means smaller nucleationcontact angle (Eq. (10)) and smaller nucleation radius (Eq. (11)),and thus higher superheat can be expected (Eq. (13)). This kindof velocity effect is shown in Fig. 5(b). Considering our dynamic ef-fect theory, it can also be expected that the superheat will notchange if the velocity is large enough, making the dynamic effectdisappear because of very low temperature ramp. The velocity ef-fect found in the experiment conducted by Kikuchi et al. [11]where the temperature ramp was reported to be not very largemay be explained with it.

It should be mentioned that clear effects of heat flux and veloc-ity was found by Dwyer et al. [1] even if the temperature ramp waskept constant (6.48 � 10�3 K/s). Some parameters about theseexperiments are shown in Table 4, and other parameters like thetest section size are the same with those shown in Table 3.

Fig. 5. Heat flux and velocity effects on IBS due to dynamic effect (a) a schematic ofthe relationship between required nucleation superheat and actual wall superheatwith one-step heat flux and (b) a schematic of the velocity effect on nucleationsuperheat with heat flux varying slowly.

Fig. 6. Effects of velocity and heat flux on IBS.

Z. Ma et al. / International Journal of Heat and Mass Transfer 70 (2014) 526–535 533

Based on our theory about dynamic effects, it can be seen thekey factor influencing wall superheat is variation of wall tempera-ture rather than heat flux or velocity. Thus for small temperatureramp when a quasi-steady state might be anticipated, the com-bined parameter qv�1 (Eq. (14)) may be used to analyze the heatflux and velocity effect together approximately.

In Fig. 4, the values of a were changed according to the heat flux(velocity 0.244 m/s), and C was taken to be 1. Based on these data,the relationship between qv�1 and a was fitted with powerfunction:

a ¼ 6:12ðqv�1=1000Þ�0:364 ð15Þ

Fig. 6 shows the comparison of the calculated values gained withEqs. (10), (11), (13), and (15) and experimental data of Dwyeret al. [1]. It can be observed that generally the experimental datacan be well predicted with the combined parameter qv�1. This im-plies that the effects of velocity and heat flux with constant temper-ature ramp on nucleation superheat are identical in essence, andconsidering our dynamic effect theory, this may be explained withthe variation of surface tension coefficient. With small surface ten-sion coefficient, it should be easier for the interface to deform, andthis means the nucleation superheat may be related with the value

Table 4Experimental conditions of velocity and heat flux effect [1].

Parameters Heat flux effect Velocity effect

Pressure (MPa) 0.027 0.027Heat flux (kW/m2) 70–940 157.4, 629.5Velocity (m/s) 0.244 0.08–0.5Temperature ramp (K/s) 6.48 � 10�3 6.48 � 10�3

of wall temperature for sodium if the temperature ramp is not verylarge, and higher initial wall temperature may result in highernucleation superheat.

3.4. Trend of Parameter effects on superheat

Form above analysis, it can be seen that the effect trends of thethree basic factors mentioned in this paper can be determined in arelatively clear way. However, trends of some apparent parametersresulting from the three basic factors may not be so easily deter-mined, and their trends can also be very different under differentconditions. This may be an explanation of the inconsistence

Fig. A1. Estimation of inert gas concentrations (a) effect of initial inert gasconcentration and (b) effect of boundary inert gas concentration.

534 Z. Ma et al. / International Journal of Heat and Mass Transfer 70 (2014) 526–535

observed in different experiments in the literature such as the datapresented by Kottowski and Grass [38] where no definite conclu-sion was drawn about the effects of temperature–pressure history.

4. Summaries and conclusions

The effects of inert gas, oxide layer and dynamic process onnucleation superheat in liquid metals were studied in this paper,and some experimental results were explained with them.

The existence of inert gas may reduce the nucleation superheatsignificantly, but its effect may be very dependent on practicaloperation process and thus difficult to be determined accurately.However, in some cases, simplification is possible.

The non-wetting properties of oxide layer may reduce thenucleation superheat by keeping large cavities un-wetted andincreasing nucleation contact angle. Considering the surface condi-tions and oxide layer effects, the ratio of the deactivation radius tothe nucleation radius can be less than, equal to, or larger than 1.

The dynamic effects may be used to explain the temperatureramp effect. With temperature ramp effect, the heat flux effectand velocity effect found in many experiments may be explained,but the trend may be very different if the effects of inert gas areconsidered.

Acknowledgements

This work is supported by the Nuclear Energy DevelopmentProject of China (Grant No. HK�KD1003-4) and the NationalMagnetic Confinement Fusion Science Program (2010GB111007).

Appendix A. Estimation of inert gas concentration

As previously mentioned, careful consideration of inert gas isvery important to predict the IBS accurately, and the initial inertgas concentration and boundary gas concentration are very impor-tant factors for analysis of inert gas effects. In principle, the twofactors can be determined by measurement, and it can be expectedthat boundary inert gas concentration may be time-varying, andthat initial gas concentration may be very dependent on operationhistory. However, for theoretical calculation, these parameters areunknown in most cases.

Study of Winterton [39] showed that generally when a flowingsystem reaches equilibrium, the inert gas concentration is constantand the value is very dependent on the minimum saturationsolubility of the whole system. According to this, in some casesthe calculation of initial concentration can be simplified. If thedeactivation time is long enough, the initial concentration can thenbe estimated with the pressure and minimum temperature of thesystem.

For Chen’s experiment, because the temperature of the cool partof the system is not provided, and considering the aging effectcaused by inert gas diffusion, the inert gas concentration in fact in-volves much uncertainty and can only be estimated.

For single liquid, physical properties related with heat transferdo not change much when pressure varies. So for the series ofexperiment in Fig. 4 with constant deactivation temperature, highchances are that the temperature of cool part does not changemuch at deactivation state. Besides, in view of the existence oflow nucleation superheat when deactivation pressure is low, theinert gas diffusion aging effect may not be strong, so in ourcalculation the heating time was set to be very short to excludethe aging effect. Thus the initial inert gas concentration can beapproximately determined (see Fig. A1(a), and note that the oxidelayer effect has been involved).

When our model is used to predict the experimental data ofChen [19], the effect of boundary inert gas concentration withd = 3 mm is assumed to be negligible. To verify this assumption,different values of boundary concentration are employed to seethe possible effect (see Fig. A1(b)). It can be seen the effect ofboundary concentration is very small, and thus it is safe to neglectthe boundary inert gas concentration.

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