A Comprehensive Model for Liquid Film Boiling in Internal ... A Comprehensive Model for Liquid Film Boiling in Internal Combustion Engines ... heat transfer and phase changes are of

Oil & Gas Science and Technology – Rev. IFP, Vol. 65 (2010), No. 2, pp. 331-343Copyright © 2010, Institut français du pétroleDOI: 10.2516/ogst/2009062

A Comprehensive Model for Liquid Film Boiling in Internal Combustion Engines

C. Habchi

Institut français du pétrole, IFP, 1-4 avenue de Bois-Préau, 92852 Rueil-Malmaison Cedex - Francee-mail: [email protected]

Résumé — Un modèle complet pour l’ébullition de film liquide dans les moteurs à combustioninterne — Dans cet article, les principaux processus physiques, régissant les régimes d’ébullition nuclééeet de transition d’un film liquide, ont été examinés à partir des observations expérimentales disponiblesdans la littérature. Les tendances physiques observées, ont été utilisées, pour développer un modèle phénoménologique complet pour l’ébullition de film liquide (LFB). Celui-ci permet le calcul de sa vaporisation, dans le régime d’ébullition nucléée, ainsi que, dans le régime d’ébullition de transition. Cesrégimes sont identifiés par les températures de saturation, de Nukiyama et de Leidenfrost. L’estimationdes températures de Nukiyama et de Leidenfrost, en fonction de la pression de gaz ambiante, a requis uneattention toute particulière. Plusieurs courbes de durée de vie, de gouttes de grosse taille déposées sur une surface chaude dansdiverses conditions, ont été choisies parmi celles qui sont disponibles dans la littérature récente pour lavalidation du modèle LFB. Les résultats numériques montrent que les ordres de grandeur et les tendancesobservées expérimentalement sont bien respectés. En particulier, le modèle LFB reproduit bien la disparitionprogressive du régime de Leidenfrost avec des pressions suffisamment élevées du gaz. D’autre part,l’augmentation progressive du taux de vaporisation du liquide avec la rugosité du mur, précédemmentobservée expérimentalement près du point de Leidenfrost, a été correctement prédite par le modèle LFB.

Abstract — A Comprehensive Model for Liquid Film Boiling in Internal Combustion Engines — Inthis paper, the main physical processes governing the nucleate and transition regimes of the boiling of aliquid film were reviewed from the available experimental observations in the literature. The physicaltendencies observed in most experiments have been used to develop a comprehensive phenomenologicalLiquid Film Boiling (LFB) model which allows the calculation of the vaporization of liquid films in thenucleate boiling regime as well as in the transition boiling regime. These regimes are identified by thetemperatures of saturation, Nukiyama and Leidenfrost. A particular attention has been made concerningthe estimation of Leidenfrost and Nukiyama temperatures as a function of the ambient gas pressure.Several curves of lifetime of rather bulky droplets deposited on a hot surface under various conditionsand chosen among those which are available in the recent literature have been used for the validation ofthe LFB model. The numerical results show that the orders of magnitude and the tendencies observedexperimentally are well respected. Particularly, the LFB model reproduces well the progressive disappearance of the Leidenfrost regime observed in experiments with sufficiently high gas pressures. Inaddition, the gradual increase of the vaporization rate with wall roughness which was previouslyobserved experimentally near the Leidenfrost point has been correctly predicted by the LFB model.

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Oil & Gas Science and Technology – Rev. IFP, Vol. 65 (2010), No. 2

INTRODUCTION

In automobile engines, the technology of direct injection inthe combustion chamber is used more and more because itallows controlling the consumption of combustible better incomparison to port fuel injection engines. Besides, the pres-sures used in the injection systems are increased more andmore to improve the spray atomization, mixture preparationand combustion. In these conditions, liquid spray mayimpinge the internal surface of the combustion chamber. Thisinteraction of spray with the wall involves different physicalphenomena according to the conditions of impact (speed ofthe droplets, temperature of the wall, roughness of the sur-face, etc.) and of the pressure of gases in the combustionchamber [1-7].

In diesel engines for instance, injection is often made inthe course of the compression of fresh gases. As the satura-tion temperature of fuel oil increases with the pressure ofgases, the impact of spray on the wall leads in most cases tothe formation of a liquid film. The latter evaporates ratherslowly and may even survive to the combustion [8]. In thecourse of the expansion stroke, the saturation temperature offuel oil diminishes with gas pressure to the point of attaininga value lower than the wall temperature and consequentlyleading to the boiling of the liquid film. Furthermore, theLeidenfrost and Nukiyama temperatures diminish as wellduring the expansion stroke leading to a continuous variationof the boiling regimes. Very few works in the literature werededicated to model these phenomena for shallow liquid films.Based on available experimental observations in the litera-ture, this paper proposes a synthetic vision of the physicalprocesses taking place during the different modes of boilingof a thin liquid film. The physical tendencies observed inmost cases with experiments of type “pool boiling” and “ses-sile droplet” or deposited droplet on a hot wall, are discussedin the first section. Next, the identified main physical para-meters of the boiling process are modeled as functions of thewall temperature. The equations of the Liquid Film Boiling(LFB) model are then presented. The third section is dedi-cated to the LFB model validation. A particular attention ispaid to the estimation of Leidenfrost and Nukiyama tempera-tures. Finally, experimental lifetime curves of droplets havingdiameters larger than 2 mm and which were deposited on ahot wall were used to validate the LFB model in differentconditions including the variations of wall roughness [9] andambient gas pressure [10].

1 PHYSICS OF BOILING

Among the physical processes occurring when a liquid filmis formed on the surface of a combustion chamber of a directinjection engine, heat transfer and phase changes are of pri-mary importance. The vaporization rate of the liquid film is

not only a function of the wall temperature but it is stronglyaffected by the gas pressure variation inside the combustionchamber. Indeed, processes occurring during the pistonexpansion stroke are similar to those occurring when increas-ing the wall temperature Tw. In this last case, one may distin-guish three or four regimes or modes of vaporization in theliterature [1, 11, 12], which can be classified according to theextent of superheating of the wall ΔTwsat = (Tw – Tsat) andusing as limits for these regimes, the saturation temperatureTsat, the Nukiyama temperature TN and the Leidenfrost temperature TL. The two last critical temperatures are usuallydetermined experimentally from the boiling or lifetime curveof a droplet (Fig. 1). This curve is obtained by measuring thetotal time that it takes a droplet to completely evaporate afterit has been gently deposited on a hot wall [13-15]. The differ-ent boiling regimes are discussed in the following paragraphsin order to emphasize the data available for the modeling ofthe boiling processes and their validation.

Mode I, the complete wetting regime when Tw < Tsat

The spray, impinging a wall having a temperature Tw smallerthan the saturation temperature, forms a liquid film whichevaporates slowly. In this mode, the evaporation rate stronglydepends on the turbulence level in the ambient gas [16-18].

Mode II, the nucleate boiling regime when Tsat < Tw < TN

In this regime, the liquid very close to the wall (e.g. in thethermal boundary layer) is overheated. This leads to the for-mation of vapor cavities which are born starting from germs(or sites of nucleation) often hidden in the wall roughness.While growing, these cavities are detached and form smallbubbles. This process of phase change prevents the tempera-ture of the liquid Tl from exceeding its temperature of satura-tion by consuming most of the heat flux Qwl transferred to theliquid by the wall. The experimental observations ofCornwell and Brown and Singh et al. (quoted in the review

Figure 1

Droplet lifetime curve and schematics illustrating the boilingstates near the limits of the boiling regimes.

Tsat TN TL Tw

Regime ILiquide filmevaporation

Regime IINucleateboiling

Regime IIITransition

boiling

Regime IVFilm

boiling

Dro

plet

life

time

332


C Habchi / A Comprehensive Model for Liquid Film Boiling in Internal Combustion Engines

of Dhir [12]) showed that in the nucleate boiling case of waterunder atmospheric conditions, the number of vapor cavitiesNc is at the same time proportional to ΔT 2

wsat and inverselyproportional to the square of the diameter of the cavities, D2

c.Thus, the bubbles formed have four main effects:

• They increase the dry fraction of the wall, αdry:

(1)

where Aw is the total wetted surface in the absence of boiling. The value of αdry which corresponds to the minimallifetime duration of a droplet (defined when Tw = TN) wasestimated by several authors in order to calculate the boilingcurve and in particular the Critical Heat Flux (CHF). Themeasured value of the dry fraction by Iida and Kobayasi[19] is αdry(Tw ≈ TN) ≈ 0.9 which is approximately the sameas the value obtained by the model of Nishio and Tanaka[20]. One can also get an estimate of αdry by usingEquation (1). Indeed, assuming that the maximum valueof the cavities diameter Dc is attained when Tw = TN (i.e.just before the beginning of the collapse of the cavities):

has been obtained [20]. Furthermore, Dhir and Liaw [21]found that during pool boiling, αdry (Tw ≈ ΤΝ) variesbetween 0.68 and 0.8 when contact angles vary between27 and 90 degrees respectively.

• They accelerate the evaporation of the liquid at the wall.Indeed, several authors noted this acceleration. Dhir [12]has noticed that in the case of nucleate boiling of water,most of the evaporation is carried out in the periphery ofthe vapor cavities close to the contact lines. The lengthdensity of these contact lines is then a parameter of greatimportance as far as nucleate boiling is concerned. Thisparameter (referred to below as Clld) corresponds to thesum of the perimeters of the cavities or dry zones.

While following the same reasoning, Nishio and Tanaka[20] considered that the CHF point and the point of mini-mum lifetime (Tw = TN) correspond to the same conditionsfor which contact line length density is maximum. Thismaximum contact line length density value (denotedClld

max) was found close to 3000 m/m2 at Tw = TN. As a matter of fact, Clld increases between Tsat and TN, thendecreases between TN and TL in a way similar to the boil-ing curve. It worth noting that Nishio and Tanaka haveused an optical technique to visualize the liquid-wall con-tact zones and the boiling structures for high heat-fluxpool boiling conditions. The test liquid was R141b at

atmospheric pressure and the boiling surface is a sapphireplate coated with an electro-conductive film. Althoughthese experimental conditions are not really representativeof those encountered in piston engines, the paper ofNishio and Tanaka [20] is the only reference that theauthor has found in the literature which provides experi-mental values of Clld.

• In the case of an horizontal wall, the bubbles can go uptowards the liquid-gas interface. In this case, the collapseof these bubbles at the liquid surface leads to the produc-tion of tiny droplets which are ejected in the gas andwhich are characteristic of this boiling regime. Dhir [11]also underlines that the bubbles act as a pump whichmoves away the hot liquid from the wall. These phenom-ena are important in the case of “pool boiling” and do notseem to be important for shallow liquid films which are ofinterest in this study.

• They cool the wall by pumping the latent heat of vaporiza-tion necessary for the phase change especially when thetemperature of gas is lower than the temperature of thewall. This process is of course desired for the coolingtechnologies. Indeed, most of the references cited previ-ously have aimed at better understanding the CHF condi-tions in order to optimize cooling systems.

Mode III, the transition boiling regime (TN < TW < TL)In this regime, the small cavities on the wall and the bubblesmay coalesce and form vapor columns and/or a larger vaporpocket. Several authors (see [11, 20, 22]) explained thedecrease of the heat flux provided by the wall and theincrease of the droplet lifetime, by the coalescence betweenthem of the vapor cavities on the wall. This process can beregarded as the initial phase of the subsequent Leidenfrostregime during which a vapor cushion completely preventsthe contact of the liquid with the wall. As a consequence, inthis transition boiling regime, the heat flux passes more andmore through a vapor cushion between the wall and the liq-uid, and therefore strongly increases the droplet lifetime.

Mode IV, the Leidenfrost regime when Tw > TL

This regime which is often called “film boiling regime” is theresult of intense evaporation which leads to the formation of avapor cushion and prevents the contact of the liquid with thewall as just said above. The lifetime curve (Fig. 1) shows thatthe evaporation rate in this regime is less intense than in thepreceding boiling regimes. This observation is all the moretrue as the ambient gas pressure is low, particularly when it islower than 1 bar. Indeed, Temple-Pediani [23] and morerecently Stanglmaier et al. [10] studied the effect of the ambi-ent gas pressure on the evaporation of a droplet posed on a hotwall. They showed that the Leidenfrost regime is less and lessimportant as the gas pressure increases. Furthermore, theexperimental visualizations of Chaves et al. [2] and Moita andMoreira [7] show a cloud of droplets above the wall, which

C Dlld c

i

Nc

i≈

=

∑π1

αdry w NT T( ) .≈ = ≈π

40 8

αdrydry

w

c

i

Nc

w

A

A

D

A= = =

∑ π 2

1 4

333



levitates under the effect of the Stefan flow induced by theevaporation. Therefore, in this boiling regime, the wall heatsthe vapor cushion which in its turn heats the liquid. The heatflux can be written in this case as follows:

(2)

where λv and δv are respectively the thermal conductivity of thevapor and the thickness of the vapor cushion. In addition, Fardadand Ladommatos [9] underlined the influence of the nature ofthe wall (roughness, adsorption, wettability, color, type of metal,etc.) on the Leidenfrost regime. In particular, they observed thatroughness decreases the lifetime duration of the droplet. Theyexplained this phenomenon by saying that the peaks of rough-ness contribute to increase the heat flux by wall-liquid directcontact and thus lead to a larger evaporation rate (Fig. 2).Moreover, the increasing importance of the wall roughnesswith surface temperature has been underlined by Moita andMoreira [7]. From their experiments on the disintegrationmechanism of impinging droplets onto heated rigid surfaces inthe Leidenfrost regime, they also reported that the wall rough-ness is as important as the dynamic vapor pressure (i.e. theStephan flow) associated with the rate of vaporization.

2 LIQUID FILM BOILING MODEL

According to the experimental evidence summarized above,the intensity of the boiling phenomenon seems to be gov-erned by two main parameters: – the dry fraction of liquid film area due to boiling αdry;– the length of the contact lines of the liquid film Clld.

The new model of Liquid Film Boiling (referred to below asLFB) which is presented in this paper is based on different phe-nomenological correlations. The two first describe the behav-iour of the above parameters according to each boiling regime.

First, two dimensionless functions have been defined: thefunction for the dry fraction αdry and the function for thelength density of the contact lines of the liquid film, kclld. Theway according to which these functions were obtained isgiven below in Sections 2.1 and 2.2, respectively.

Figure 2

Schematics illustrating how the surface roughness promotesthe liquid-wall contact in the Leidenfrost regime.

2.1 Function of Dry Fraction

The dry fraction αdry is defined by Equation (1). It variesinevitably between 0 and 1 when the wall temperatureincreases from the saturation temperature to the Leidenfrosttemperature respectively. Moreover, the experimental obser-vations mentioned in Section (1) showed that αdry increasesquickly during the nucleate boiling regime. It reaches a valueof about 0.7 to 0.9 when Tw = TN. This behavior of the func-tion αdry(Tw) has been expressed by the following relation:

(3)

where T* is defined by:

(4)

and αPLdry is the value of αdry at the Leidenfrost point (i.e.

when Tw = TL ). A value of αPLdry = 0.98 is used as we

considered that the liquid takes off completely from the wallshortly after the Leidenfrost point.

2.2 Function of Length Density of Contact Lines

In the nucleate regime of boiling, the length of contact linesincreases as the bubbles diameters increase at the wall surface. Hence, the function of length density of contact lines(referred to below as Clld (Tw)) will be assumed to follow thesame evolution as the function of dry fraction αdry(Tw). Themaximum value of Clld (Tw) function is assumed to take placeat the Nukiyama point (i.e. when Tw = TN). In these condi-tions, Nishio and Tanaka [20] have measured values more orless equal to Clld

max ≈ 3000 m/m2. Then, in the transitionregime of boiling, Clld (Tw) decreases towards a minimalvalue at the Leidenfrost point (i.e. when Tw = TL), by effectof coalescence of the various nucleation cavities and by theprogressive formation of a vapor cushion (see the corre-sponding diagrams in Fig. 3). This minimal value Clld (TL)depends on the nature of the wall and its roughness in partic-ular. To model this behavior according to the temperature ofthe wall, a dimensionless function of length density of contact lines kclld is defined in the following way:

(5)

This function increases from 0 to 1 during the nucleateregime of boiling. Then, it decreases in the transition regimeof boiling up to a minimal value denoted kmin

clld . This behav-iour is specified in the nucleate boiling regime by the follow-ing relation:

(6)

where αdry is given by Equation (3) and αPNdry = αdry (Tw = TN).

k TT

clld wdry w

dryPN

( )( )

=α

α

k TC T

Cclld wlld w

lld

( )( )max

=

TT T

T Tw sat

L sat

* =−−

α αdry w dryPLT T( ) *=

14

a) Smooth wall b) Rough wall

QT T

wvl vw l

v

=−

λδ

( )

334



Figure 3

The function of dry fraction αdry and the function of lengthdensity of contact lines kclld drawn in the iso-octane case atatmospheric pressure. The schematics illustrate the boilingstates near the limits of the boiling regimes.

In the transition boiling regime, kclld is supposed todecrease quickly towards kmin

clld . This behavior is specified bythe following expression:

(7)

where:

(8)

The kclld function (Eq. 6 and 7) is drawn in Figure 3 alongwith the αdry function (Eq. 3) using data of iso-octane (Tab. 1). The diagrams in the lower part of this figure showthe physical behaviours assumed to take place at the limits ofthe boiling modes, in particular close to the critical points ofsaturation, of Nukiyama and of Leidenfrost. The value of kmin

clld is estimated according to surface roughness in the following paragraph.

2.2.1 Influence of the Surface Roughness on Boiling

Most of previous works on this issue (for instance those ofBernardin et al. [3] and Fardad and Ladommatos [9]) havereported that surface features promote liquid-wall contact anddecrease droplet lifetime during the Leidenfrost regime. Thissuggests that rough surfaces require a thicker vapor cushionbetween the droplet and the surface to sustain film boiling.Thus, the influence of the average roughness of the wall Ruhas been introduced into the LFB model via the value of kmin

clld

in Equation (7). For perfectly smooth walls, we assumed atotal levitation of the liquid film above a vapor cushion at theLeidenfrost point. This assumption leads to kmin

clld = 0 when

Tw = TL. For rough surfaces, the observed increase of liquid-wall heat transfer has been modelled by setting:

(9)

where kRu1 and kRu2 are two constants to be adjusted usingexperimental data (cf. Sect. 3.2).

2.2.2 Influence of the Gas Pressure on Boiling

The gas pressure is an important parameter in the evaporationof liquid films as we underlined above in the bibliographicalreview (cf. for instance [9] and [23]). The increase in thepressure of gas induces two main effects:

1st gas pressure effectTemple-Pediani [23] noted the absence of the vapor cushionwhich is the main characteristic of the Leidenfrost regime, ifthe gas pressure exceeds the critical pressure of the fluid. Heexplained the absence of the vapor cushion by the fact thatthe vapor pressure becomes insufficient to raise the liquid farfrom the wall. In other words, an increase in the gas pressureleads to a reduction of the rate of vaporization. In the LFBmodel, the reduction of the thickness of the vapor cushion asa function of the gas pressure is expressed by the followingrelationship:

(10)

where p0 is taken equal to the atmospheric pressure (~1 bar),p is the ambient pressure of gas, and kpres is the thickness ofthe vapor cushion when p = p0. Its value shall come fromavailable experimental results (cf. Sect. 3.2). The squareexponent of the pressure ratio was selected empirically usinga fitting process ensuring a decrease of the vapor cushionthickness δv slower than the growth of the gas pressure inaccordance with the experimental tendencies (cf. Sect. 4.3).

2nd gas pressure effectA shift of the lifetime curve towards the right side (i.e.towards the high wall temperatures) is observed when the gaspressure is increased. This shift involves Tsat, TN and TL at thesame time. On the one hand, the variation of the saturationtemperature with pressure can be easily obtained using theClausius-Clapeyron formula, at least for single-componentliquid film. On the other hand, the shifts of TN and TL must beformulated according to the experimental observations avail-able. This is going to be discussed in the following paragraph.

2.3 Estimation of TN and TL Temperatures

On the contrary of the saturation temperature, Tsat, the temperatures of Nukiyama TN and Leidenfrost TL are not thermodynamic temperatures. Let us recall that TN corre-sponds to the point of maximum heat flux (CHF) or to the

δv preskp

p=

⎛

⎝⎜

⎞

⎠⎟0

2

kk

RuclldRu

dryPN

kRumin = 1 2

α

TT T

T TL w

L sat

** =−−

k T k T kclld w clld clld( ) min**

min= −( ) +1 4

Wall temperature (K)

1.0

0.8

0.6

0.4

0.2

0380 400 420 440 460

Tsat TN TL

TN = 395 K

0.68

Dry

frac

tion,

αdr

y an

d co

ntac

t lin

e le

ngth

den

sity

(k c

lld)

αdrykclld

Iso-octane case

335



minimum point of the lifetime duration; and TL corresponds tolocal minimum of the curve of the heat flux or to the maxi-mum of the lifetime duration curve separating the regimes ofLeidenfrost and transition boiling, as shown in Figure 1. Asindicated above, these critical temperatures depend on severalfactors [9, 11] including the properties of the wall (roughness,presence of impurity, wettability, thermal conductivity, spe-cific heat, etc.) as well as the gravity, the gas pressure, the liquid flow rate, etc. One can find in the articles of Bernardinet al. [4-6] a complementary bibliographical review on thissubject. In fact, previous research works have generallyignored the effects of the gas pressure on the TN Temperatureand the available Leidenfrost temperature models have notmet reasonable success. Moreover, few theoretical worksdealt with the evaluation of the TL temperatures at high pres-sures. First, let us recall the low pressure (P/Pc << 1)metastable liquid model of Speigler et al. [24] in which theycorrelated the critical temperature to the limiting temperatureof homogeneous nucleation of an overheated liquid, assumedclose to the TL temperature, as follows:

(11)

In addition, for conditions of high pressures up to the criticalpressure, Lienhard [25] suggested the following correlation:

(12)

This relationship describes the convergence of the point ofLeidenfrost towards the critical temperature. Indeed, thisbehaviour was observed in experiments by Emerson andSnoek [26] who studied the effects of the pressure on theLeidenfrost point using Freon and water droplets. Theyshowed that when the pressure increases, the Leidenfrostpoint of water on brass grows up to a value higher than thecritical temperature, and then it decreases until the criticaltemperature when the pressure reaches the critical pressure.This tendency was confirmed experimentally by Breuer et al.

[27]. These authors showed that the temperature ofLeidenfrost increases with the gas pressure. Moreover, itbecomes equal to the saturation temperature as soon as thepressure exceeds half of the critical pressure.

Another interesting experimental work was publishedrecently by Fardad and Ladommatos [9]. They showed that(TL – TN) is very close to (TN – Tsat) for several liquidsincluding a gasoline and Diesel. Moreover, the differences(TL – Tsat) and (TN – Tsat) decrease when the gas pressureincreases. In addition, they seem to be constant when the gaspressure decreases under the atmospheric pressure. Thisbehavior are formulated by the following expressions:

(13)

where Tcr represents either TN or TL and ΔT is calculatedaccording to the gas pressure P in the following way :

(14)

where Tb and Tc are respectively, the normal boiling tempera-ture and the critical temperature. For the high pressure case,the value of ΔT is assumed to tend linearly towards the valueA = Max (1, Tcr⎢1bar – Tc) when the gas pressure tends towardsthe critical pressure. Finally, let us note that the values ofTcr⎢P = 1 bar (i.e. TN and TL at P = 1 bar) are supposed to be provided experimentally by lifetime curves of fuel droplets.Table 1 gathers the whole of the TN and TL data which wehave found in the literature for fuels at atmospheric pressure.Whenever experimental values are not available, TL|1 bar maybe estimated using Equation (11) but this may lead to a pooraccuracy, and TN|1 bar could be estimated as follows:

(15)TT T

Nb L=

+2

ΔT

T Tcr b

=

−1 bar if 1 bar

bar

P

T T A

T TT T

cr b

c b

c sa

≤

−( ) −

−−

1

tt A P( ) +

⎧

⎨⎪⎪

⎩⎪⎪

if > 1 bar

T T Tcr sat= + Δ

T TT

TL csat

c

= +⎛

⎝⎜

⎞

⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

0 905 0 0958

. .

T TL c=27

32

336

TABLE 1

Nukiyama and Leidenfrost Temperatures obtained at atmospheric pressure. The experimental conditions given in the original papers are also recalled

Fuels TN (K) TL (K) References and experimental conditions

n-pentane 325 390

iso-octane 395 463 [10]: Impact of drops (d0 = 2 mm) on aluminum heated plate

n-decane 473 526

n-heptane 423 483[9]: Impact of drops (d0 = 2.34 mm) on aluminum heated plate. Average

Roughness Ru = 0.22

Diesel 643 733 [9]: Impact of drops on aluminum heated plate. Average roughness Ru = 0.22

Diesel 629 708 [13]: Impact of drops (d0: 0.19 to 0.42 mm) on stainless steel heated plate



2.4 Equations for the Vaporized Mass and the WallHeat Flux

The vaporized mass per unit area is calculated by the follow-ing energy balance:

(16)

where Lsat is the fuel latent heat of vaporization at the temperature Tsat. Qwl is the heat flux per unit area betweenthe wall and the liquid in direct contact. Qwvl is the heat fluxper unit area between the wall and the liquid through thevapor cushion. This last becomes dominant compared to thefirst when the temperature of the wall tends towards theLeidenfrost temperature. On the one hand, Qwl is calculatedby the following relation:

(17)

where λl, sat is the thermal conductivity of fuel at the tempera-ture Tsat and δth is a characteristic length of order of the thick-ness of the thermal boundary layer in the liquid film near thewall. δth is the most important dependent parameter and isdefined in Section 3.2. On the other hand, Qwvl is calculatedby Equation (2) with a thickness of vapor cushion given byEquation (10).

In Equation (16), the β1 parameter represents the fractionof the wetted area (1 – αdry) where most liquid vaporization isproduced. In order to estimate β1, let us recall that the vapor-ization of a liquid film in boiling modes occurs in a preferen-tial way by an area band along the contact lines. At the CHFpoint, this area band around the film is proportional toClld

max⋅ hf if the contact angle θ is equal to 90 degrees. Hence,the coefficient of proportionality β1 mainly depends of θ. Inaddition, by assuming that β1 is proportional to the square ofkclld, this parameter can be written in the following way:

(18)

The second parameter β2 in Equation (16) represents the fraction of dry area αdry where the liquid film is separatedfrom the wall by a vapor cushion. As the theoretical determi-nation of β2 is difficult, it will be regarded as an adjustableparameter of the LFB model (see Sect. 3.2).

3 VALIDATION OF THE LFB MODEL

The determination of the parameters of the LFB model andits validation were carried out mainly on the basis of experimental data resulting from the articles of Stanglmaier

et al. [10] and Fardad and Ladommatos [9]. In these experiments, the lifetime durations of rather bulky droplets (d0 > 2 mm) deposited on a hot surface were measured undervarious conditions. On the one hand, we retained from thearticle of Stanglmaier et al. [10] the lifetime duration curvesthat were measured using droplets of n-pentane, n-decaneand iso-octane. For this last hydrocarbon, lifetime curvesobtained using several ambient pressures are also available.On the other hand, we retained from [9], the lifetime curvesof droplets of n-heptane deposited on surfaces of differentaverage roughnesses.

3.1 Initialization Method of the Liquid Film

Once deposited gently on the wall, a droplet spreads andforms a liquid film of a maximum diameter (denoted Dmax)up to 3 to 5 times the initial diameter d0 depending on thevalue of the droplet Weber number [9, 28, 29]. In this work,we assumed:– The droplet after impact takes the form of a cylinder with

radius rf— and height hf

—.

– The vaporized mass before the droplet takes the cylindri-cal form is negligible relatively to the mass of the droplet.In order to calculate rf

— and hf—

, we first used the model ofNagaoka et al. [30] which can be summarized by the follow-ing system of equations:

(19)

while denoting r0 = d0/2 and hf max the height of the cylinderwhich has a radius rf = Dmax/2 (see the diagram of Fig. 4 forthe notations). Then, one can evaluate the minimal height by:

Finally, hf—

and rf— are obtained by:

(20)

In Table 2, we summarise the initial characteristics of liquid films corresponding to the droplets used byStanglmaier et al. [10] for n-pentane, iso-octane and

hh h

rV

h

ff f

f

f

=+

=

max min

π

2

0

hV

rV rf

f

min πwith π= =0

2 0 034

3

rr

h r

f

h f

3 03 3

3

4

2 3

1

=− +

=− (

sin

cos cos

cos/max

φφ φ

φ ))+ ( )

=

2

2

00 51

1

0 0354

h r

h

r E

h f

h

f

and

max

max.

/

. E

g f g

f

0

4=

−( )ρ ρ

σ

βθ1

2=( )

Ch

klldf

clldmax

sin

QT T

wl l satw sat

th

=−( )

λδ,

�m

Q Q

Lvdry wl dry wvl

sat

=− +β α β α1 21( )

337



Figure 4

Numerical initialization of a droplet deposited on a wall. a) Schematic and notations for Nagaoka model [30]. b) Minimal film thickness, hf min definition.

n-decane, and by Fardad and Ladommatos [9] for n-heptane.The ratio of the maximum spreading radius to the initialdroplet radius just prior to impact, rf

—/r0 is more or less equalto 4. This large value may indicate that Stanglmaier et al.[10] and Fardad and Ladommatos [9] have probably usedsimilar and relatively large Weber numbers [28]. The experi-mental Weber numbers were not given in their articles.

TABLE 2

Initial conditions for the calculations of lifetime durations for 4 liquid fuel droplets using the LFB model

n-pentane iso-octane n-decane heptane

V0 (μL) 5 5 5 6.7

r0 (mm) 1 1 1 1.17

rf— /r0 4.1 4.28 4.32 4.1

Af = πrf—2 (mm2) 53.3 57.4 58.7 72.3

hf—

(μm) 94 87 85 93

3.2 Choice of the LFB Model Parameters

The parameters of LFB model which remain to be determinedare: δth , β2, kRu1, kRu2 and kpres.

δδth estimationδth is a characteristic length for the calculation of the heatflux between the wall and the liquid in direct contact (Eq. 17). Its value is of the order of the thickness of the ther-mal boundary layer in the liquid on the wall. It dependsmainly on the thermo-physical properties of the liquid. Forthe estimation of δth, we considered that the Critical HeatFlux (CHF) comes primarily from the heat flux between thewall and the liquid in direct contact. Indeed, Temple-Pediani[10] showed that the heat flux dominating is that whichcomes from the wall by direct contact with the liquid,

especially near the CHF. Under these conditions, the vapor-ization rate at Tw = TN can be written as follows:

(21)

In addition, Stanglmaier et al. [10] has provided an estimation of this vaporization rate: ≈ 20 mg/s in thecase of iso-octane. Moreover, Equation (3) gives αdry ≈ 0.68and kclld = 1 at Tw = TN (Fig. 3). Then, using Equation (18)and assuming an average contact angle θ equal to 150 degrees [29, 31], one can obtain β1 ≈ 0.5. We finallyused the physical properties of iso-octane at atmosphericpressure (Tab. 3) in Equation (21) to obtain δth ≈ 3 μm. Inpractice, we used δth = min (hf, 3 μm) to remain coherentwith film thicknesses lower than 3 μm.

TABLE 3

Physical parameters of iso-octane at atmospheric pressure used to calculate δth using Equation (21)

λl, sat (W/(m.K)) Lsat (J/kg) Tsat (K) TN (K)

0.1 2.67 × 105 372 395

ββ2 estimationβ2 is the fraction of αdry where the liquid film is separated fromthe wall by a vapor cushion in Equation (16). This means that(1 – β2)αdry is the fraction of the initial liquid film area Af whichis already completely dry. In this work, β2 is supposed to beconstant during the boiling transition regime. We adjusted itsvalue on the experimental lifetime curve duration of iso-octaneat atmospheric pressure from Stanglmaier et al. [10]. The valueβ2 = 0.055 gives the best fit as shown in Figure 5.

Figure 5

Results for β2 parameter adjustment using the curve ofexperimental lifetime of iso-octane. The experimental datawas taken from [10].

20

15

10

5

0380 400 420 440 460

Wall temperature, Tw (K)

Tota

l dro

p lif

etim

e (s

)

Experiment(iso-octane: P = 101 kPa)β2 = 0.045β2 = 0.055β2 = 0.065

�mCHF

�m

L

T T ACHF

dry

sat

l sat N sat f

th

=− −β α λ

δ1 1( ) ( ),

�mCHF

a) b)

φ φ rfrf

hf max hf min

338



kRu1 and kRu2 estimationThey are the two parameters in Equation (9) relating thedimensionless function of length density of contact lines kclldto the average roughness of the wall. For the determination of these two coefficients, we used, once again the experi-mental curve of lifetime duration of the iso-octane droplet.We proceeded by trial and error and by supposing that thealuminium block on which Stanglmaier et al. [10] depositedthe droplets is relatively smooth (Ru = 0.2 μm). The resultspresented in Figure 5 were obtained using kRu1 = 1 and kRu2 = 0.2.

kpres estimation

kpres is a characteristic thickness of the vapor cushion δvseparating the liquid from the wall in the Leidenfrostregime (see Eq. 2). Actually, few theoretical works havebeen devoted to the effect of pressure on the thickness ofthe vapor cushion δv in the Leidenfrost regime. Using a the-oretical model, Rein [1] showed that the characteristic valueof the thickness of the vapor cushion in the Leidenfrostregime is between 10 and 100 μm. In the same way, Yangand Fan [32] have recently developed a numerical modelfor the simulation of the impact of a droplet on a plane platein the Leidenfrost regime. They calculated the profile thick-ness of the vapor cushion during the impact. They found inparticular that the value of δv varies during the impactbetween 120 μm and 10 μm. In addition, the experiments ofChandra and Aziz [33] and Kistemaker [34] do corroboratethis order of magnitude. Thus, we considered in the LFBmodel, δv = 100 μm for a pressure p0 = 1 bar. This valueleads to kpres = 100 μm in Equation (10). For a supercriticalgas ambient pressure (saying for a fuel, p = 20 bars),

Equation (10) gives δv = 0.25 μm. This small value is inconformity with the experimental observations of Temple-Pediani [23].

3.3 Sensitivity of LFB Model to the Values of TN and TL

We already presented in Section 2.3 Equations (13) and (14)following the estimation of the temperatures of Nukiyama TNand Leidenfrost TL according to the ambient pressure valueP. The experimental values of these temperatures are notavailable for all fuels. In the following two subsections, thesensitivity of the results of the LFB model to the given valuesto these two temperatures is studied.

3.3.1 Sensitivity to the TN Value

Two simulations were carried out with the LFB model in thecase of an iso-octane droplet at atmospheric pressure. Thefirst simulation uses the experimental value of the tempera-ture of Nukiyama (TN,exp = 395 K) and the second uses thefollowing estimation: TN = (Tsat + TL,exp)/2 = 419 K, asshown in Figure 6. One can see that the increase by 6% ofthe value of TN results in a delaying of the nucleate boilingand the transition boiling regimes. For the latter regime, thisdelaying has led to an overestimation of the vaporized mass.

3.3.2 Sensitivity to the TL Value

In the same way as described previously, two simulationswere carried out with the LFB model in the case of a n-heptane droplet at atmospheric pressure. The first simula-tion uses the experimental value of the temperature ofLeidenfrost (TL,exp = 483 K) and the second uses TL = 456 K

339

20

15

10

5

0380 400 420 440 460


Tota

l dro

p lif

etim

e (s

)

Experiment(iso-octane: P = 101 kPa)TN,exp = 395 KTN = (TL,exp + Tsat)/2 = 419 K

20

15

10

5

0380 400 420 440 480460


Tota

l dro

p lif

etim

e (s

)

Experiment(heptane: P = 101 kPa)TL,exp = 483 KTL,exp = 456 K = 27 Tc/32

Figure 6

Sensitivity of the results of LFB model according to the valueof the temperature of Nukiyama TN in the case of an iso-octane droplet at atmospheric pressure. The experimentaldata was taken from [10].

Figure 7

Sensitivity of the results of LFB model according to the valueof the temperature of Leidenfrost TL in the case of a n-heptane droplet at atmospheric pressure. The dashed curvewas obtained using Equation (11). The experimental data wastaken from Fardad and Ladommatos [9].



obtained using Equation (11). This equation results in anunderestimation of more than 5% on TL values. This errorleads to a shortening of the transition boiling regime due to anunderestimation of the vaporized mass as shown in Figure 7.

4 DISCUSSION OF THE RESULTS OF THE LFB MODEL

Initially, let us consider again the case of the iso-octanedroplet studied in experiments by Stanglmaier et al. [10].Figure 8 shows the curves of total wall-liquid heat flux andof vaporized mass obtained by the LFB model in the atmos-

pheric pressure case. It is worth noting that the maximumvaporized mass (~20 mg/s) obtained at the Critical Heat Flux(CHF) is obviously in a satisfactory agreement with theexperimental value of Stanglmaier et al. [10]. Moreover, theorder of magnitude of the CHF is more or less equal to 120 kW/m2. This value is in line with the experimental obser-vations met in the open literature.

In addition to the classical heat flux in the wetted wallzone, the main characteristic of the LFB model lies in theexplicit consideration of the heat flux between the wall andthe liquid through the vapor cushion. Figure 9 shows that

340

150

100

50

0380 400 420 440 460


Iso-octane case, P = 101 kPa

TN

Total wall heat flux, kW/m2

Mass flow rate, mg/s

CHF

1.0

0.6

0.2

0.8

0.4

0380 400 420 440 460


TN = 395 K

β 2αdryQ

wvl/Q

total Iso-octane case

Figure 8

Total heat flux given by the wall to the liquid film (solid line)and vaporized mass (dashed line) curves in the case of an iso-octane droplet with atmospheric pressure.

Figure 9

Fraction of the total heat flux that passes through the vaporcushion during the nucleate and transition boiling modes.Results obtained in the case of iso-octane with atmosphericpressure.

20

15

10

5

0380 400 420 440 480460


Tota

l dro

p lif

etim

e (s

)

Experiment: Ru = 0.22 μmLFB model: Ru = 0.22 μmExperiment: Ru = 2 μmLFB model: Ru = 2 μmExperiment: Ru = 5 μmLFB model: Ru = 5 μm

1.0

0.6

0.8

0.4

0.2

0380 400 420 440 480460


Con

tact

line

leng

th d

ensi

ty fu

nctio

n (k

clld

)

Ru = 0.2 μmRu = 2 μmRu = 5 μm

TN

Figure 10

Influence of surface roughness on the lifetime of an iso-octane droplet. Comparison of the LFB results with themeasurements of Fardad and Ladommatos [9].

Figure 11

Influence of the roughness on the function of contact lineslength density kclld(Tw).



the percentage of this last heat flux increases during thetransition boiling regime, in a nonlinear way up to 90% ofthe total heat flux (Qtotal = mv Lsat) at the Leidenfrost point.In fact, the value of this percentage at the Leidenfrost pointdepends on a multitude of factors among which roughnessthat we are going to study in the following paragraph.

4.1 Influence of the Surface Roughness

We used here the experimental results of Fardad andLadommatos [9]. These results depicted again in Figure 10,show a correct sensitivity of the numerical results of the LFBmodel with the average roughness of the wall. Indeed, thenumerical results show a gradual increase of the vaporizationrate before the Leidenfrost point as the wall roughness isincreased, in accordance with the Fardad and Ladommatos [9]experiments. However, the model overestimates the vaporiza-tion rate increase slightly for small roughnesses which leadsto the presence of a plateau before the point of Leidenfrost. Inaddition, the LFB model does not reproduce the effect of wallroughness observed in the nucleate boiling zone.

Besides, Figure 11 shows the evolution of the dimension-less function kclld expressed by Equations (6), (7) and (9).One may note that in the LFB model roughness increases thecontact of the liquid with the wall near the Leidenfrost point,in accordance with the experimental observations [7, 9].

4.2 Influence of the Fuel Properties

We carried out here calculations using the LFB model forthe 4 following fuels: n-pentane, iso-octane, heptane and

the n-decane at atmospheric pressure. The curves of dropletlifetime obtained numerically are compared with experi-ments in Figure 12. The LFB model correctly renders thegeneral S shape of the curves of lifetime in both nucleateand transition regimes of boiling. The minimal value of thedroplet lifetime durations is in agreement with the measure-ments for all the fuels tested. In addition, the LFB modelqualitatively reproduces the experimental observations inthe zone of passage from the transition boiling regimetowards the Leidenfrost regime. The orders of magnitude ofthe lifetime durations at Leidenfrost point are also correctlypredicted.

4.3 Influence of the Gas Pressure

A second series of calculations were carried out to test theresponse of the LFB model to the pressure of ambient gas(see Sect 2.2.2). Figure 13 compares the numerical resultswith the experiments of Stanglmaier et al. [10]. One maynote that the dependence of the temperatures of Nukiyamaand Leidenfrost with the pressure (Eq. 13 and 14) is correctlypredicted. In addition, the LFB model reproduces well theprogressive disappearance of the regime of Leidenfrostobserved in experiments [10, 23]. It is however worth notingsome dissension between the experimental and numericalcurves, in particular at the beginning of the nucleate boilingregime (with low pressure, P = 50 kPa) and towards the endof the transition boiling regime (with moderated pressure, P = 242 kPa). These dissensions may be due to both experi-mental and numerical uncertainties; for example, someuncertainties could arise from the approximations made in

341

20

15

10

5

0300 350 400 450 500


Tota

l dro

p lif

etim

e (s

)n-pentane: Experimentn-pentane: LFB modelIso-octane: ExperimentIso-octane: LFB modelHeptane: ExperimentHeptane: LFB modeln-decane: Experimentn-decane: LFB model 20

30

25

15

10

5

0360 380 400 460 480440420 500


Tota

l dro

p lif

etim

e (s

)

Experiment: P = 50 kPaLFB model: P = 50 kPaExperiment: P = 101 kPaLFB model: P = 101 kPaExperiment: P = 242 kPaLFB model: P = 242 kPaExperiment: P = 444 kPaLFB model: P = 444 kPaExperiment: P = 717 kPaLFB model: P = 717 kPa

Iso-octane case

Figure 12

Comparisons of the LFB model results with the measurementsof the lifetimes of droplets with atmospheric pressure for 4 pure fuels: n-pentane, iso-octane, n-heptane and n-decane.The experimental data was taken from [9, 10].

Figure 13

Influence of ambient gas pressure P on the lifetime of an iso-octane droplet. Comparison of the LFB model results withthe measurements of Stanglmaier et al. [10].



the method of initialization of the numerical liquid film onthe wall (see Sect. 3.1). Nevertheless, the LFB model seemsto behave correctly and the orders of magnitude and the ten-dencies observed experimentally are well respected.

CONCLUSIONS

A comprehensive Liquid Film Boiling (LFB) model has beendeveloped using available experimental results from the liter-ature. The main characteristics of the LFB model are:

It allows the calculation of the vaporization of liquid filmsin the nucleate boiling regime as well as in the transition boil-ing mode. These modes are identified by the temperatures ofsaturation, Nukiyama and Leidenfrost.

It takes into account the effects of the ambient gas pres-sure as well as the wall roughness on the evaporation rate ofliquid films.

Relationships representing the variation of the tempera-tures of Nukiyama and Leidenfrost with the gas pressurewere proposed and validated. It is shown that values of thesetemperatures at ambient pressure must result from the experi-mental curves of droplets lifetimes in order to ensure a cor-rect behaviour of the LFB model.

The numerical results of the LFB model compare wellwith the experiments under various conditions including thevariations of the wall roughness and of the ambient gas pres-sures which are of considerable interest in several deviceslike internal combustion engines.

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28 Chen R.H., Chiu S.L., Lin T.F. (2207) On the collisionbehaviors of a diesel drop impinging on a hot surface, Exp.Therm. Fluid Sci. 32, 2, 587-595.

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Final manuscript received in August 2009Published online in April 2010

343

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