a new model for nucleate boiling heat transfer.pdf

W~irme- und Stofffibertragung 30 (1994) 119-125 Springer-Verlag 1994

A new model for nucleate boiling heat transfer P. Stephan, J. Hammer

Abstract A new model to calculate heat transfer coefficients in f nucleate boiling is presented. Heat transfer and fluid flow around K a single bubble are investigated taking into account the i n f luence ~/evap of meniscus curvature, adhesion forces and interfacial thermal Pc resistance on the thermodynamic equilibrium at the gas-liquid Pv interface. The model requires only bubble site densities and departure diameters. Further quantities except the thermophysical (~ properties are not needed. From the results bubble growth rates R i can be derived. As an example nucleate boiling heat transfer co- R efficients of R-n4 were calculated. They agree with experimental r values within the experimental accuracy, s

T Ein neues Modell fCir den W~irmeiJbergang beim Blasensieden Zusammenfassung Es wird ein Modell zur Berechnung des W~ir- meiibergangskoeffizienten beim Blasensieden vorgestellt. Das c~ Modell berficksichtigt den Einflufl der Oberfl~ichenspannung, der /~ AdMsionskr~ifle und des W~irmewiderstands zwischen Flfissig- V keit und Dampf auf das thermische Gleichgewicht an der Phasen- grenze. Zur Berechnung des Wiirmefibergangskoeffizienten r/ mfissen lediglich die Anzahl der Blasen je Fl~icheneinheit und die p Abreifldurchmesser bekannt sein. Weitere GrGgen auger den cy thermophysikalischen Stoffeigenschaften werden nicht benGtigt, q~ Aus den Ergebnissen wird auf das Blasenwachstumsgesetz ge- Ah~ schlossen. Die aus dem Modell berechneten W~irmefibergangs- koeffizienten ffir das K~iltemittel R-n4 stimmen mit experimen- tellen Werten innerhalb deren Meflgenauigkeit tiberein.

Nomendature A dispersion constant h heat transfer coefficient

Received on August 3, 1994

Dr.-Ing. P. Stephan Joint Research Centre of the E.C. Ispra, Italy now with Daimler-Benz Research Centre Ulm D-89o13 Ulm, Germany

Dipl.-Ing. J. Hammer Institut ffir Technische Thermodynamik und Thermische Verfahrenstechnik Universit~it Stuttgart Postfach 8o 114o D-7o5n Stuttgart, Germany

The authors thank Prof. K. Stephan, director of the Institut ffir Techni- sche Thermodynamik und Thermische Verfahrenstechnik of the Uni- versitat Stuttgart and Prof. C. A. gusse, retired division head at the Joint Research Centre of the E.C., Ispra, Italy for fruitful discussions.

condensation coefficient curvature of the meniscus evaporation rate capillary pressure vapour pressure heat flux heat flow interfacial thermal resistance gas constant bubble radius wall thickness temperature

Greek symbols thickness of the liquid layer thermal conductivity kinematic viscosity coordinate parallel to the wall coordinate normal to the wall density surface tension apparent contact angle heat of vaporization

Subscripts 0 interline adsorbed film - micro region o overall eq equilibrium ad adsorbed film bub bubble in input i liquid-vapour interface L liquid s solid mic micro region out outside surface of the wall sat saturation v vapour w heated wall

1 Introduction In the recent decade many authors studied the phenomena of nucleate boiling heat transfer. Nevertheless the physical background is not sufficiently understood, especially the infuence of formation and growth of bubbles during nucleate boiling heat transfer. Heat transfer coefficients often cannot be evaluated with satisfactory accuracy. Instead, they must be determined experimentally,

n9

120

because correlations are only valid in the range of parameters they were developed for. The main reason for this is, that the heat transfer is governed by a great number of physical phenomena They are interconnected with each other and are not sufficiently well understood.

According to a recent publication of Dhir [1], which sum- marizes the state of the art, four different mechanisms are deci- sive for the heat transfer. These are: Time dependent heat conduction, forced convection close to a growing bubble, evaporation at the liquid-vapour interface and natural convection on the heated wall in the liquid pool area. The contribution of each of these mechanisms to the overall heat transfer depends on the input heat flux, the physical properties of the fluid and the geometry of the heat source.

As the heat flux is rising dramatically with the formation and growth of vapour bubbles, these phenomena are of special im- portance. For this reason several authors developed models for the heat flow in the micro layer under a vapour bubble These models are mainly based on those of Hart and Griffith [2] and van Strahlen [3]. A summary is published in the book of van Strahlen and Cole [4]. According to the micro layer theory of van Strahlen the overall heat transfer is the sum of the heat transfer in the micro layer and of the neighbouring convective heat transfer. The model of van Strahlen includes however, several fitting parameters, particularly for the micro layer thickness and the size of the area influenced by a single bubble. Hence, the results apply only for those conditions where the parameters were fitted

Non of the published models takes into account that a liquid film is adsorbed between wall and bubble 1. This liquid film consists normally only of a few molecular layers and cannot be evap- orated due the adhesion forces. Wayner et al. [5] showed that evaporation starts in a micro region close to the adsorbed film where extremely high heat fluxes occur. For the evaporation in capillary grooves P. Stephan and Busse [6] found that the heat and mass transport phenomena in the micro region can have a significant influence on the overall macroscopic heat transfer.

In the following paragraphs a model to calculate the heat transfer coefficient in nucleate boiling is presented. It includes the phenomena in the micro region. The only parameters neces- sary for the model are bubble site density and departure radius for a single bubble. These parameters are partly known or can be determined from experiments or correlations.

2 Basis of the model Figure 1 shows the basic geometry considered. The heat transfer is investigated in a axial symmetrical system around a single vapour bubble. Boundary conditions are the vapour pressure Pv and the outside temperature Tou t of the heated wall. At a given time t, where the radius of the bubble is r(t) and the heat flux qin (t), the heat transfer coefficient h o of the system is defined as

ho _ qin (1) Tou t - Tsat'

After submission of this paper, Lay and Dhir [14] published a model assuming a liquid film to be adsorbed between wall and vapour stems. Contrary to our paper their model is valid for high heat fluxes and under the assumption of constant wall temperature and a steady state boiling process.

Micro region

Liquid I ~'-

t)

Fig. 1. Heated wall with vapour bubble and surrounding liquid

where T~a t is the saturation temperature corresponding to the vapour pressure Pv and qin is the average heat flux through the heated wall. The overall heat transfer coefficient h o includes the thermal resistance of the wall. The heat transfer coefficient on the vapour side, which is often used in literature, is given by h = (1/h o- s/Z s)-t In order to calculate the heat flux qin (t) one must consider the heat flow in the liquid adjacent to the micro region and inside the wall.

The liquid adjacent to the micro region and the wall are called macro region. From the heat flux into the bubble at time t, the related evaporation rate ghevap (t), the bubble radius r ( t+At) after a small time step At and hence the bubble growth rates can be pre- dicted.

3 The model in the micro region The curved line in Fig. z represents the interface between bubble (G) and liquid (L). In the micro region the heat transfer is signifi-

K:Kbubbte

Adsorbed I Tsar Ti X i /

I Transverse I J . _ - . - / ~ flow I Micro region I L

I Tw, mic I

I "~ Interfocio[ curvature I m,~ I

I Adhesion forces I

I ~ w,- I Interfacia[ I thermal resistance '

Fig. 2 Significant phenomena in the micro region

cantly influenced by the interface curvature, the adhesion forces and the molecular kinetic thermal resistance at the interface.

Outside the micro region, in Fig. 2 on the right hand side, mi- croscopic phenomena are still negligible, so that the liquid-vapour interface has practically a constant curvature, K=Kbubble.

Because of the thin liquid film in the micro region, heat fluxes can be very high, provided that the thermal conductivity of the working fluid is much lower than that of the wall material. Usu- ally this is the case for combinations of a metallic wall with or- ganic or inorganic working fluids. The high evaporation rates in the micro region then create a significant transverse liquid flow. This flow is driven by two mechanisms, that both imply varia- tions ofKand Ti:

In the thicker part of the micro region the driving pressure gradient is generated in the liquid-vapour interface by the capillary forces due to an increase of the curvature in the flow direction. This in turn, leads to an increase of the interface temperature at a given vapour pressure and then reduces the volatility of the liquid.

In the micro region where the meniscus comes close to the wall the short range attractive forces between the liquid mole- cules and the wall atoms become significant and produce an "adhesion pressure" in the liquid. Its gradient is normal to the wall, similarly to gravity that generates a hydrostatic pressure gradient normal to a horizontal surface. However, while the hydrostatic pressure gradient is constant, the adhesion pressure gradient is (in the range of interest) inversely proportional to the 4th power of the distance from the wall and can only be noticed very close to the wall. Similarly as a hydrostatic pressure causes liquid to spread out over a horizontal surface, also the adhesion pressure forces causes the liquid to spread out over the wall. It tends to steadily reduce the curvature of the interface in the flow direction until at the end of the micro region the meniscus levels out in a stationary adsorbed film on the wall. The attractive force between liquid and wall also reduces the volatility of the liquid, i.e., it increases the interface temperature.

The interfacial thermal resistance leads to a further increase of the interface temperature. In most evaporation processes this interfacial thermal resistance is negligible. It is, however, significant in the central part of the micro region, because of the small thermal resistance due to the thin liquid film and the high heat flux.

3.1 Heat transfer in the micro region Assuming one-dimensional heat conduction in the liquid film the heat flux is

~L q = ~- (rw,mi c - T/), (2)

where 5 is the thickness of the liquid layer, "~L is the thermal conductivity of the liquid, Tw, mic the wall temperature and r i the temperature at the vapour-liquid interface.

The heat flux at the interface is given by [7]:

0 = ~/ / (T / - Ti,eq) (3)

r Tsat "~" 27gRTsat (2 - f)

Ri = (4) Ah2 Pv 2 f

with the interfacial thermal resistance R i [8] and the equilibrium interface temperature Ti, eq. It is different from saturation temperature due to the capillary pressure Pc:

( pc Zi,eq ---= Zsa t ]1 + Ahv PL \

(5)

In Eq. (3) the capillary pressure is defined as

pc=yK +A/5 3 . (6)

It consists of the term o'K due to the interface curvature and the adhesion pressure A/5 3, also known as "disjoining pressure" [9], wherein A is the dispersion constant. Values of A depend on the wall material and the liquid. They can be determined theoreti- cally or from experiments [lO].

According to the rules of differential geometry the curvature Kis

5" 1 5 ' K - 5 ' 2)312 ~ ' (7) (1-1- ~ (1+5'2) 1/2

where ~ is the distance to the axis of the rotation symmetric system and 5' and 5" are the first and second derivatives of the film thickness 5with respect to q.

Eliminating T i from Eqs. (2) to (5) the heat flux in the micro region follows as

rw,mic- Tsat (1 + Pc/Ahv PL ~) - (8)

3.2 Liquid transport in the micro region For the modelling of the transverse liquid flow in the micro region a one-dimensional laminar boundary layer is assumed. Ac- cording to P. Stephan [7] the convective terms in the momentum balance are neglected. The resultant velocity profile is

vL PL a~ )

Together with the mass balance

1 ~(rhS~) mewp- ~ a~ ( lo)

with 5

1 /n=pLU ~ =~ ~ U~(~,t/) dr/ 0

(n)

one obtains the evaporation mass flux

mewp-- Ah~ 3VL~ d~ ~53 " (12)

121

122

3.3 Differential equations for the micro region Combining (8) and (i2) yields a fourth-order non-linear differential equation

(Tw,mic- rsat (1 + Pc/Ahv PL))

R i + 6//~ L

/" "x _ kh~ d [dpc483 /

3 VL 4 d~ ~. d~ ) (13)

for the film thickness 6(~) in the micro region. As Eq. (13) is difficult to solve and to interpretate, we transform it into four first order differential equations. Introducing the total heat flow 0 transferred in the area between the adsorbed film and the radius

0(4) = ~//(4) dA = ~//(~) 2z~ d~ (14) A ~a

Eq. (13) can be written as

da; a, d~

d~ Pc -

dpc-=- 3VL O. / Ahva3 d~

d(04) -4 d4

1 5' [ 6,2)3/2 4 (1+6'2) 1/2 ) (1+

Tw,mi c_ Tsa t (1 + Pc/Ahv PL )

R i + 6/~ L (15)

This system of differential equations is equivalent to Eq. (13) and describes the heat and mass transfer in the micro region.

4 The model in the macro region In the macro region - defined as the liquid adjacent to the micro region and the wall - we assume a two-dimensional heat conduction

4~. q- 4.~L =0, (16)

where ,~ is the thermal conductivity of the liquid or the wall, re- spectively. We neglect energy storage. This assumption is justified because the phase conversion number Ph=cAT/Ahv, defined as the ratio of energy storage and heat of evaporation, is very small and for liquid and wall of the order to 10 -2. Nevertheless one obtains a transient temperature field because of the moving boundary r(t). Convective heat transport in the liquid layer is also neglected.

The boundary conditions as shown in Fig. 3 are:

T ~ Tou t aT/a~=O

OT/Orl=O

at the outside of the evaporator wall, along the symmetry axis and the outside cylinder of the system, for ~_< ~ad at the interface between wall and liquid, because no heat is transferred through the adsorbed film,

rl ~ "Tsat

I - '~ Tout

Fig. 3. The macro region and the boundary conditions for the computation

T= rw, mic

T= Tsa t (1 + aK/pL Ah~)

T= rsa t

at the interface between wall and film in the micro region, along the vapour-liquid interface in the macro region and in the liquid on the top of the system, at a distance 7/sufficiently far away from the heated wall.

5 Numerical treatment The differential equations of the micro region Eq. (15) and the macro region Eq. (16) require different numerical methods. Both systems are coupled with each other. The boundaries between the systems are given by the radius of the adsorbed film ~ad and the length of the micro region A~mic. These values and in addition the temperature Tw, mi c have to be determined by iteration in such a way that the mass and energy balance at the interface from macro to micro region are correctly fulfilled. The temperature Tout, the vapour pressure Pv and the initial bubble radius r(to) at a time t o are given values.

The differential equations of the micro region, Eq. (15), are integrated using a Runge-Kutta method. For the integration the initial values of 6, 6', Pc and 0 have to be specified at 4= ~ad where the meniscus is connected to the non-evaporating film. The thickness 6 o of the adsorbed film follows from Eq. (6), because its curvature Kis zero, 6o=(A/pc, ad) 113 with pc, ad=( Tw, mic/Tsat-1) . Ahvp L from Eq. (8). The slope is set to zero, 6~=0. The initial value Pc, o is equal to Pc, ad, Oo is chosen so that the integration ends in a meniscus with a constant radius r(to). The adhesion pressure A/6 3 is normally negligible at a distance ~ where the radius becomes constant, whereas the interfacial thermal resistance R i still can be important. The micro region therefore can be subdivided into two parts. The first starts at the adsorbed film. Here the complete system of differential Eqs. (15) has to be considered. The second starts when the meniscus curvature becomes constant. Then the adhesion forces can be neglected. The micro region ends at a radius (~ad+A~mic) where according to Fig. z the interface curvature is constant, K=Kbubble, and the adhesion forces, as well as the interfacial thermal resistance are negligible.

0.15

0.12-

0.09-

0.06.

i = i i , , , , i i

0.10 0.20 0.30

~- ~ad' ~.l,m

6 Results 0.03. For assessment of the model heat transfer in nucleate boiling of refrigerant R-114 on a copper plate was studied. We chose this 0 . . . . J ~ substance because heat transfer coefficients and the related bub- 0 ble site densities were measured simultaneously by Barthau [11]. He determined cumulative bubble site densities over a period longer than the inverse of the bubble frequency. However, he stated as others that for low heat fluxes in the region of isolated bubbles the cumulative bubble site densities are equal to the average number of active sites at a given instant.

We calculated heat transfer coefficients for vapour pressures 308.0 . pv=l.91barand2.47barcorrespondingtosaturationtempera-ll tures of Tsar=294.60 K and 302.63 K and different wall superheat- 307.0 ings Tou t - rsa t between 3.5 K and 5.2 K. In this temperature range measured bubble site densities were between 77 cm -2 and 306.0-1 -, 990 cm -2. v

The copper plate was 0.5 mm thick and the dispersion con- 305.0 stant A=2.0 10 -21 ] was taken from [10]. In the liquid layer adjacent to the bubble, where the temperature drops from its value 304.0. T w at the wall saturation temperature Ts~ t, conduction prevails and convective heat transfer as usually in a boundary layer is 303.( comparatively small. In this thin layer we assumed pure conduction,The boundary layer thickness was estimated from the rela- 302.0 tions [12] for natural convective heat transfer. This approxima- 0 tion procedure seems to be justified because the boundary layer turned out to be very thin namely of the order of 0.3 mm and because only a smaller part of the total heat flux is transfered through this layer.

The partial differential equations of the macro region Eqs. (16) are integrated using a finite element method. The finite ele- /~ ments are distributed depending on the temperature gradient in / the macro region. Triangular elements with quadratic functions are used. The grid width in the neighbourhood of the micro region is extremely small and the number of elements therefore becomes very high. E --t

0.40 0,50 ~ '1 ;>

Fig. 4. Meniscus shape 6 in the micro region (Rn4/Cu, p~= 2.47 bar, Tou t- Tsat=3.5 K, r=0.125 ram)

\ \

\

Tw,mic _ i , '~ '

Tsat (1 +~)

' ' ' ' l ' ' ' ' I ' ' ' ' i ' ' ' ' I ' ' ' '

0.10 0.20 0.30 0.40 0.50

Fig. 5. Interface temperature T i in the micro region (R114/Cu, p~=2.47 bar, Tou t- Tsat=3.5 K, r=0.125 ram)

123

6.1 The single bubble The results for a single bubble a radius r=0.125 mm are shown in Figs. 4 to 6.

The wall superheat Tou t - Tsa t is 3.5 K. The wall temperature in the micro region and the radius of the adsorbed film obtained by iteration are Tw, mic=306.08 K and ~aa=0.0563 mm. Figure 4 shows the meniscus shape in the micro region, Fig. 5 the interface temperature Ti. At ~= ~aa the meniscus is connected to the adsorbed film, which is about 10 ~ thick and has a temperature r i equal to the wall temperature rw, mic. The liquid film becomes thicker with the distance ~ from the adsorbed film and thus the adhesion forces decrease. The curvature of the meniscus, Fig. 4, undergoes a pronounced maximum and approaches a constant value, corresponding to the bubble radius r=0.125 mm. From a macroscopic point of view one has the impression that a meniscus of constant curvature hits the wall under a finite contact angle. This apparent contact angle in our example is ~=27 . This value corresponds to experimental values for refrigerants. The temperature of the vapour-liquid interface Ti decreases rapidly with the distance ~ and approaches the values

Ti= rsa t (1 + 2 a/(rAhv PL)). According to Fig. 6 the heat flux reaches a maximum of q= 1473 W/cm 2, which is approximately 100 times larger than the burn out heat flux. Close to the adsorbed film, where ~- ~aa--~ 0, the heat flux is dominated by the adhesion pressure.

The finite element grid and the macroscopic temperature distribution in the wall and in the liquid bulk are plotted in Fig. 7. The heat flux at the outside copper wall is c)in= 1901 W/m 2. With the wall superheat Tou t - rsat=3.5 K one obtains an overall heat transfer coefficient of ho=543 W/m 2 K. The thermal resistance s/CL s of the wall is comparatively small and the vapour side heat transfer coefficient h almost identical with the overall heat transfer coefficient h o. The isotherms in Fig. 7 indicate, that the heat flow is concentrated in the micro region. About 38% of the total heat flow passes through this region, although it covers only 0.16% of the copper plate. This holds for the temperature differ- ence Tou t - Zsa t = 3.5 K of the present example. At higher temperature differences and therefore higher bubble site densities the macro region becomes smaller. Then the heat flux through the micro region becomes even more important. For a temperature different Tou t - Tsar=4.2 K, for example, about 60% of the total

0 ' ' ' I '

0 0.40

1.60

I 1.20-

% o

0.80

124

0.40.

I*" , , ' I ' ' ' ' I ' ' ' ' I '

0.10 0.20 0.30 ~" ~ad' gm

Fig. 6. Heat flux c) in the micro region (Rn4/Cu, pv=2.47 bar, Tou t- rsat =3.5 K, r=0.125 ram)

i = ,

0.50 - - I : : :>"

They depend linearly on the radii. For r---~ 0 mm the heat transfer coefficient calculated with the present model approaches the value h=210 W/m 2 K. This corresponds to the thermal resistance of the liquid layer in natural convection without bubbles.

Related to each value r(t) is a heat flow 0bub (r) into the bubble due to the evaporating liquid

0bub(r) = ~t V Pv Ahv AV = At t Pv Ahv (17)

where V(t) is the volume of the vapour bubble. Integration of Eq. (17) over small time steps At yields the function r(t). Figure 9 shows calculated values r(t). They can be correlated by a relation r(t)=Ct 1/2. Such a relation is typical for bubble growth as shown by many other authors, see e.g. [4], however by different methods.

~ _ 298,92 K ~

J

299,21 K "

299,30 K - -

Fig. 7. Finite element distribution and isotherms in the macro region (R114/Cu, pv=l.91 bar, Tou t- Tsat=4.7 K, r=0.15 ram)

heat flow passes through the micro region. This clearly demonstrates that a correct modelling of the micro region is essential for a prediction of heat transfer coefficients.

6.2 Bubble growth The previous heat transfer coefficient of h = 543 W/m 2 K belongs to a bubble of r 1 =0.125 mm in diameter. Such a bubble size is reached after a certain growth time tl. In other words, we calculated the heat transfer coefficient h (tl) for a given time. For smaller bubble diameters r_< rl or growth times t_< tl on obtains from the model heat transfer coefficients h (t)_ , -

Fig. 8. Heat transfer coefficient h for different bubble radii (Rll4/Cu, p~=2.47 bar, Tou t- Tsar=3.5 K)

0.125

I 0.100.

0,075 - E E

"" 0.050

0025-

0 0

u

' . . , , . . . . , . . . . , . . . . , . . . . , . . . . , . . . . , u t ; . ,7 ' . _ , . . .

1 2 3 4 5 6 7 8 9 10 t, ms - -{ : : :>-

Fig. 9. Bubble growth radius versus time (Rl14/Cu, pv= 2.47 bar, Tou t- Tsat=3.5 K)

6.3 Comparison with experimental data In order to compare with experimental heat transfer coefficients one has to consider that experiments deliver only mean, time in- dependent heat transfer coefficients. From our model we evaluated instead the heat transfer coefficients h (r(t)) varying with time. Therefrom one obtains the mean heat transfer coefficients

h- 1 tt = - - ~ h( r ( t ) ) dt

t] o (18)

where t 1 is the growth time for a single bubble. The calculated mean heat transfer coefficients h and the mean

heat fluxes qi~ are presented in Table 1. Figure lO compares the results with experimental values of

Barthau In] and the results from a correlation of K. Stephan and Abdelsalam [13]. The agreement is satisfactory. The deviation between experiment and the results of our model is less then -+8%.

Table 1. Mean heat transfer coefficient h and mean heat flux Tin for a bubble life cycle. In brackets experimental h - values of Barthau [n] R114/Cu, pv-= 1.91 bar and 2.47 bar

Tou t- Tsat, K h, W/m2K ~in, W/m2

p~= 1.91 bar 4.3 469 (495) 2017 4.7 762 (800) 3581 5.0 1433 (1447) 7165 5.2 2095 (2092) 10892

pv=2.47 bar 3.5 432 (468) 1510 3.7 687 (715) 2542 4.0 1810 (1686) 7240 4.2 3095 (3102) 12997

_Natural convection_,_

104- I 7 ~P I I

6. A,/k this work

4 ' O, O Barthau GI, [] K.Stephan,

2 . Abde lsa lam

10 3 _ a

tx f 6

4 s'

2 " "

10 2.

Nucleate boiling

A

, , '7 ,/

Y

/~ ,O, i-] 1,91 bar

A ,O ,~ 2,47 bar

i t 1 1 102 2 4 6 8 103 2 4 6 8 104 2 4 6 8 0 5

~in, W / m 2 [~

Fig. lO. Comparison with experiments (Barthau) and correlations (Ste- phan, Abdelsalam): h- mean heat transfer coefficient, ~in mean heat flux (Rn4/Cu, pv = 1.91 and 2.47 bar)

7 Summary The model for nucleate boiling heat transfer presented here clearly demonstrates that the phenomena in the micro region are of significant influence on the overall heat flow. Due to the surface tension and the adhesion forces near the solid wall the curvature of the liquid-vapour meniscus and its temperature are not constant in the micro region. The molecularkinetic thermal resistance at the interface is not negligible. The heat flux reaches a high maximum in the micro region, which is about 100 times larger than the burn-out heat flux. The corresponding high evaporation rate leads to a local temperature drop in the heated wall. Thus, the wall temperature cannot be assumed as constant.

With increasing film thickness ~ the adhesion forces rapidly decrease. The interface curvature exhibit a sharp maximum close to the wall. The micro region ends where the adhesion forces and the molecularkinetic thermal resistance are negligible and the interface curvature becomes constant.

The apparent macroscopic contact angle, often used as a pa- rameter in previous correlations, is a result of the computation.

As an example the nucleate boiling heat transfer ofR-114 was investigated. For this fluid heat transfer coefficients and the related bubble site densities were known from Barthan's experiments [11]. The agreement of the results of the model and the experiments is quite satisfactory and within the experimental accuracy.

References 1. Dhir, V. K.: Nucleate and transition boiling heat transfer under

pool and external flow conditions. Proc. 9 th Int. Heat Transl. Conf. 1 (199o) 129-155

2. Han, C. Y.; Griffith, P.: The mechanism of heat transfer in nucleate pool boiling. Int. Journal of Heat and Mass Transfer 8 (1965) 887-904, 9o5-914

3. Van Strahlen, S. J. D.: The mechanism of nucleate boiling in pure liquids and in binary mixtures. Int. Journal of Heat and Mass Transfer 9 (1966) 955-1o2o, lO21-1o46, lO (1967) 1469-1484, 1485-1498

4. Van Strahlen, S. I. D.; Cole, R.: Boiling phenomena. Vol. i and 2, Hem- isphere, Washington, 1979

5. Wayner, P. C.; Kao, Y. K.; Lacroix, L. V.: The interline heat transfer coefficient on an evaporating wetting film. Int. Journal of Heat and Mass Transfer 19 (1976) 487-492

6. Stephan, P. C.; Busse, C. A.: Analysis of the heat transfer coefficient of grooved heat pipe evaporator walls. Int. Journal of Heat and Mass Transfer 35 (1992) 383-391

7. Stephan, P. C.: W~irmedurchgang bei Verdampfung aus Kapillarrillen in W~irmerohren. Fortschr. Ber. VDI Reihe 19 Nr. 59. Dfisseldorf: VDI Verlag 1992

8. Stephan, K.: W~rmefibergang beim Kondensieren und beim Sieden. Springer Verlag, Berlin 1988

9. Derjaguin, B. V.: Definition of the concept of and magnitude of the disjoining pressure and its role in the statics and kinetics of thin layers of liquid. Kolloidnyi Zhurnal 17 (1955) 191-197

lO. DasGupta, 84 Schonberg, 1. A.; Kim, I. Y.; Wayner, P. C., Jr.: Use of the augmented young-laplace equation to model equilibrium and evaporating extended menisci. I. Colloid and Interface Science (1992) 1-39

11. Barthau, G.: Active nucleation site density and pool boiling heat transfer - an experimental study. Int. Journal of Heat Mass Transfer 35 (1992 ) 271-278

12. VDI-W~rmeatlas. VDI Verlag GmbH, Dtisseldorf 1991 13. Stephan, K.; Abdelsalam, M.: Heat - transfer correlations for natural

convection boiling. Int. Journal of Heat and Mass Transfer 23 (198o) 73-87P

14. Lay, J. H.; Dhir, V. K.: A nearly theoretical model for fully developed nucleate boiling of saturated liquids. Proc. loth Int. Heat Transfer Conference, Brighton UK, I994, lo-PB-17, p. lO5-11o

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