Heat and Mass Transfer 32 (1997) 385–391  Springer-Verlag 1997

Mixed convection film condensation from downward flowing vapors onto a sphere with uniform wall heat flux C. H. Hsu, S. A. Yang

385 Abstract A model is developed for the study of mixed convection film condensation from downward flowing vapors onto a sphere with uniform wall heat flux. The model combined natural convection dominated and forced convection dominated film condensation, including effects of pressure gradient and interfacial vapor shear drag has been investigated and solved numerically. The separation angle of the condensate film layer, /s is also obtained for various pressure gradient parameters, P3 and their corresponding dimensionless Grashof ’s parameters, Gr 3 . Besides, the effect of P3 on the dimensionless mean heat transfer, Nu…Re†ÿ1 2 will remain almost uniform with increasing P3 until P3 ˆ 29 Gr 3 for various corresponding available values of Gr 3 . Meanwhile, the dimensionless mean heat transfer, Nu…Re†ÿ1 2 is increasing significantly with Gr 3 for its corresponding available values of P3 . For pure natural-convection film condensation, Nu…Rew =Gr †1 3 ˆ 1:706 is obtained. =

=

=

Mischkonvektion bei Filmkondensation von Da¨mpfen, die an einer Kugel unter gleichfo¨rmigen Wa¨rmefluß abwa¨rts stro¨men Zusammenfassung Es wird ein Modell zur Untersuchung der Mischkonvektion bei Filmkondensation von Da¨mpfen an einer Kugel entwickelt, die unter gleichfo¨rmigen Wa¨rmefluß daran abwa¨rts stro¨men. Das Modell verbindet die durch natu¨rliche und durch erzwungene Konvektion bewirkte Filmkondensation unter Einschluß von aus einem Druckgradienten resultierenden Effekten sowie von Dampfschubspannungen an der Phasengrenze. Die numerische Lo¨sung liefert den Separationswinkel des Kondensatfilms Us fu¨r verschiedene DruckgradientenParameter P3 und zugeho¨rige Grashof-Parameter Gr 3 : Der Einfluß von P3 auf den mittleren Wa¨rmeu¨bergangsparameter Nu…Re†ÿ1 2 bleibt bis P3 ˆ 29 Gr 3 ziemlich gering, auch wenn Gr 3 zwischen 0.01 und 100 variiert. Fu¨r reine natu¨rliche Filmkondensation erha¨lt man: Nu…Rew =Gr †1 3 ˆ 1:706: =

List of symbols Cp specific heat of condensate at constant pressure D diameter of sphere g acceleration due to gravity Gr Grashof number ˆ q…q ÿ qv †gD3 =l2 3 Gr dimensionless parameter defined as Reÿ3 2 Gr =Rew h condensing heat transfer coefficient at angle /  h mean value of condensing heat transfer coefficient hfg latent heat of condensation k thermal conductivity of condensate m00 local condensate mass flux (per unit area) Nu local Nusselt number ˆ hD k  Nu mean Nusselt number ˆ hD k p static pressure of condensate P3 dimensionless parameter defined as …qv =q†Re1 2 =Rew q local heat flux r radius of sphere Re two-phase Reynold number defined as qv U1 D=l Rew the film Reynold number defined as 4pqD lhfg Tsat saturation temperature of vapor Tw wall temperature U1 vapor velocity of main stream ue tangential velocity at ‘edge’ of vapor boundary layer u velocity component in x direction x coordinate measuring distance along circumference from top of sphere y coordinate normal to the wall surface d thickness of condensate film d3 dimensionless thickness of condensate film, defined in Eq. (14) / angle measured from top of sphere q density of condensate qv density of vapor l absolute viscosity of condensate =

=

=

Received on 16 July 1996 C. H. Hsu S. A. Yang Department of Mold and Die Engineering National Kaohsiung Institute of Technology Kaohsiung 807, Taiwan Correspondence to: Chao–Ho Hsu

Subscripts s condition for separation flow v vapor (Note: when not subscripted, a property is taken as that of the liquid phase.) 1 Introduction Filmwise condensation heat transfer of pure vapor flowing onto a body, like a plate, cone, and tube has been widely

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studied in view of the practical importance in the design of the condensers for power plants, air-conditioning equipment, and many other chemical industrial process equipment. For laminar filmwise condensation with constant properties and ignoring the vapor velocity, the assumption of the simple Nusselt theory [1] have been proved in later and more complete studies [2, 3] to be basically accurate. Basically, condensation of flowing vapor on a body differs from the Nusselt’s analysis due to the two major factors existing as follows: firstly, effect of flowing vapor velocity, for a circular tube explored by a number of researchers [4–6] and proved by experimental study from Michael et al. [7]. Secondly, as a matter of fact, by several investigations [8–10], the tube wall temperature has been observed to vary around the tube by amounts comparable with mean temperature difference across the condensate film, the agreement between experiment and theory for mean heat transfer coefficient might seem somewhat surprising, although the local film thickness and local heat flux depend strongly on the amplitude of the wall temperature variation. However, for laminar filmwise condensation with vapor flow velocity and inclusion of pressure gradient effect, the mean heat transfer coefficients are influenced significantly with increasing the wall temperature variation amplitude, see Memory et al. [11]. Memory and Rose [8] observed that for condensation of flowing vapor onto a horizontal tube, the temperature variation can be approximated by a cosine trigonometric function for the uniform surface heat flux conditions of the heat removal. For the case of natural convection condensation with a uniform wall heat flux, Fujii et al. [12] found that when the thermal resistance of the cooling side is larger than that of the condensing side as encountered in the condensation of steam, the experiments agree approximately with the theory for the uniform surface heat flux. From a practical point of view the thermal loading is generally prescribed and in this regard Lee and Rose [13] presented a correlation for the case of uniform wall heat flux. More recently, Yang and Chen [14] investigated the same topic for a porous elliptical tube. In practice, the spherical body is a common element frequently employed in heat-transfer process and its isothermal wall case for natural convection condensation problem is ever studied by Yang [15] and Karimi [16] and Yang and Chen [17]. As for forced convection condensation on a sphere using constant properties analysis, Hu and Jacobi [18] and Jacobi [19] recently investigated the problem and developed a generalized model by employing the vapor shear approximation of Shekriladze and Gomelauri [20] model. However, the analysis of Shekriladze and Gomelauri employed a concept that the condensation process in effect is similar to suction applied around the tube on which condensation takes place and hence vapor boundary separation can be assumed to be absent. Hence, the major aim of the present work is to investigate the combined natural and forced convection condensation on a sphere subjected to uniform surface heat flux with inclusion of effects of vapor shear stress, and pressure gradient. A simple numerical predictive model is developed which provides a more convenient basis for engineering application without solving two-phase

boundary layer tedious work. The present approach is acheived by simple fourth-order Runge–Kutta numerical technique. 2 Analysis Consider a sphere immersed in a downward flowing vapor which is at its saturation temperature Tsat and moves at uniform velocity U1 . The wall temperature Tw may be non-uniform and below the saturation temperature. Thus, condensation occurs on the wall and a continuous film of the liquid runs downward over the sphere under the combined action of gravity force, the vapor shear stress and the pressure gradient. Meanwhile, it is assumed that the thermal resistance of the cooling side is larger than that of the condensing side as encountered in the condensation of vapor and peripherally uniform and will cause a uniform wall heat flux. The physical model and coordinate system under consideration is illustrated in Fig. 1 where the curvilinear coordinates (x; y) are aligned along the spherical wall surface and its normal. The conservation of mass, momentum and energy for the steady laminar flow with negligible viscous dissipation and neglecting the advective terms are described by the following equations:

q m00 ˆ

d sin / rd/

l

Z

d



sin / u dy

0

o2 u dp ‡ … q ÿ qv †g sin / ÿ ˆ0 2 oy rd/

hfg m00

ˆq

…1 † …2† …3†

Employing Bernoulli’s equation to the pressure gradient term of Eq. (2) along the interface yields

l

o2 u e ˆ ÿ…q ÿ qv †g sin / ÿ qv ure du d/ oy2

Fig. 1. Physical model and coordinate system

…4†

with the following boundary conditions:

l

ou ˆ sd oy

and

uˆ0

where,

p  d Re D Gr3  Reÿ3 2 Gr Rew 3 qv 1 2  …6† P  q Re Rew

at y ˆ d

…5 †

at y ˆ 0

…7 †

This approximation has limitations, and other models may be available [21]; however, Eq. (7) is useful in its simplicity and is known to yield good results [4]. By potential flow theory, the vapor velocity outside the vapor boundary layer for uniform vapor velocity U1 past a sphere can be derived as

3 ue ˆ U1 sin / …8 † 2 Thus, inserting Eqs. (3) and (8) into Eq. (7) and then substituting the resultant Eq. (7) into Eq. (5), one may obtain the interfacial vapor shear drag as follows: ou q 3 ˆ U1 sin /=l oy hfg 2

at y ˆ d

…9 †

Consequently, the momentum equation, Eq. (4) can be integrated, applying boundary conditions, to yield the velocity profile within the condensate film layer:

q 1 uˆ ue y=l ÿ hfg l 

y2 ÿ dy 2







due …q ÿ qv †g sin / ‡ qv ue rd/

…10†

By utilizing Eq. (8) respectively, the pressure gradient term may be expressed respectively:

qv u e

due 9 ˆ q U 2 sin 2/=D rd/ 4 v 1

…11†

Substituting Eqs. (8) and (11) into Eq. (10), and then inserting the resultant Eq. (10) into Eq. (1), and finally integrating the updating Eq. (1) and introducing the dimensionless parameters, one may obtain the first order nonlinear ordinary differential equation for the film thickness as follows: 

d 3 32 4 2 d sin2 / ‡ pd33 Gr3 sin2 / d/ 4 3

‡ 3pd33P3 sin / sin2/



ˆ sin / with the boundary condition:

d3

=

=

ˆ m00ue

2

= =

The interfacial boundary condition, i.e., the vapor shear, remains to be modeled. This could be carried out by numerical approach to the two-phase conservation equations; however, the usefulness, simplicity, and generality, of the present study would likely be vitiated through such an approach. A good approximation for high condensate rates, is given by Shekriladze & Gomelauri [20] model as

sd

d3

ˆ finite value or dd3

d/ ˆ 0 at / ˆ 0

=

…12† …13†

…14† …15† …16†

The first term inside the derivative in Eq. (12) results from the interface shear stress while the third term involving P3 is due to the inclusion of the pressure gradient. When both of these terms are ommited, Eq. (12) reduces to the gravity-dominated flow, i.e. Nusselt-type condensation problem. Before proceeding to obtain solutions of Eq. (12), and thence to calculate the heat transfer for the sphere, it is to be noted that the condensate film flow may happen to separation when there exists the following condition:

ou=oy†yˆ0

0

…17†

Thus, this condition may also be obtained by Eq. (12) in the following relationship: ÿ 1 3 3 9 …18† d ‡ d32 4p Gr3 ‡ P3 cos / ˆ 0 2 2 If / ˆ /s satisfies the above equation, /s is called the singular point or separation angle, i.e.

dd3 =d/ ! 1 as / ! /s

…19†

However, since Eq. (12) is first order non-linear ordinary differential equation of d3 , it can be solved by means of a fourth-order Runge–Kutta integration that uses a step size D/ ˆ 0:05 deg., and then substituted into Eq. (19) by interpolation to determine the position or value of /s . In general, the accumulated error of the present numerical calculation is O…D/4 †, thus, the relative numerical error is quite small. By the way, it is very unstable and sensitive to calculate d3 at / close to /s , so we are required to check if the condensate film thickness will become abruptly extra thick, i.e.,

d3

! 1 as / ! /s …20† 9 3 3 3 Obviously, when P ˆ 0 or Gr  2 P it satisfies, /s ˆ p.

These cases mean that the condensate film will separate or drip off at the bottom of sphere. Otherwise, Gr 3 < 92 P3 ; the critical angle becomes p=2 < /s  p. Theses cases indicate that the condensate film will drip off before reaching the bottom of sphere. For the latter cases, the temperature difference is going infinitely as d is approaching infinity so as to maintain the uniform surface heat flux, thus, these cases will be excluded in the present work, because DT is always finite so that its surface heat flux can not maintain constant after / > /s . As in the Nusselt’s theory, by neglecting the convection term, the surface temperature distribution Tw …x† is given as

Tw

ˆ Tsat ÿ qk d

…21†

The mean wall temperature coefficient is calculated based on the area average value of the surface temperature on the whole sphere as follows:

387

DT

ˆ 12

Z

p

0

DT …/† sin / d/

…22†

Furthermore, the local heat transfer coefficient can be shown to be:

Nu ˆ 388

hD ˆ Re1 2 =d3 …23† k Next, the expression for the overall mean heat transfer coefficient i.e. the mean Nusselt number is given by =

Nu ˆ qD=…kDT † Nu…Reÿ1

=

2

†ˆ2

Z

p

…24† …25†

d3 sin / d/

0

As can be seen from Eqs. (12) and (15), the dimensionless Grashof number group, Gr 3 , represents the ratio of the gravity force and inertial force. Hence, two asymptotic cases Gr 3  1 and Gr 3  1, will be investigated as follows: For the case of Gr 3  1, i.e., for very slow flow vapor, the gravity force is much larger than the pressure gradient and vapor shear force. Therefore, the problem reduces to the gravity-dominated flow condensation, i.e. naturalconvection film condensation. After omitting the first term and the third term, Eq. (12) becomes

Further, its /s profile for wide range of P3 is discussed in the latter section. 3 Discussion of results In this section, numerical results for the mixed convection film condensation of vertical downward vapor flow onto a sphere are presented in two parts. In the first part, numerical results of the local condensate film thickness and its corresponding singular or separation angle are obtained and discussed for a practical range of Gr 3 and its corresponding P3 . Then, the second part will indicate the numerical results of mean heat transfer coefficient, or Nusselt number for a practical range of Gr 3 and its corresponding available P3 …or P3  29 Gr 3 † in order to maintain the uniform surface heat flux condition. 3.1 Characteristics of flow dynamics: condensate film thickness d3 ; separation angle /s

3.1.1 Distribution of condensate film thickness Firstly, the results of numerical solutions from Eq. (12) are obtained for various pressure gradient effects P3 and dimensionless Grashof ’s effects Gr 3 and shown in Fig. 2. It is seen that the film thickness increases continuously with /. 3 8p d 2 3 3 3 d Gr sin2 / sin / 26 The effect of P3 on the dimensionless local film thickness is 3 d/ appreciably affected for mixed-convection condensation The dimensionless local film thickness can be obtained by for Gr 3 ˆ 1:0, while for Gr 3 ˆ 100, i.e. natural-convection film dominated condensation, that on d3 is negligible. As applying the separation of variable to above equation, for Gr 3 ˆ 0:01; i.e. forced-convection film dominated  1 3 3 condensation, the only one applied for the uniform surface d3 Gr 3ÿ1 3 sinÿ2 3 / 1 cos / 27 heat flux condition on the entire sphere is P3 ˆ 0. Basi8p cally, in the upper half of sphere, the positive or favorable Furthermore, one can also obtain the local Nusselt number pressure gradient effect makes the film thinner; however, as follows in the lower half of sphere, the negative or opposite D pressure gradient effect tends to retard the liquid conNu Re1 2 =d3 28 densate film flow down and makes the film thicker than d

… †

ˆ

… †

ˆ

=

… ÿ

=

ˆ ˆ

†

=

… † … †

=

Hence, from Eq. (28), the overall mean Nusselt number is integrated over the entire spherical surface area

Nu…Reÿ1

=

2

ˆGr31 32

†ˆ2

Z

Z

p

=

(

p

sin / d3 d/

0

1 = 3 )



sin

1 =3

0

3 / …1 ÿ cos /† 8p

d/ …29†

At last, the overall Nusselt number can also be presented as follows:

Nu‰Rew =Gr Š1 Z

ˆ2

p

=

3

(



sin 0

1=3

1=3 )

3 / …1 ÿ cos /† 8p

d/

…30†

It is to be noted that, for a uniform surface heat flux sphere, the result of above equation is evaluated to be Nu‰Rew =Gr Š1 3 ˆ 1:706: On the other hand, for Gr 3  1 case and when Gr 3  92 P3 , the other asymptotic case-forced convection dominated film condensation will be also investigated. =

Fig. 2. Dependence of dimensionless local film thickness on parameters Gr 3 and P3 around periphery of sphere

that without considering the pressure gradient effect P3 ˆ 0. Hence, the effect of pressure gradient makes the condensate film drop off more ahead in the lower half of sphere than that in the case P3 ˆ 0; which can be seen in the next figure too. Further, d3 decreases as Gr 3 increases at the same position /, which can be seen from Eqs. (14) and (15), because Re comes down so that Gr 3 increases while d3 decreases. 3.1.2 Phenomena of film flow separation; values of the separation angles Fig. 3 shows the dependence of separation angle of condensate film on parameter Gr 3 for the mixed convection film condensation. In general, /s increases at very small pace for smaller Gr 3 or forced convection dominated condensation case. As Gr 3 is increasing near 92 P3 ; /s is going up abruptly until /s ˆ p. For the cases /s < p, the uniform surface heat flux can not be maintained, thus, these cases are excluded in the present study. In fact, they belong to the variable surface heat flux problem. Fig. 4 shows the effect of the pressure gradient on the separation angle i.e. the dependence of /s on parameter P3 for different values of Gr 3 when Gr 3 ˆ 0:01 and 0.1. It is obviously seen that for Gr 3 ˆ 0:01 case, /s remains p continuously with increasing P3 until P3 > 29 Gr 3 around, i.e. /s begins to decrease and becomes less than p when P3 > 29 Gr 3 . The larger Gr 3 increases, the wider range of P3 satisfies /s ˆ p, for instance, Gr 3 ˆ 0:1, its /s ˆ p always during 0  P3  0:022 cases.

389

Fig. 4. Dependence of separation angle on parameter P3 for mixed convection film condensation

the local heat transfer coefficient also increases as P3 increases in the upper half of sphere because there exists an additional favorable action of the pressure gradient effect, especially for Gr 3 ˆ 1:0 which plays an appreciable varying. However, the local heat transfer decreases with increasing P3 because of an opposite effect of the pressure gradient in the lower half of the sphere. Therefore, the pressure gradient force has significant influence on the local heat transfer coefficient, above all in the mixed3.2 convection condensation; however, the effect of the presPerformance of local and mean heat transfer sure gradient upon the local heat transfer coefficient for natural-convection dominated condensation is nearly 3.2.1 negligible. As for the forced-convection dominated conProfile of the local heat transfer In Fig. 5, the results from Eq. (23) show that, the local heat densation, e.g. Gr 3 ˆ 0:01, the available range of P3 for transfer decreases continuously around the sphere due to d uniform surface heat flux condition is more limited, thus, increasing with / when Gr 3 ˆ 0:01, 1 and 100. In general, P3 ˆ 0 is the only one can apply.

Fig. 3. Dependence of separation angle on parameter Gr 3 for mixed Fig. 5. Dependence of dimensionless local heat transfer coefficient on parameters Gr3 and P3 around periphery of sphere convection film condensation

390

Fig. 6. Dependence of dimensionless mean Nusselt number on Gr 3 for mixed convection film condensation

Fig. 7. Dependence of dimensionless mean heat transfer coefficient on P3 for mixed convection film condensation

3.2.2 the film, thus the separation point /s of vapor Performance of the mean heat transfer boundary-layer must occurs at the more downstream The following sections regarding the mean heat transfer than the liquid film layer /s does. Owing to very fast coefficient or mean Nusselt number are obtained nuflow of vapor, the separation of vapor were accommerically from Eq. (24). Firstly, it is seen that the lower P3 panied by a sharp pressure rise this could initiate an is, the wider and lower range of Gr 3 extends. Basically, for instability in the condensate film near /s . the natural-convection dominated condensation case, the (3) The mean heat transfer coefficient is also nearly dimensionless mean heat transfer coefficient is increasing unaffected by the pressure gradient term for all cases with …Gr 3 †1 3 . Next, as for the pressure gradient effect on of Gr 3 . Further, the smaller Gr 3 is or the more part of the mean heat transfer coefficient, Fig. 7 indicates the forced-convection film condensation dominates, the dependence of Nu Reÿ1 2 the dimensionless mean Nusselt smaller range of P3 available is to maintain the uniform 3 number on P , the pressure gradient effect for the mixed surface heat flux on the whole sphere. convection film condensation when Gr 3 =0.01, 0.1, 1.0, 10.0 (4) Once the two-phase Reynolds number becomes larger, and 100, and shows that the dimensionless mean heat the effect of ripple waves upon the hydrodynamics transfer coefficients for all curves, no matter what Gr 3 is, characteristics and heat transfer rates should be taken remain nearly uniform with increasing P3 until P3 > 29 Gr 3 . into account in the present problem. Hence, the model In other words, the effect of the pressure gradient on the should be cautiously applied when Re > 3 2 105 . mean heat transfer coefficient can be neglected. References 1. Nusselt, W.: Die Oberfla¨chen-Kondensation des Wasserdampfes. 4 Z. Ver. Dtsch. Ing. 60 (1916) 541–546 Concluding remarks 2. Rohsenow, W. M.: Heat transfer and temperature distribution in This is the first analytical approach for solving the mixedlaminar film condensation, Trans. ASME. 78c (1956) 1645–1648 convection laminar film condensation on a sphere with 3. Sparrow, E. M.; Gregg, J. L.: A boundary-layer treatment of launiform surface heat flux. The obtained results apply to minar film condensation. J. Heat Transfer 81 (1959) 13–18 both forced-convection dominated and natural-convection 4. Rose, J. W.: Effect of pressure gradient in forced convection film condensation on a horizontal tube. Int. J. Heat Mass Transfer 27 dominated film condensations on a sphere. Four major (1984) 39–47 conclusions are drawn here: 5. Jacobi, A. M.: Filmwise condensation from a flowing vapor onto (1) The present numerical results for the mixed convecisothermal, axisymmetric bodies. J. Thermophys. Heat Transfer 6 tion film condensation with further inclusion of both (2) (1992) 321–325 effects of pressure gradient and the interfacial vapor 6. Yang, S. A.; Hsu, C. H.: Combined forced and natural convection shear can easily apply to the engineering calculation, film condensation from downward flowing vapors onto a horizontal tube with variable wall temperature (submitted to Int. J. compared with the two-phase boundary layer techniHeat Mass Transfer) ques. (2) The separation angle of condensate /s can be obtained 7. Michael, A. G.; Rose, J. W.; Daniels, L. C.: Forced convection condensation on a horizontal tube-experiments with vertical by delicate numerical approach. Noted that, when downflow of steam. J. Heat Transfer 111 (1989) 792–797 9 3 3 3 Gr  2 P or P ˆ 0, the separation angle /s ˆ p 8. Memory, S. B.; Rose, J. W.: Free convection laminar film conalways. Since the flowing vapor not only passes a densation on a horizontal tube with variable wall temperature. moving condensate liquid film but also condenses into Int. J. Heat Transfer 34 (1991) 2775–2778 =

=

9. Yang, S. A.; Chen, C. K.: Laminar film condensation on a horizontal elliptical tube with variable wall temperature. J. of Heat Transfer 116 (1994) 1046–1049 10. Shklover, G. G.; Semyonov, V. P.; Usachyev, A. M.: Condensation on a horizontal tube with a spatially non-uniform temperature distribution. Heat Transfer-Soviet Research 21 (1989) 29–33 11. Memory, S. B.; Rose, J. W.: Forced convection film condensation on a horizontal tube-effect of surface temperature variation. Int. J. Heat Transfer 36 (1993) 1671–1676 12. Fujii, T.; Uehara, H.; Oda, K.: Filmwise condensation with uniform heat flux and body force convection. Heat Transfer Japanese Research 4 (1972) 76–83 13. Lee, W. C.; Rose, J. W.: Film condensation on a horizontal tubeeffect of vapor velocity, Proc. of 7th int. Conference, Mu¨nchen, FRG (1982) 101–106 14. Yang, S. A.; Chen, C. K.: Laminar film condensation on a horizontal elliptical tube with uniform surface heat flux and suction at the porous wall. J. CSME 14, No. 1, (1993) 93–100 15. Yang, J. W.: Laminar film condensation on a sphere. J. Heat Transfer 95c (1973) 174–178

16. Karimi, A.: Laminar film condensation on helical reflux condensers and related configurations. Int. J. Heat Mass Transfer 20 (1977) 1137–1144 17. Yang, S. A.; Chen, C. K.: Effects of surface tension and nonisothermal wall temperature variation upon filmwise condensation on vertical ellipsoids/sphere, Proc. R. Soc. Lond. A 442 (1993) 301–312 18. Hu, X.; Jacobi, A. M.: Vapor shear and pressure gradient effects during filmwise condensation from a flowing vapor onto a sphere. Exp Therm. Fluid Sci. 5 (1992) 548–555 19. Jacobi, A. M.: Filmwise condensation from a flowing vapor onto isothermal, axisymmetric bodies. J. Thermophy. Heat Transfer 6 (2) (1992) 321–325 20. Shekriladze, I. G.; Gomelauri, V. I.: The theoretical study of laminar film condensation of a flowing vapor. Int. J. Heat Mass Transfer 9 (1966) 581–591 21. Blangetti, F.; Naushahi, M. K.: Influence of mass transfer on the momentum transfer in condensation and evaporation phenomena. Int. J. Heat Mass Transfer 23 (1980) 1694–1695

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