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STEADY STEAM CONDENSATION ON AN EXTENDED SURFACE WITH SUCTION OF CONDENSATE O.A. KABOVa,c*, D.R. KOLYUKHINa,d I.V. MARCHUKa, and J.-C. LEGROSb,c a – Institute of Thermophysics SB RAS 1 Lavrentyev Ave., Novosibirsk 630090, Russia b – Euro Heat Pipes S.A., Brussels, Belgium c – Microgravity Research Center, Université Libre de Bruxelles, Belgium d – Weierstrass Institute for Applied Analysis and Stochastics, Germany * E-mail: [email protected]

T

he condensation of stationary steam on curvilinear fins is investigated. The fins are described by an equation for a spiral with a variable direction of rotation when the capillary pressure has a profound effect on the condensate film motion. To obtain optimal fins, it is proposed to extend the well-known condensation surfaces of Gregorig (1954) and Adamek (1981). Mathematical analysis and some numerical calculations are performed for the special case when the condensate is drawn away from the center of a flute. The properties of these surfaces and the effect of the geometrical parameters of fins on the enhancement of heat transfer are analyzed. The extended surface proposed can be manufactured by traditional methods, and can produce an amount of condensate per unit length of the film surface projection by 62% greater than that produced by the surface proposed by Adamek (1981). Key words: film condensation, heat transfer enhancement, surface tension.

1. INTRODUCTION The use of complex-shaped surfaces for steam condensation. The use of surfaces with fins of various shapes (Bergles, 1978; Marto and Numm, 1983; Kabov, 1993; Webb, 1994) represents the most effective method of steam condensation enhancement. Transverse fins are used for horizontal tubes, and longitudinal fins are used for vertical tubes. With this method of heat transfer enhancement, in addition to an increase in the area of the surface, the average integral thickness of the film decreases considerably. The horizontal tubes used in industrial machines and laboratory investigations can be divided into the following three groups: 1) tubes with transverse continuous fins of “canonical” shape, which are, for instance, rectangular (Wanniarachchi et al., 1986; Gogonin and Kabov, 1991), trapezoid (Gogonin et al., 1993), or cylindrical (Rifert and Trokoz, 1987); 2) tubes with transverse continuous fins of special shape (Kedzierski and Webb, 1990; Honda and Kim, 1995; Gogonin and Kabov, 1996; Anisimov and Smirnov, 1997); 3) surfaces with three-dimensional fins and spikes (Arai et al., 1997; Sukhatme, 1990; Kabov et al., 1993; Cheng and Tao, 1994). Journal of Engineering Thermophysics, Vol. 12, No. 1, P. 1–24 http://rjet.itp.nsc.ru; e-mail: [email protected] © 2003, Institute of Thermophysics SB RAS

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O.A. KABOV, D.R. KOLYUKHIN, I.V. MARCHUK, J.-C. LEGROS

Models of heat transfer on a fin. The regularities of hydrodynamics and heat transfer at steam condensation on non-smooth surfaces were considered in some review papers (Marto, 1988; Webb, 1988; Sukhatme, 1990; Webb, 1994; Fujii, 1995). It was shown that the most part of the liquid is condensed at the fin. Therefore, heat transfer models of steam condensation at fins were thoroughly studied. Gregorig (1954) was first to propose a theoretical model of condensation at a curvilinear fin with allowance for the forces of surface tension. A detailed analysis of this model was made by (Webb, 1979). The influence of gravitation on the process of film motion was ignored. A solution was obtained within the framework of the statement proposed by Nusselt (1916). The liquid is moving along a fin under the action of the gradient of capillary pressure in the condensate film: dp d1 =σ   ds ds  R 

(1)

The pressure in the film decreases as the curvilinear coordinate along the fin increases:

p = ps +σ / R

(2)

The following equation was obtained for the film thickness at the fin: d  3 d  1   3ηλ∆T 1 δ   = ρr σ δ ds  ds  R  

(3)

Integration of this equation yields s

1 1 3ηλ∆T = − R R0 ρr σ ∫0

 s ds  ds ∫  3  0 Φ ( s)  Φ ( s)

(4)

Here δ = Φ(s) is the film thickness as a function of the curvilinear coordinate. Equation (4) makes it possible to obtain the shape of the film surface and the fin for a given Φ(s). Since Φ(s) is known, the distribution of local coefficients of heat transfer at the fin can also be calculated, because, in accordance with Nusselt (1916), α = λ/δ. The following solution was obtained at the condition Φ(s) = δ = δ0 = const: 1 1 3ηλ∆Ts 2 − = R0 R 2ρr σδ04

(5)

Here R0 is the initial radius of curvature of the film at the fin top. In the case being considered, it coincides with the initial radius of curvature of the solid surface. It follows from (5) that the pressure and grad p in the film vary in accordance with the following law: p = ps +

σ 3ηλ∆Ts 2 − R0 2ρr δ04

(6)

3

STEADY STEAM CONDENSATION ON AN EXTENDED SURFACE ...

dp 3ηλ∆Ts =− ds ρr δ04

(7)

The motion along the fin under the action of capillary forces continues at the condition p–ps > 0. At the point where p = ps, for the film and fin the curvature radius R = ∞. The coordinate of this point can be determined from (5): 12

 2r σδ04 ρ S1 =   3ηλ∆TR0 

  

(8)

The surface proposed by Gregorig (1954) is described by the following equation: R0 s2 =1− 2 R S1

(9)

The local and average coefficient of heat transfer at the fin is described by  2λ3 r ρσ α=  3η∆TR S 2 0 1 

14

  

(10)

The surface proposed by Gregorig (1954) was used in Bromley et al. (1966) to enhance steam condensation on a rotating disk with radial flutes. For a simpler representation of the surface, the authors introduced an angular coordinate θ between the axis of symmetry of the fin and the radius to the surface at point s as follows:

s = ∫ Rdθ

(11)

The maximal value of the angle at R = ∞ is this: w=

2 S1 3 R0

(12)

The surface of the fin and the coefficient of heat transfer are described by the following equations: 12

θ  R0  = 1− w  R 

s 4  s   1 R0  1 + 2 R  = R w − 27  R w    0  0 

3

(13)

14

 8λ 3rρσ  α=  27η∆TR 3w2  0  

(14)

Equation (13) is an equation for a spiral with a variable direction of rotation. To calculate the shape of the surface, it is convenient to represent it in a parametric form (Figure 1):

O.A. KABOV, D.R. KOLYUKHIN, I.V. MARCHUK, J.-C. LEGROS

Y

R0

Θ φ w S1

S2 X Figure 1. Extended surface of finning (system of coordinates).

0.80

2.50

0.60 R0 = 0.3 mm

0.40

0.6

0.20

0.00 −0.40

2.00

0.8

R0 = 1.0 mm

Y, mm

Y, mm

4

1.00

0.4 0.3

−0.20

1.50

0.50 0.00

0.00 X, mm

0.20

0.40

−1.00

0.00 X, mm

1.00

Figure 2. The shape of a surface with a constant thickness of the condensate film proposed in Gregorig (1954).

STEADY STEAM CONDENSATION ON AN EXTENDED SURFACE ...

5

dX = sin ( φ ) ds = sin ( π 2 − θ ) ds = cos ( θ ) ds dY = − cos ( φ ) ds = − cos ( π 2 − θ ) ds = − sin ( θ ) ds S

X ( s ) = ∫ cos ( θ (t ) ) dt 0

S

Y ( s ) = − ∫ sin ( θ (t ) ) dt 0

Here θ = θ(s), s∈[0, S2], t∈[0, s], θ(0) = 0. The results of a calculation of the shape of the Gregorig surface are presented in Figure 2 for values of the initial radius R0 = 0.3 and 1 mm. Extended Gregorig surface. An extended Gregorig surface was proposed in Kabov (1999). The pressure in a film at s > S1 is described by Eq. (6), which can be written in the following form: σ σ  s  p = ps + −   R0 R0  S1 

2

(15)

It is seen from (15) that as the coordinate s increases after the point s = S1, the pressure in the film continues to decrease below the steam pressure. This provides further motion of the film under the action of the pressure gradient determined by Eq. (7). It is reasonable to extend the surface of the fin to a certain point where the derivative dY(s)/ds changes its sign, that is, (dY/ds = 0). At this point, the tangent to the curve is perpendicular to the axis of symmetry of the fin, and the radius of curvature is parallel to the axis of symmetry, that is, the condition w = w2 = 0 is satisfied. To determine the coordinate of this point S2 , it is convenient to write Eqs. (5), (9), and (13) for the Gregorig surface in the following form: θ=

s s3 − 2 R0 3S1 R0

It follows that S2 = 3 S1

(16)

The length of the extended Gregorig surface is by 73% greater than that of the surface in Figure 2 with similar parameters R0 and S1. Zener and Lavi (1974) studied steam condensation on a curvilinear surface for the case when the radius of curvature varies so that the gradient of capillary pressure in the film remains constant (dp/ds = const). The following expression was obtained from Eq. (3) for the average coefficient of heat transfer:

6

O.A. KABOV, D.R. KOLYUKHIN, I.V. MARCHUK, J.-C. LEGROS

14

8  λ 3rρσ  α=   3  η∆TR0 s 2 

(17)

An analytical investigation of steam condensation for a wider class of curvilinear surfaces described by the equation 1/R = (w/S1)[(ξ+1)/ξ][1–(s/S1)ξ] was performed in Adamek (1981). Here ξ is a nondimensional parameter that characterizes the fin shape (–1 ≤ ξ ≤ ∞). The solutions obtained in Gregorig (1954) and Zener and Lavi (1974) are particular cases of the solutions obtained in Adamek (1981). There also exist other models of steam condensation on fins, for instance Markowitz et al. (1972), Hirasawa et al. (1980), and other papers. There are some models of steam condensation for the entire finned surface, these models, however, will not be considered in this paper. For a current review of the field see book by Webb (1994) and Kandlikar et al. (1999). The purpose of this paper is to investigate theoretically and numerically film condensation of stationary steam on extended surfaces of condensation introduced in Gregorig (1954) and Adamek (1981). The geometrical parameters (shape) of such surfaces are optimized. For this, the mass of the liquid condensed per unit length of the condensation surface is maximized. Our calculations were made for the case when the condensate is drawn away from the center of a flute. The condensate film is assumed to be moving only under the action of capillary pressure and friction on the fin. This problem statement is valid for condensation under conditions of microgravitation (g→0) or condensation on fins of much smaller size than the capillary constant of the liquid. For most liquids at room temperature, the capillary constant varies from 2.7 mm (water) to 0.87 mm (FC-72) at a temperature of the liquid of 20 °C. 2. PROBLEM STATEMENT AND DERIVATION OF BASIC RELATIONS We consider the process of film condensation on a fin (Figure 1) with a constant temperature Tw. The steady flow of the condensate film on the surface of the fin under the action of capillary forces in the approximation of lubrication theory with allowance for heat balance and absence of heat conduction along the film is described by the following system of equations: η

∂ 2u ∂y 2

∂ 2T ∂y 2 s

∫ 0

= σκ′( s )

(18)

=0

λTy ( s, δ ( s ) ) r

(19) δ( s )

ds =

∫

ρu ( s, y ) dy

0

with boundary conditions

(20)

STEADY STEAM CONDENSATION ON AN EXTENDED SURFACE ...

u ( s,0 ) = uy ( s, δ ( s ) ) = 0, T ( s,0 ) = Tw , T ( s, δ ( s ) ) = Ts

7

(21)

The boundary conditions represent non-slip and absence of friction on the free surface of the film. The wall temperature Tw is also known. The temperature at the film surface is equal to the temperature of saturation Ts. The gravitational forces are assumed to be negligibly small. For a section of small length, the film flow is assumed plane, that is, the condition δ R 1

(22)

must be satisfied. Integration of Eqs. (18), (19) with allowance for boundary conditions (21) gives expressions for the profiles of velocity and temperature in the condensate film: u ( s, y ) = −

σκ′  y ∆T y  δ( s ) −  , T ( s, y) = Tw + y 2 η  δ( s )

(23)

Here ∆T = Ts − Tw . Substituting expressions (23) for u(s,y) and T(s,y) into Eq. (20), we obtain s

λ∆T ρσκ′δ( s)3 −1 ( s ) ds δ = − r ∫0 3η

(24)

λη∆T δ( s ) 4 3 (25) + δ ( s ) δ ( s )′ κ′ + κ′′ = 0 , r σρ 3 Analysis of this equation was made in Adamek (1981). In accordance with Adamek (1981), we introduce the following designations: c = ∆Tλη σrρ and l = δ4. Then Eq. (25) takes the following form:

or

1 1 κ′l′ + κ′′l + c = 0 4 3

(26)

Multiplying it by 4 ( κ′ )

13

, we obtain

4 κ′′ ( κ′ ) 1 3l + 4c ( κ′ ) 1 3 = 0 3 Using the identity

( κ′ ) 4 3 l ′ +

(( κ′) l )′ = l′ ( κ′) 43

43

4 13 + l ( κ′′ ) 3

we get l = 4 ( −κ′ ( s ) )

−4 3

s

∫ c ( −κ′ ( τ ) )

13

dτ

(27)

0

δ ( s ) = ( −κ′ ( s ) )

−1 3 

s

14

 4 ∫ c ( −κ′ ( τ ) )   0

13

 d τ  

(28)

8

O.A. KABOV, D.R. KOLYUKHIN, I.V. MARCHUK, J.-C. LEGROS

The following expression for the condensate flow along the fin can be obtained from (24) and (28): s s ( −κ′ ( s ) ) λ∆T λ∆T −1 ( ) m= s ds ds δ = 14 r ∫0 r ∫0  s  13  4c ∫ ( −κ′ ( τ ) ) d τ     0  13

(29)

This equation can be integrated if the integral in the denominator of the right-hand side is denoted as an unknown function F(s) and the following identity is used:

F′

∫F

4

=

4 ( F 3 4 )′ 3∫

As a result, we obtain the following expression for the condensate mass flow: s  13 ρσ   4 ∫ c ( −κ′ ( τ ) ) d τ  m(s) =  3η  0 

34

(30)

Expression (30) is simpler than expression (29), which was obtained by Adamek (1981). It is easier to analyze the problem of steam condensation on a curvilinear fin by using expression (30). 3. STEAM CONDENSATION ON A FAMILY OF ADAMEK SURFACES Let us introduce, by analogy with Adamek (1981), a family of curvilinear surfaces to fin the condensation surface. We write, however, all expressions in terms of an arbitrary coordinate s. This is necessary to analyze the properties of the surface and to change over to an extended surface: κ ( s ) = κ0 − a s ξ ,

0≤ξ 0, κ0 and ξ are geometrical parameters of the surface. Note that, in contrast to Adamek (1981), we exclude the case where ξ = 0 from the family (31). Let S1 be a point of inflexion of the film surface (κ(S1) = 0), and w is the angle between the vertical axis of symmetry of the fin and the radius of curvature of the film surface at the point S1(w = θ(S1)). Using these relations, we pass to the model parameters S1, w, ξ. Then  ξ + 1  −( ξ+1) a = ±w   S1  ξ 

(32)

 ξ + 1  −1 κ0 = ±w   S1  ξ 

(33)

STEADY STEAM CONDENSATION ON AN EXTENDED SURFACE ...

9

and

(

 ξ + 1  −1 −( ξ+1) ξ s κ ( s) = w  S1 − S1  ξ 

)

(34)

From (28), we obtain the following relation between the film thickness and the new model parameters: 14

δ ( s ) = 12c ( w ( ξ + 1)( ξ + 2 ) ) S1ξ+1s 2−ξ    −1

(35)

Then the average coefficient of heat transfer and the condensate flux on the surface are represented as follows: α ( w, S1 , ξ, s ) = s

1 λ d τ = 2.149λ  c−1 = ∫  s 0 δ ( τ)

14

( ξ + 1)( ξ + 2 )−3 wS1−( ξ+1) s( ξ−2) 

(36)

,

m ( w, S1 , ξ, s ) = 14

 λ ∆T 3 ρσ ( ξ + 1)  λ ∆T −( ξ+1) ( ξ+ 2 )  d τ = 2.149  wS s =∫  1 3 r r δ τ η ( )   2 ξ +   ( ) 0  s

(37)

The shape of the surface under consideration and the condensate flux as a function of the s-coordinate are shown in Figure 3. It is seen from (34) and (37) that the shape of the surface and the condensate flux are determined by three geometrical parameters of the surface. The parameters w and S1 are fixed in the calculations, and the effect of the parameter ξ alone is shown. In what follows, the (a)

(b) 1 2 3 4 5 6

Y/S1

0.8 0.6 0.4

0.00025 0.00020 m, kg/m ⋅ s

1

0.00015

1 2 3 4 5 6

0.00010 5E-005

0.2 0 0.2

0.4 X/S1

0.6

0

0.0004 s, m

0.0008

Figure 3. The shape of a surface proposed by Adamek (1981) (a); the condensate mass flux on the fin surface (b). S1 = 10–3 m; w = π/2; ξ = –0.9 (1), –0.5 (optimal value in Adamek’s statement) (2), –0.01 (3), 1 (4 ), 2 (Gregorig surface) (5), 4 (6 ).

10

O.A. KABOV, D.R. KOLYUKHIN, I.V. MARCHUK, J.-C. LEGROS

numerical calculations are performed for the condensation of water steam. The following values of physical characteristics are used: Tw = 368.15 K, Ts = 373.15 K, ∆T = Ts – Tw = 5 K, λ = 0.678 J/(m⋅s⋅K), η = 0.000294 kg/(m⋅s), r = 2270381 J/kg, σ = 0.0598 N/m, ρ = 961.7 kg/m3 at atmospheric pressure. At ξ → –1, an Adamek surface represents narrow high fins most often used in industrial condensers (Webb 1994). It was shown in Adamek (1981) that at ξ = –0.5 the condensate flux at the point with coordinate s = S1 is maximal. One can mention a further important property of Gregorig and Adamek surfaces. Expression (34) can be rewritten in the following form: ξ  ξ + 1  −1   s    κ ( s) = w  S1 1 −      S1   ξ    

(38)

Let us introduce non-dimensional coordinates s =

s S1

(39)

 s  κ ( s ) = κ   S1  S1 

Then, in the new coordinates, the equation for a surface from the family introduced above takes the following form:

(

 ξ +1 ξ κ ( s ) = w   1 − s  ξ 

)

(40)

It is seen from (40) that at fixed values of the parameters w and ξ all the surfaces with various values of the parameter S1 are geometrically similar. 4. EXTENDED CONDENSATION SURFACE It was assumed in Adamek (1981) that intensive condensation of steam takes place on the section from the fin top to the point of inflexion of the film surface S1, and a channel for the discharge of condensate under the action of gravity is located below this. In this paper, we considered an extension of this family. The liquid is continuously drawn off by a pump through a flute of width A on a section of the condenser surface located between fins. Owing to this, intensive condensation takes place on the entire fin surface, that is, up to the point of contact of the fin surface with the base S2 of the condenser, and not to the point S1, as it was assumed in the previous papers. It should be mentioned that such steam condensers can be realized in the conditions of microgravitation, where the condensate is removed due to the pressure difference caused by a pump. The size of a fin along the s-coordinate is designated as length and the size across the s-coordinate is designated as width. Let us determine the quantities S1 and S2 as functions of the model parameters a, κ0, and ξ.

STEADY STEAM CONDENSATION ON AN EXTENDED SURFACE ...

κ=

dθ = κ0 − a s ξ , ds

0 0.42 the point sopt coincides with S2. It is seen from Figures 10 and 14 that the value closest to –1 is an optimal value of the parameter ξ for an extended surface at w ≥ π/2. It is, however, noted in Webb (1988) that (34) yields κ0 = ∞ at ξ ≤ 0. Therefore, for reasons of feasibility it was suggested to restrict the consideration to values of ξ > 0. It was also assumed in the paper that the minimum radius of curvature that can be realized in practice at the fin top is this: Rmin = 5⋅10–5 m or κmax = 2⋅104 m–1.

STEADY STEAM CONDENSATION ON AN EXTENDED SURFACE ...

19

0.5 1 2

m1, kg/s.m2

0.4

0.3

0.2

0.1

0

0.4

0.8 w

1.2

1.6

Figure 13. The condensate mass flux vs. the parameter w with allowance for the finning coefficient. ξ = –0.9 (1), –0.8 (2); P = 10–3 m.

m1, kg/s ⋅ m2

0.6

1 2 3

0.5 0.4 0.3 0.2 0.1

−1

0

1

ξ

2

3

Figure 14. The condensate mass flux m1 vs ξ at w = π/2 at Adamek surface extended to the points: sopt (1), S1 (2), S2 (3); P = 10–3 m.

Figure 15 shows the fin surface shape at ξ = 0.03. In this case S1 = 1.84⋅10–3, sopt = 3.62⋅10–3 m, κ0 = 1.87⋅104 m–1, and the liquid flux m1 = 0.374 kg/(m2⋅s). The fin surface shape at ξ = 0.03 and κ0 = 1.87⋅104 m–1 extended to the point S2 is also shown in Figure 15. This surface is only 3% less efficient. It was assumed at the construction of the model (18)–(20) that the gravitational forces are negligibly small in comparison to the capillary forces. This condition is satisfied if the Bond number Bo is much smaller than unity. The following expression was obtained in Webb (1988): ρ g S1ξ+1 ρg 1 (56) Bo = = ξ−1 ′ w 1 s σ ξ + ( ) σκ ( )

O.A. KABOV, D.R. KOLYUKHIN, I.V. MARCHUK, J.-C. LEGROS

1 2

Y, m

0.003

0.002

0.001

0

0.0008 X, m

Figure 15. The film surface shapes at extension to the points: sopt(1), S2 (2), ξ = 0.03.

Expression (56) is valid also for extended surface. It is seen from (56) that this condition is always satisfied in the microgravity condition. If Bo ≅ 1, the force of gravity affects the flow considerably. Besides, to use the model (18)–(20) correctly, it is necessary to satisfy condition (22). It is shown in Figure 16, with s varying from 0 to the point S2, that under Earth’s gravitation conditions (56) and (22) are satisfied satisfactorily at S1 ≅ 10–4 m, that is, for surfaces about 1 mm in length. Recall that R = ∞ at s = S1. Therefore, in Figure 16, δ/R at this point is equal to zero. (a)

(b)

0.003

0.4 0.3 |δ/R|

0.002 Bo

20

0.2

0.001 0.1

0

0.0001

0.0002 s, m

0.0003

0

0.0001

0.0002

0.0003

s, m

Figure 16. Criteria of applicability of the model under Earth’s gravitation; ξ = 0.01, w = π/2, S1 = 4⋅10–4m.

STEADY STEAM CONDENSATION ON AN EXTENDED SURFACE ...

21

8. CONCLUSIONS The theoretical investigation of stationary steam condensation on curvilinear fins described by an equation for a spiral with a variable direction of rotation started in Gregorig (1954) was continued in this paper. It was proposed to extend the wellknown surfaces of condensation proposed in Gregorig (1954) and Adamek (1981) to obtain more optimal fins. The fin shape obtained has made it possible to increase the length of the surface at which the condensate film moves under the action of the capillary pressure gradient by 73% and more. Mathematical analysis and numerical calculations were carried out for the case when the condensate was drawn away from the center of a flute. The properties of these surfaces, which are described in the general case by four parameters, have been analyzed. The effects of these parameters on the intensity of heat transfer have been analyzed analytically and numerically. The condensate mass flux per unit length of film surface projection has been optimized. The shapes of fins that can be realized in practice have been calculated. At the use of an extended surface that can be manufactured by traditional methods, the condensate flux can be by 62% greater than at the use of the surface proposed in Adamek (1981). For extended surfaces in a rather wide range of geometrical sizes of fins, the condensate flux varies only slightly. This fact makes it possible to choose “the most technological” fins for practical use. The calculations performed have shown that fins of minimum length should be used. In the best case, for any fin shape from the family under consideration, the ratio between the width P of a fin and the width A of a flute between the fins through which the condensate is drawn off is 1:3. Thin fins are most optimal in this statement. In subsequent studies, for a more correct description of heat transfer at steam condensation on finned surfaces, it is necessary to consider heat conduction in the fin body. Conditions of applicability of the model at Earth’s gravitation were considered in the last section of this paper. From the point of view of practical application, this problem statement most fully corresponds to condensation under the microgravity condition. At steam condensation on conventional vertical and horizontal tubes without removal of the condensate, and with a given kind of fins, calculations will provide the greatest possible intensification of heat transfer, i.e., a certain theoretical limit. In this way, this paper can be used to construct condensation surfaces for industrial devices. Acknowledgements The authors gratefully acknowledge the support of this work by the Russian Foundation for Basic Research (Grant for Support of Leading Scientific Schools No. 00-15-96810) and financial support by Euro Heat Pipes S.A. (Brussels). NOMENCLATURE A Bo g m

width of a flute to draw the condensate off, m Bond number gravitational acceleration, m/s condensate flux, kg/(m⋅s)

22

O.A. KABOV, D.R. KOLYUKHIN, I.V. MARCHUK, J.-C. LEGROS

m1 P p R r S1 S2 s sopt T Ts Tw ∆T u(s,y) w y

condensate flux per unit length of fin projection surface, kg/(m2⋅s) length of film surface projection, m pressure, N/m2 radius of film surface curvature, m, radius of tube, m heat of phase transition, J/kg coordinate of inflexion point, m coordinate of contact of film surface with condenser surface, m coordinate along the film surface, m optimal coordinate along film surface to which the fin is extended, m film temperature, K steam saturation temperature, K wall temperature, K temperature difference Ts.– Tw, K film flow velocity, m/s angle between the vertical and radius of curvature at inflexion point S1 coordinate

Greek Symbols α α α1 δ εA η θ κ κ0 λ λm ξ ρ σ φ

coefficient of convective heat transfer, W/(m2⋅K) average coefficient of convective heat transfer, W/(m2⋅K) average coefficient of convective heat transfer with allowance for the finning coefficient, W/(m2⋅K) film thickness, m coefficient of finning (surface area ratio) coefficient of dynamic viscosity, kg/(m⋅s) angle between the vertical and the radius of curvature curvature of the film surface, l/m curvature of the film surface on the fin top, l/m heat conductivity coefficient of the liquid, J/(m⋅s⋅K) heat conductivity coefficient of the fin, J/(m⋅s⋅K) geometrical parameter of the surface density of liquid, kg/m3 coefficient of surface tension, N/m angle between the vertical and the tangent to the surface

Indices 0 opt s w

initial value optimal saturation wall

REFERENCES Adamek, T., Bestimmung der Kondensationgrossen auf Feingewellten Oberflachen zur Auslegung Optimaler Wandprofile, Waerme- und Stoffuebertrag., 15, P. 255–270, 1981. Anisimov, S.V. and Smirnov, Yu.B., Heat transfer at steam condensation on horizontal tubes with fins of complex shapes, Therm. Eng., (11), P. 38–41, 1997.

STEADY STEAM CONDENSATION ON AN EXTENDED SURFACE ...

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Arai, N., Fukushima, T., Arai, A., et al., Heat transfer tubes enhancing boiling and condensation in heat exchangers of a refrigerating machine, ASHRAE Trans., 83, P. 2, P. 58– 70, 1977. Bergles, A.E., Enhancement of heat transfer, Heat Transfer 1978, Proc. Sixth Int. Heat Transfer Conf., Toronto, Vol. 6, P. 89–108, 1978. Bromley, L.A., Hemphris, R.F., and Murray, V., Condensation and evaporation on rotating disks with radial flutes, Proc. Amer. Soc. of Eng.-Mech., Ser. C, Heat Transfer, 88(1), P. 87–96, 1966. Cheng, B. and Tao, W.Q., Experimental study of R-152a film condensation on single horizontal smooth tube and enhanced tubes, J. Heat Transfer, 116, P. 266–270, 1994. Gogonin, I.I. and Kabov, O.A., Enhancement of heat transfer at vapor condensation by finning, Russ. J. Eng. Thermophys., 1(1), P. 51–57, 1991. Gogonin, I.I. and Kabov, O.A., An experimental study of R-11 and R-12 film condensation on horizontal integral-fin tubes, J. Enhanced Heat Transfer, 3(1), P. 43–53, 1996. Gogonin, I.I., Kataev, А.I., and Sosunov, V.I., Investigation of Heat Transfer and Hydrodynamics at Steam Condensation on the Banks of Smooth and Finned Tubes, Condensation and Condenser Design, ASME 1993, Florida, P. 443–452, 1993. Gregorig, R., Hautkondensation an Feingewellten Oberflächen bei Berüksichtigung der Oberflächenspannungen, Zeitschrift für angewandte Mathematik und Physik, 5(1), P. 36–49, 1954. Fujii, T., Enhancement to condensing heat transfer – New developments, J. Enhanced Heat Transfer, 2(1–2), P. 127–137, 1995. Hirasawa, S., Hijikata, K., Mori, Y., and Nakayama, W., Effect of surface tension on condensate motion in laminar film condensation (Study of liquid film in a small trough), Int. J. Heat Mass Transfer, 23(11), P. 1471–1478, 1980. Honda, H. and Kim, K., Effect of fin geometry on the condensation heat transfer performance of a bundle of horizontal low-finned tubes, J. Enhanced Heat Transfer, 2(1–2), P. 139–147, 1995. Kabov, O.A., Enhancement of heat transfer at vapor condensation on horizontal tubes in evaporative cooling systems of electronic equipment, Proc. Int. Sem. Evaporative Cooling Systems of Electronic Equipment, Novosibirsk, P. 209–235, 1993. Kabov, O.A., Capillary Effects Influence on Vapor Condensation and Heat Transfer in Falling Liquid Films, Dr. Sci. Thesis, Institute of Thermophysics SB RAS, Novosibirsk, Russia, 1999. Kabov, O.A., Shamovskii, A.E., and Sippolainen, V.K., Experimental investigation of film condensation on horizontal tubes with spine fins, Sibirskii Fiz.-Tekh. Zh., 1, P. 9–18, 1993. Kandlikar, S. G., Shoji, M., and Dhir, V.K., Handbook of Phase Change: Boiling and Condensation, Taylor and Francis, London, P. 738, 1999. Kedzierski, M.A., Webb, R.L., Practical fin shapes for surface-tension-drained condensation, ASME J. Heat Transfer, 112, P. 479–485, 1990. Markovich, A., Mikich, V.V., and Bergles, A.E., Condensation on a horizontal wavy surface directed downward, Proc. Amer. Soc. of Eng.-Mech., Ser. C, Heat Transfer, 94(3), P. 62–69, 1972. Marto, P.J., An evaluation of film condensation on horizontal integral fin tubes, J. Heat Transfer, 110, P. 1287–1305, 1988. Marto, P.J. and Nunn, R.H., The Potential of Heat Transfer Enhancement in Surface Condensers, Proc. Condensers: Theory and Practice 1983, Pergamon Press, P. 23–39, 1983. Marto, P.J., Zebrowski, D., Wanniarachchi, A.S., and Rose, J.W., An experimental study of R-113 film condensation on horizontal integral-fin tubes, J. Heat Transfer, 112, P. 758–767, 1990.

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O.A. KABOV, D.R. KOLYUKHIN, I.V. MARCHUK, J.-C. LEGROS

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