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BOOKCOMP, Inc. — John Wiley & Sons / Page 725 / 2nd Proofs / Heat Transfer Handbook / Bejan FILM CONDENSATION ON LOW FINS 725 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [725], (7) Lines: 303 to 342 ——— 13.24428pt PgVar ——— Long Page * PgEnds: Eject [725], (7) the interface induces a pressure decrease within the film. The expression for this pressure variation, known as the surface tension pressure gradient, is obtained by differentiating eq. (10.13) with respect to the fin arc length(s) dP ds = σ d(1/r 1 ) ds = σ dκ i ds (10.14) where 1/r 2 = 0 for a two-dimensional surface. Figure 10.2 shows the coordinate system for a condensate film on a convex fin surface profile. The coordinate measured along the liquid–vapor interface is s  . The coordinate measured along the fin surface is s. The location s = 0 is the point of symmetry and is referred to the fin tip. The film has thickness δ. The condensate surface turns through a maximum angle of θ m and a maximum arc length S m . The coordinate measured perpendicular to the base-metal surface is y. The curvature of the liquid–vapor interface, shown in Fig. 10.2, decreases for increasing values of the coordinate s  . In general, decreasing pressure gradients can be achieved with fin tips of small curvature. A general function for the liquid–vapor interface curvature (κ i ) can be represented as a function of s  : κ i = C 1 − C 2 s  ζ (10.15) where C 1 ,C 2 , and ζ are arbitrary constants. As illustrated in the following section, specification of the interface curvature allows the fin designer to investigate the influence of the fin shape on the condensation performance. 10.3.3 Specified Interfaces Gregorig (1954) proposed to increase Nusselt condensation by shaping a convex condensate surface such that surface tension forces alone would produce a film of constant thickness: δ =  k l µ l (T sat − T w )S 3 m σλρ l θ m  1/4 (10.16) By using h = k l /δ, the local heat transfer coefficient for Gregorig’s surface becomes h =  σλθ m k 3 l ν l (T sat − T w )S 3 m  1/4 (10.17) Zener and Lavi (1974) proposed a convex shape that gives a constant-pressure gradient along the convex surface. The local heat transfer coefficient for the Zener and Lavi profile is h =  σλθ m k 3 l 2ν l (T sat − T w )S 2 m s   1/4 (10.18) BOOKCOMP, Inc. — John Wiley & Sons / Page 726 / 2nd Proofs / Heat Transfer Handbook / Bejan 726 CONDENSATION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [726], (8) Lines: 342 to 375 ——— 0.88617pt PgVar ——— Normal Page PgEnds: T E X [726], (8) The average heat transfer coefficient for the Zener and Lavi (1974) profile is 15% larger than that for the Gregorig (1954) surface. Adamek (1981) showed that there is an entire family of convex shapes that utilize surface tension to drain the film. His curvature is defined as κ i = (ζ + 1)θ m ζS m  1 −  s  S m  ζ  (10.19) Figure 10.3 shows five different Adamek profiles for ζ values within the range −0.9 ≤ ζ ≤ 2. The profiles of the liquid–vapor interface, as shown in Fig. 10.3, start at s  = 0 at the fin tip and rotate through equal lengths of S m . The local heat transfer coefficient for the Adamek profile is h =  σλθ m k 3 l (ζ + 1)(ζ + 2) 12ν l (T sat − T w )S ζ+1 m s 2−ζ  1/4 (10.20) where ζ =−0.5 gives the maximum heat transfer coefficient. The region of surface tension influence is confined to a few molecular thicknesses at the liquid–vapor interface (Freundlich, 1922). The thinness of the film permits the base-metal shape to influence the shape of the liquid–vapor interface. Therefore, fins can be designed for large pressure gradients by carefully considering the fin size and the base-metal fin curvature (κ b ). ␨ = 2 Gregorig (1954) ␨ = 0 ␨Ϫ= 0.9 ␨Ϫ= 0.5 ␨ = 1 Zener & Lavi (1974) Figure 10.3 Family of Adamek (1981) liquid–vapor interface fin profiles. BOOKCOMP, Inc. — John Wiley & Sons / Page 727 / 2nd Proofs / Heat Transfer Handbook / Bejan FILM CONDENSATION ON SINGLE HORIZONTAL FINNED TUBES 727 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [727], (9) Lines: 375 to 408 ——— -0.87085pt PgVar ——— Normal Page PgEnds: T E X [727], (9) It is not convenient to specify fins base on curvature. A designer would rather specify the geometry of the fin by controlled machining parameters. Kedzierski and Webb (1990) have defined an alternative set of condensate surface profiles (referred to here as the K-W profiles), where the fin tip radius (r o ), the fin thickness at the root (t r ), the fin height (e), the fin angle (θ m ), and the shape factor Z are independently specified. Smaller values of Z produce narrower fin tips. Appendix A presents the equations that describe the fin profiles. The thickness of the condensate film for the K-W profile is given by (Jaber and Webb, 1996) δ = k l h = 4Cp −1 r  3C 2  e Zθ m − 1  + 4Cp −1 r  C 1/3 3 Z  0.5ln N m N o + √ 3  tan −1 Y m − tan −1 Y o    (10.21) where C = 3µ l k l (T sat − T w )/ρ 2 l gλ, and C 2 ,C 3 ,p,N m ,N o ,Y m , and Y o are given in Appendix A. 10.3.4 Bond Number Care must be taken to ensure that the fin height (e) is not so large that the surface tension pressure gradient has dissipated over a significant portion of the fin. The Bond number (Bd), which is the ratio of gravity forces to surface tension forces, can be used to test the strength of the surface tension pressure gradient. If the surface tension forces are dominant over gravity forces, the condensate drainage is determined by surface tension. The strength of the surface tension pressure gradient weakens as the film approaches the base of the fin. The Bond number at the base of the fin can be approximated by (Kedzierski and Webb, 1990) Bd = (ρ l − ρ g )ge 2 σθ m (10.22) Here, Bd = 1 implies that surface tension forces are equal to gravity forces at the end of the fin and that surface tension forces are greater than gravity forces for the remainder of the fin. Equation (10.22) should always be used to check first if surface tension forces are truly dominant (Bd < 1) over gravity forces before performing an analysis that assumes so. Equation (10.22) predicts that small fin heights and large θ m give strong pressure gradients. 10.4 FILM CONDENSATION ON SINGLE HORIZONTAL FINNED TUBES 10.4.1 Introduction Advances in metal-forming processes have enabled the use of surface tension drain- age theory in the design of special finned surfaces for horizontal tubes. In this section BOOKCOMP, Inc. — John Wiley & Sons / Page 728 / 2nd Proofs / Heat Transfer Handbook / Bejan 728 CONDENSATION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [728], (10) Lines: 408 to 440 ——— -0.95297pt PgVar ——— Long Page PgEnds: T E X [728], (10) we review the condensing performance of the trapezoidal and sawtooth fin geometries for vapor space condensation. The trapezoidal fin is a relatively simple and inexpen- sive fin geometry. However, a greater heat transfer performance can be obtained with a more complicated and expensive sawtooth fin geometry. 10.4.2 Trapezoidal Fin Tubes Figure 10.4 shows a sketch of the cross section of an integral finned tube with an envelope diameter of D o and a root diameter of D r . During manufacturing, the outer fin surface of the tube is lifted from the surface of a plane tube via a rolling process that leaves one side of the fin tip with a rounded corner. The sketch shows the key ge- ometric parameters of the tube: the spacing between fins at the tip (b) and root (b r )of the fin, fin pitch (p), fin height (e), fin tip thickness (t), and half-angle at the fin tip (β). Compared to many passively enhanced condenser tubes, integral finned tubes are relatively inexpensive and can significantly improve the heat transfer performance over that of a plain tube. The heat transfer of a low fin (< 1.5 mm) tube is greater than that of a plain tube per unit length because the finned tube exhibits (1) additional surface area over a plain tube per unit tube length, (2) a short condensing length over the fin compared to the tube diameter, and (3) surface tension drainage forces along the fins. Surface tension forces also cause a degradation in heat transfer through the reten- tion of a relatively thick condensate film between the fins of the lower part of the tube. Honda et al. (1983) derived the following expression for the condensate reten- tion angle (φ f ), defined as the angle between the top of the tube and the point where the tube begins to flood with condensate: φ f = cos −1  4σ cos β ρ l gbD 0 − 1  for e>2b(1 − sin β)/ cos β (10.23) Equation (10.23) was also derived by Rudy and Webb (1981) for the case of β = 0. Figure 10.4 Cross section of integral-fin tube. BOOKCOMP, Inc. — John Wiley & Sons / Page 729 / 2nd Proofs / Heat Transfer Handbook / Bejan FILM CONDENSATION ON SINGLE HORIZONTAL FINNED TUBES 729 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [729], (11) Lines: 440 to 463 ——— 8.57927pt PgVar ——— Long Page PgEnds: T E X [729], (11) Typically, heat transfer in the flood zone is neglected. Analysis of the unflooded region focuses on calculating the condensate film thickness along the fin, that is, the solution to eq. (10.8). The key obstacle to solving eq. (10.8) is the determination of the appropriate expression for the surface tension pressure gradient (dP /ds) and the component of gravity (ρ l g) that acts to drain the condensate from the fin. It is common to assume some relationship for dP/ds as a function of fin arch length. Researchers such as Webb et al. (1985), Adamek and Webb (1990), Rifert (1980), and Karkhu and Borovkov (1971) have assumed a linear pressure gradient along the fin length. Even with a linear assumption for the pressure gradient, an explicit solution that includes the effect of gravity has yet to be derived. Toward this end, Honda et al. (1987) have managed a numerical solution that couples the effects of surface tension and only the component of gravity along the fin arch. Consequently, this solution is strictly valid only at the top and bottom of a horizontal tube. Considering that it is crucial to include the effects of gravity as the Bond number increases above 0.1, the explicit semiempirical calculation method of Rose (1994) is much welcomed. Rose (1994) couples the effects of surface tension and gravity through an expression for the average condensate thickness on the unflooded portion of the tube: δ =  µV A(ρ l − ρ g )g/l g + Bσ/l 3 σ  1/3 (10.24) Here V is the mean volume of condensate flux per area of surface; A and B are constants representing the influence of gravity and surface tension, respectively; and l g and l σ are characteristic lengths for gravity- and surface tension–driven flows, respectively. The characteristic lengths and the constants are each assigned a value for the fin tip, fin root, and fin side regions for the unflooded portion of the tube. The gravity constant A is given as either 0.728 for tubelike surfaces or 0.943 for fin sides [see the leading coefficients in eqs. (10.11) and (10.12), respectively]. The B constants are obtained through regression against measured condensation heat transfer data on finned tubes. By using V = q  /h fg ρ l , eq. (10.24), and q  = k l (T sat − T w )/δ, the heat flux may be written for the fin tip, fin side, and tube surface between the fins. The flooded portion of the tube is assumed to be inactive for heat transfer. The heat fluxes for the three regions are summed and weighted by the appropriate surface areas and rearranged to give the heat transfer coefficient for the finned tube (h f ). Rose (1994) obtained the enhancement ratio by normalizing h f by the heat transfer of the smooth tube (h s ) with an outside diameter equal to the root diameter of the finned tube: h f h s = Ψ 1 + Ψ 2 + Ψ 3 0.728(b +t) Ψ 1 =  D o D r  3/4 t  0.281 + BσD o t 3 g(ρ l − ρ g )  1/4 BOOKCOMP, Inc. — John Wiley & Sons / Page 730 / 2nd Proofs / Heat Transfer Handbook / Bejan 730 CONDENSATION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [730], (12) Lines: 463 to 503 ——— 5.84865pt PgVar ——— Normal Page PgEnds: T E X [730], (12) Ψ 2 = φ f π 1 − f f cos β D 2 o − D 2 r 2e 1/4 v D 3/4 r  0.791 + Bσe v e 3 g(ρ l − ρ g )  1/4 Ψ 3 = φ f π B 1 (1 − f r )b r   ξ(φ f )  3 + BσD r b 3 r g(ρ l − ρ g )  1/4 (10.25) where ξ(φ f ) = 0.874 + 0.1991 ×10 −2 φ f − 0.2642 ×10 −1 φ 2 f + 0.5530 ×10 −2 φ 3 f − 0.1363 ×10 −2 φ 4 f (10.26) f f = 1 − tan(β/2) 1 + tan(β/2) 2σ cos β ρ l gD r e tan(φ f /2) φ f (10.27) f r = 1 − tan(β/2) 1 + tan(β/2) 4σ b r ρ l gD r tan(φ f /2) φ f (10.28) e v = φ f sin φ f e φ f ≤ π 2 (10.29) e v = φ f 2 − sin φ f e π 2 ≤ φ f ≤ π (10.30) Here f f and f r are the fraction of the fin side and root of the tube that are flooded with condensate, respectively, and e v represents the mean vertical fin height of the tube. The enhancement ratio is valid where h f and h s have the same driving temperature difference. The agreement that Rose (1994) obtained between the experimental data available and the fit of those data to eq. (10.25) was approximately ±20% for B = 0.143 and B l = 2.96. 10.4.3 Sawtooth Fin Condensing Tubes Figure 10.5 compares the cross sections of a sawtooth or notched tube (Turbo-CDI) to that of a tube with trapezoidal fins (Turbo-Chil). The Turbo-CDI has 1575 fins per meter, a 1-mm fin height before notching on the outside tube surface, with 35 ridges with a 0.5-mm ridge height on the inside tube surface. The fins on the outside of the Turbo-Chil (1024 ft/min) are 1.4 mm in fin height and the 10 ridges on the inside of the tube are 0.4 mm high. Both tubes have a wall thickness of 0.7 mm, the same envelope diameter (D o ), and internal fins. Coolant flows inside the tube while condensation takes place on the outside tube surface. The sharp tips of the sawtooth provide the large curvature and curvature gradients that are necessary to induce large surface tension pressure gradients, which act to thin the condensate and enhance condensation heat transfer. BOOKCOMP, Inc. — John Wiley & Sons / Page 731 / 2nd Proofs / Heat Transfer Handbook / Bejan FILM CONDENSATION ON SINGLE HORIZONTAL FINNED TUBES 731 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [731], (13) Lines: 503 to 511 ——— 0.36098pt PgVar ——— Normal Page PgEnds: T E X [731], (13) Figure 10.5 Cross sections of the Turbo-CDI and Turbo-Chil. (Courtesy of Wolverine Tube, Inc.) According to Webb (1994), no model exists for condenser tubes with sawtooth fin shapes. Consequently, the heat transfer performance of select tubes is presented here graphically. Figure 10.6 provides the condensation heat transfer coefficient for a sawtooth and trapezoidal fin tube versus the heat flux, both based on the envelope area of the tube. Figure 10.6 illustrates that the heat transfer performance of the sawtooth tube is approximately twice that of a trapezoidal integral fin tube of the same envelope diameter. Both area increase and surface tension effects contribute to the performance increase. BOOKCOMP, Inc. — John Wiley & Sons / Page 732 / 2nd Proofs / Heat Transfer Handbook / Bejan 732 CONDENSATION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [732], (14) Lines: 511 to 513 ——— -1.26302pt PgVar ——— Normal Page PgEnds: T E X [732], (14) 5000 10000 15000 20000 25000 30000 qЉ (Btu/hr-ft ) 2 qЉ (kW/m ) 2 0 5 0 1000 10 2000 15 3000 20 4000 25 5000 30 6000 Btu/hr-ft -°F 2 (kW/m . K) 2 Turbo-CDI 1575 fpm, = 0.97 mme Turbo-Chil 1024 fpm, = 1.4 mme 20 30 40 50 60 70 80 90 R134a Vapor Space Condensation, = 314 K, 19.1 mm OD single copper tubes with 0.71 mm thickness T s h 0 h 0 Figure 10.6 Condensation heat transfer performance of standard and sawtooth tubes. (Cour- tesy of Wolverine Tube, Inc.) 10.5 ELECTROHYDRODYNAMIC ENHANCEMENT 10.5.1 Introduction The electrohydrodynamic (EHD) enhancement technique uses a high-voltage, low- current electric field to mix the condensate film or to remove it from the tube surface. EHD enhancement requires a fluid that has low electrical conductivity, such as re- frigerants. Consequently, most of the EHD studies have been with refrigerants and refrigerant mixtures. BOOKCOMP, Inc. — John Wiley & Sons / Page 733 / 2nd Proofs / Heat Transfer Handbook / Bejan ELECTROHYDRODYNAMIC ENHANCEMENT 733 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [733], (15) Lines: 513 to 563 ——— 0.6131pt PgVar ——— Normal Page PgEnds: T E X [733], (15) The magnitude and nature of the enhancement is a function of the (1) electric field, (2) flow parameter, and (3) heat transfer surface (Ohadi, 1991). The field potential, its polarity, and the electrode geometry and spacing determine the electric field. The Reynolds number and working fluid properties fix the flow parameters. The EHD technique is more effective for low Reynolds numbers. Bologa et al. (1987) reported enhancements as large as 2000% for film condensation on a plate. 10.5.2 Vapor Space EHD Condensation Typically, the electrode is a screen wrapped around the tube and spaced a certain distance from the tube. The gap between the tube and the electrode and the dielectric strength influence the enhancement. Currently, there are no correlations that predict vapor space EHD condensation. However, it is possible to enhance plain tube and enhanced tube performance with EHD by ten- and threefold, respectively (Ohadi, 1991; Da Silva et al., 2000). Enhancements result from condensate removal from the tube. In fact, Yabe (1991) shows that the extraction of liquid from the tube surface can be effective enough to promote pseudo-dropwise condensation. In general, the heat transfer coefficient is directly proportional to the applied voltage. 10.5.3 In-Tube EHD Condensation Typically, a helical or a straight-rod electrode is centered within the tube to enhance in-tube condensation. The tube is grounded to create an electric field between the electrode and the tube wall. EHD can be used to increase smooth tube condensation by a factor of nearly 6.5. Gidwani et al. (1998) developed R-404a and R-407c con- densation heat transfer correlations for a 3.17-mm straight-rod electrode placed in a 11.1-mm-diameter smooth tube and a 10.60-mm-diameter corrugated tube with a 7.1-mm pitch and a 1-mm corrugation. The ratio of the in-tube condensation heat transfer coefficient with EHD (h E ) to that with no EHD (h) for R-404a is h E h = 1 + cEs n l  G T 300  n 2  1 − x q x q  n 3 Ja n 4 (10.31) where the ranges for which the correlation holds are 70 ≤ Es = κε o E 2 D i σ ≤ 120 5kg/m 2 · s ≤ G T ≤ 300 kg/m 2 · s 0.06 ≤ Ja ≤ 2.8 950 ≤ Re lo ≤ 13,000 0.1 ≤ x q ≤ 0.9 where the constant c and the exponents are given in Table 10.1 and h is calcu- lated from the smooth correlation given in Section 10.5. Hence ε 0 is the dielectric BOOKCOMP, Inc. — John Wiley & Sons / Page 734 / 2nd Proofs / Heat Transfer Handbook / Bejan 734 CONDENSATION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [734], (16) Lines: 563 to 614 ——— 12.04817pt PgVar ——— Long Page PgEnds: T E X [734], (16) TABLE 10.1 Values of Coefficients for R-404a EHD In-Tube Condensation Coefficient Smooth Tube Corrugated Tube c 0.00174 0.00868 n 1 0.98833 0.70733 n 2 −0.54505 −0.33768 n 3 0.59805 0.37158 n 4 −0.27722 −0.15107 Source: Gidwani et al. (1998). permittivity for a vacuum and κ(= ε/ε 0 ) is the dielectric constant of the fluid. Most of the data were correlated to within ±30%. The electric field (E) for use in eq. (10.31) is estimated from the applied voltage (V )as E = 2V D i ln(D i /D el ) (10.32) where D i is the inner diameter of the tube and D el is the outer diameter of the inner electrode. The ratio of the in-tube condensation heat transfer coefficient with EHD (h E ) to that with no EHD (h) for R-407c is h E h = 1 + c(log Es) n 1  G T 300  n 2  1 − x q x q  n 3 Ja n 4 (10.33) TABLE 10.2 Values of Coefficients for R-407c EHD In-Tube Condensation Smooth Tube Coefficient x q 1 − x q G T P r 300 ≤ 0.05 x q 1 − x q G T P r 300 > 0.05 c 0.23173 0.0586 n 1 1.6711 0.46622 n 2 0.07777 −0.26072 n 3 0.22064 −0.09885 n 4 −0.16572 −1.77379 Corrugated Tube c 0.06221 0.018457 n 1 2.53152 2.88447 n 2 0.10825 −0.06351 n 3 0.47615 0.25078 n 4 −0.00954 −0.70633 Source: Gidwani et al. (1998). . assumed to be inactive for heat transfer. The heat fluxes for the three regions are summed and weighted by the appropriate surface areas and rearranged to give the heat transfer coefficient for the. to thin the condensate and enhance condensation heat transfer. BOOKCOMP, Inc. — John Wiley & Sons / Page 731 / 2nd Proofs / Heat Transfer Handbook / Bejan FILM CONDENSATION ON SINGLE HORIZONTAL. the heat transfer performance of select tubes is presented here graphically. Figure 10.6 provides the condensation heat transfer coefficient for a sawtooth and trapezoidal fin tube versus the heat

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