Heat Transfer Handbook part 74 pdf

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 tr, t

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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/r1)

dκ i

where 1/r2= 0 fora 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 iss The

coordinate measured along the fin surface iss The location s = 0 is the point of symmetry and is referred to the fin tip The film has thicknessδ The condensate

surface turns through a maximum angle ofθm and a maximum arc lengthS m The coordinate measured perpendicular to the base-metal surface isy.

The curvature of the liquid–vapor interface, shown in Fig 10.2, decreases for increasing values of the coordinates In general, decreasing pressure gradients can

be achieved with fin tips of small curvature A general function for the liquid–vapor interface curvature (κi) can be represented as a function ofs:

whereC1, C2, andζ 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 lµl (Tsat− T w )S3

m

σλρlθm

1/4

(10.16)

By usingh = k l /δ, the local heat transfer coefficient for Gregorig’s surface becomes

h =

m k3

l

νl (Tsat− T w )S3

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 =

m k3

l

2νl (Tsat− T w )S2

m s

1/4

(10.18)

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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

κi = (ζ + 1)θ ζS m

m



1−

s

S m

ζ

(10.19)

Figure 10.3 shows five different Adamek profiles forζ 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 ofS m The local heat transfer coefficient forthe Adamek profile is

h =

 σλθ

m k3

l (ζ + 1)(ζ + 2)

12νl (Tsat− T w )S mζ+1s2−ζ

1/4

(10.20)

whereζ = −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 (κb)

␨ = 2

Gregorig (1954)

␨ = 0

␨ ⫺= 0.9 ␨ ⫺= 0.5

␨ = 1

Zener & Lavi (1974)

Figure 10.3 Family of Adamek (1981) liquid–vapor interface fin profiles

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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 (tr), the fin height (e), the fin angle (θm), and the shape factorZ 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−1r 3C2

e Zθ m− 1

+ 4Cp−1r



C1/3

3

Z



0.5 ln N N m

o +√3

tan−1Y m− tan−1Y o

(10.21) whereC = 3µ l k l (Tsat− T w )/ρ2

l gλ, and C2, C3, p, N m , N o , Y m, andY oare given in Appendix A

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− ρg )ge2

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

θmgive strong pressure gradients

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

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we review the condensing performance of the trapezoidal and sawtooth fin geometries forvaporspace 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 diameterofD oand a root diameter ofD 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 (br) 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 overthat of a plain tube The heat transferof a low fin (< 1.5 mm) tube is greater than that of a plain tube perunit 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 (φf), defined as the angle between the top of the tube and the point where the tube begins to flood with condensate:

φf = cos−1

4σ cos β

ρl gbD0

− 1



fore > 2b(1 − sin β)/ cos β (10.23) Equation (10.23) was also derived by Rudy and Webb (1981) for the case ofβ = 0

Figure 10.4 Cross section of integral-fin tube

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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 (ρl g) that acts to drain the condensate from the fin It is common

to assume some relationship fordP /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− ρg )g/l g + Bσ/l3

σ

1/3

(10.24)

HereV 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 andlσ 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 constantA is given as either0.728 fortubelike surfaces or0.943 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 usingV = q/h fgρl, eq (10.24), andq = k l (Tsat− 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 (hf) Rose (1994) obtained the enhancement ratio by normalizingh fby the heat transfer of the smooth tube (hs) with an outside diameterequal to the root diameterof the finned tube:

h f

h s =

Ψ1+ Ψ2+ Ψ3

0.728(b + t)

Ψ1=



D o

D r

3/4

t



0.281 + BσD o

t3g(ρ l− ρg )

1/4

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Ψ2 =φf π

1− f f

cosβ

D2

o − D2

r

2e1/4

v D3/4 r



0.791 + Bσe v

e3g(ρ l− ρg )

1/4

Ψ3 =φπf B1(1 − f r )b r ξ(φ f ) 3

+b3 BσD r

r g(ρ l− ρg )

1/4

(10.25)

where

ξ(φ f ) = 0.874 + 0.1991 × 10−2φf − 0.2642 × 10−1φ2

f + 0.5530 × 10−2φ3

f

− 0.1363 × 10−2φ4

f f =1− tan(β/2)

1+ tan(β/2)

2σ cos β

ρl gD r e

tan(φf /2)

f r =1− tan(β/2)

1+ tan(β/2)

4σ

b rρl gD r

tan(φf /2)

e v = φf

sinφf e φf ≤

π

e v = φf

2− sin φf e

π

Heref fandf rare the fraction of the fin side and root of the tube that are flooded with condensate, respectively, ande vrepresents the mean vertical fin height of the tube

The enhancement ratio is valid whereh fandh shave 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 Bl = 2.96.

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(Do), 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

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Figure 10.5 Cross sections of the Turbo-CDI and Turbo-Chil (Courtesy of Wolverine Tube, Inc.)

According to Webb (1994), no model exists forcondensertubes 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

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q⬙ (Btu/hr-ft )2

q⬙ (kW/m )2

0

5

0 1000

10 2000

15 3000

20 4000

25

5000

30 6000

K)

Turbo-CDI

1575 fpm,

= 0.97 mm

e

Turbo-Chil

1024 fpm,

= 1.4 mm

e

R134a Vapor Space Condensation,

= 314 K, 19.1 mm OD single copper tubes with 0.71 mm thickness

T s

h0 h0

Figure 10.6 Condensation heat transfer performance of standard and sawtooth tubes (Cour-tesy of Wolverine Tube, Inc.)

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

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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 vaporspace 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

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 factorof 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-diametersmooth tube and a 10.60-mm-diametercorrugated tube with a 7.1-mm pitch and a 1-mm corrugation The ratio of the in-tube condensation heat transfer coefficient with EHD (hE) to that with no EHD (h) forR-404a is

h E

h = 1 + cEs n l



G T

300

n2

1− x q

x q

n3

where the ranges for which the correlation holds are

70≤ Es = κεo E2D i

5 kg/m2· s ≤ G T ≤ 300 kg/m2· s

0.06 ≤ Ja ≤ 2.8

950≤ Relo≤ 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 Henceε0 is the dielectric

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TABLE 10.1 Values of Coefficients for R-404a EHD In-Tube Condensation

Source: Gidwani et al (1998).

permittivity for a vacuum andκ(= ε/ε0) is the dielectric constant of the fluid Most

of the data were correlated to within±30%

The electric field (E) foruse in eq (10.31) is estimated from the applied voltage (V ) as

whereD i is the innerdiameterof the tube andDelis the outerdiameterof the inner electrode The ratio of the in-tube condensation heat transfer coefficient with EHD (hE) to that with no EHD (h) forR-407c is

h E

h = 1 + c(log Es) n1



G T

300

n2

1− x q

x q

n3

Jan4 (10.33)

TABLE 10.2 Values of Coefficients for R-407c EHD In-Tube Condensation

Smooth Tube

1− x q

G T P r

1− x q

G T P r

300 > 0.05

Corrugated Tube

Source: Gidwani et al (1998).

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... rearranged to give the heat transfer coefficient for the finned tube (hf) Rose (1994) obtained the enhancement ratio by normalizingh fby the heat transfer of the smooth