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Universal relations for calculation of the drag of transversely finned tube bundles
International Journal of Heat and Mass Transfer 73 (2014) 293–302
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Universal relations for calculation of the drag of transversely finned tube bundles E.N. Pis’mennyi a,⇑, A.M. Terekh a, G.P. Polupan b, I. Carvajal-Mariscal b, F. Sanchez-Silva b a Heat Power Engineering Faculty, Nuclear Power Plants and Engineering Thermophysics Department, National Technical University of Ukraine ‘‘Kiev Polytechnic Institute’’ (NTUU ‘‘KPI’’), 37, Peremogy Ave., Kiev 03056, Ukraine b National Polytechnic Institute of Mexico, Av. IPN s/n, Edificio 5, U.P.A.L.M., Col. Lindavista, Del. G.A.M., Mexico D.F. CP 07738, Mexico
a r t i c l e
i n f o
Article history: Received 13 August 2013 Received in revised form 23 December 2013 Accepted 5 February 2014 Available online 6 March 2014 Keywords: Finned tube Tube bundle Aerodynamic drag Calculation method
a b s t r a c t The drag of cross flow staggered and in-line helically and annularly finned tube bundles is investigated experimentally. The widest range of the reported geometric characteristics of finned tubes and tube bundles is considered. It is shown that when generalizing the experimental data obtained the best result is achieved by taking into account the Reynolds number exponent in the similarity equation as a function of geometric characteristics. Here as dimensionless parameters the geometric similarity of the reduced extended surface and the transverse-to-longitudinal tube pitch ratio are used, and also as the determining dimension in the Reynolds number – the equivalent diameter of the minimum flow section of the tube bundle. The results obtained by this approach describe with a sufficient accuracy 90% of all reported experimental data. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction The aerodynamic drag of transversely finned tube heat exchangers (Fig. 1), which find wide use in power engineering and industry, to a considerable extent determines expenses for their maintenance. Therefore, to design heat exchangers with required construction and performance characteristics, in addition to the thermal design methods, it is necessary to have generalized relations for calculation of their aerodynamic drag. The review of the works dealing with such relations [1–43], as well as of the existing normative documents [26] has shown that the generalized relations outlined their reliably describe the experimental data over the relatively narrow ranges of the geometric characteristics S1/S2 = 1.0 . . . 1.5; w = 5.0 . . . 12.0. At the same time, the range of geometric characteristics of tube bundles being of practical and scientific interest is much wider both in finning parameters and tube arrangement characteristics. In particular, this also relates to the so-called contracted tube bundles (S1/S2 > 2.0). Of late, for the reasons of technology, operation, etc. they are widespread in constructions of generating units. To our opinion, the mentioned drawbacks of the existing generalized relations are associated with the following basic reasons.
⇑ Corresponding author. Tel.: +380 503583720; fax: +380 444068087. E-mail addresses: [email protected], [email protected] (E.N. Pis’mennyi). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.02.013 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
The generalized relations were obtained from the small number of experiments. Most of the experiments were made over the range of the finning coefficient w = 5.0 . . . 10.0, although in individual cases, tube bundles with w values attaining 21 were used. The largest number of staggered tube bundles (up to 60%) data relates to a very narrow range of tube arrangement parameters S1/ S2 = 1.0 . . . 1.5. Approximately three quarters of experiments over the given range were performed on the so-called equilateral arrangements (S1/S2 = 1.15) or on those close to them. The range S1/S2 = 1.5 . . . 2.0 was studied greatly little (about 20% of experiments). The range of tube arrangement characteristics S1/S2 > 2.0 was not practically studied. Data on in-line tube bundles are very scanty: a relatively small number of experiments is mainly associated with ‘‘square’’ tube arrangements or with those close to them (S1/S2 = 0.8 . . . 1.25). The structure of the proposed calculation expressions based on the similarity equation of the form
Eu0 ¼ C r Ren e ;
ð1Þ
does not take into account fully the influence of the finning and pitch of tube bundles on their drag. This is first of all expressed in the fact that the Reynolds number exponent in the formula for the Euler number is taken as to be constant. The search for the possibilities to generalize experimental data in many cases is directed to derive complex expressions for the determining dimension in the Reynolds number, new forms
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Nomenclature a b Cr Cz
D D dconv de Eu Euo F Fmid Fmin H H H/F
cross section width of the channel where the tube bundle is placed cross section height of the channel where the tube bundle is placed coefficient in the similarity equation for the aerodynamic drag coefficient allowing for the influence of the number of transverse rows in the tube bundle on the aerodynamic drag tube diameter in the bundle outer diameter of finning conventional diameter of the transversely finned tube equivalent diameter of the minimum flow section of the bundle Euler number Euler number based on one transverse row of tubes in the bundle area of the minimum transverse flow section of the bundle midsection of the finned tube area of the minimum flow section of the tube bundle area of the heat exchange surface of one transverse row of tubes in the bundle fin height reduced length of the extended surface
of generalized tube arrangement parameters, as well as complex dependences of the quantity Cr in equation (1). Owing to the above-mentioned, at the NTUU KPI, along with heat transfer experiments [47], study was made of the drag of staggered and in-line helically and annularly finned tube bundles over ultimately wide ranges of their geometric characteristics covering all practical and theoretical needs.
2. Investigation methods Experiments were carried out in an open-type wind tunnel under isothermal conditions at air flow temperatures ranging from 293 to 300 K with the use of the experimental methodology
DP R Re S1 S2 S02 T U z1 z2
pressure drop Euler number exponent Reynolds number transverse pitch of tubes longitudinal pitch of tubes diagonal pitch of tubes fin pitch flow velocity number of tubes in the transverse row number of transverse rows in the bundle
Greek symbols d fin thickness D deviation of the experimental values of the Euo number from those calculated by the generalized relations no = 2Euo drag based on one transverse row of tubes in the bundle q heat carrier density r1 = S1/d relative transverse pitch of tubes in the bundle r2 = S2/d relative longitudinal pitch of tubes in the bundle U friction factor uconv = (S1 dconv)/(S02 dconv) arrangement parameter of finned tubes in the staggered tube bundle W finning coefficient
presented elsewhere in [45,46]. Fifteen series of finned tubes (Table 1) were analyzed. In total, there were 77 staggered arrangements of tubes, with 33 variants of tube location in accordance with the relative tube pitches r1 from 1.69 to 6.57, r2 from 0.63 to 9.48, and S1/S2 from 0.29 to 5.28. Furthermore, 54 in-line arrangements of tubes, with 26 variants of tube location in accordance with the relative tube pitches r1 from 2.38 to 6.57 and r2 from 1.28 to 9.52 were studied. For each series, 5 to 10 different locations of tubes were investigated. They were selected in a manner to cover the widest range of tube location characteristics and to observe their influence on the thermal and hydraulic characteristics of one of the tube pitches when fixing another. Several tube locations were made in a way to get at or close values of the parameter S1/S2 for relative transverse and longitudinal tube pitch values significantly different, which at a certain level allowed assessing the possibility of using the parameter S1/S2 as the parameter for the staggered arrangement of tubes in the bundle while generalizing the data obtained. The air pressure drop DP was determined in terms of the static pressure difference in front of and behind the heat exchange surface with one, two, three, four, etc. rows of tubes. Frictional pressure loss in the wind tunnel walls was also considered. With DP values, one obtains the Euler numbers for the entire heat exchange surface
Eu ¼ DP=qU 2 ;
ð2Þ
and
Eu0 ¼ Eu=z2 :
Fig. 1. Geometric characteristics of the finned tube bundle.
ð3Þ
for a row of tubes. The experimental results are presented as the relation Eu0 = f(Ree). While generalizing the data, the flow velocity over the narrowest cross section of the tube bundle was taken as the characteristic velocity. In the staggered arrangement of tubes the minimum flow section area, Fmin, of the tube bundle can be found
295
E.N. Pis’mennyi et al. / International Journal of Heat and Mass Transfer 73 (2014) 293–302 Table 1 Geometric characteristics of studied finned tubes. Series
Tube diameter d, mm
Fin height h, mm
Fin pitch t, mm
Fin thickness d, mm
Finning coefficient w
D/d
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
16.0 16.0 16.0 16.0 16.0 16.0 21.0 21.0 21.0 21.0 42.0 21.0 21.0 16.0 16.0
6.50 4.60 2.85 1.40 0.70 6.50 30.0 24.0 18.0 12.0 14.5 7.25 7.25 3.50 1.00
7.10 7.10 7.10 7.10 7.10 3.55 4.0 4.0 4.0 4.0 8.0 4.0 8.0 3.55 3.55
1.10 1.10 1.10 1.10 1.10 1.10 1.25 1.25 1.25 1.25 1.20 1.20 1.20 1.10 1.10
3.70 2.76 2.00 1.46 1.22 6.40 38.32 27.27 18.25 10.79 5.98 6.08 3.54 3.54 1.64
1.81 1.58 1.36 1.18 1.09 1.81 3.86 3.29 2.71 2.14 1.69 1.69 1.69 1.44 1.13
both in the transverse and diagonal planes. Which of them is correct can be determined using the parameter uconv that was proposed for the first time in [44]
uconv ¼ ðS1 dconv Þ=ðS02 dconv Þ;
ð4Þ
where the quantity dy corresponds to the diameter of the reference smooth tube with the midsection equal to that of the investigated finned tube
dconv ¼ d þ 2hd=t:
ð5Þ
When uy 6 2 the minimum flow section of the tube bundle is in the transverse plane and its area is determined by the formula
F min ¼ a b z1 F mid :
ð6Þ
In [10], it was shown that the relative length of the extended surface H/F is determined as the ratio of this surface area to the area of the minimum flow section. Since the air pressure drop in staggered tube bundles in cross flow is directly proportional to the number of transverse rows of tubes z2, it is found that for one row
H p½dt þ 2hd þ 2hðh þ dÞ ¼ : F S1 t ðdt þ 2hdÞ
ð10Þ
In the experiments of this work, the Ree number range was from 2 103 to 7 104. The results for most of the characteristic tube bundles of various series are shown in Figs. 2 and 3, from which it can be seen that
When uy > 2 the minimum flow section of the tube bundle is in the diagonal plane and its area is determined by the formula
F min ¼ ða b z1 F mid Þ:
ð7Þ
The Reynolds number for both in-line and staggered arrangements of tubes when uconv 6 2 was calculated in terms of the equivalent diameter of the contracted cross section
de ¼
2½tðS1 dÞ 2hd : 2h þ t
ð8Þ
For the staggered arrangement of tubes in the bundle, when
uconv > 2, the quantity 0
de ¼
2de
uconv
;
ð9Þ
was taken as the equivalent diameter. The evaluation of the experiments accuracy showed that the error in defining the Euler number was in the range of ±3.6 to 15.1%. 3. Analysis and generalization of experimental data The preliminary analysis of our experimental data and other authors’ results on the pressure drop of air flowing across tubes with helical, annular, and square finning [10,22,39] showed that for the experimental data to be generalized considering the geometry of finned tubes as a principal parameter, it must be recommended to take the relative length of the extended surface H/F [10] instead of the finning coefficient w. Moreover, the Reynolds number is defined in terms of the equivalent diameter, de , of the minimum (perpendicular or diagonal) flow section where the characteristic flow velocity is calculated. As a parameter which considers the geometry of tube arrangement in the bundle, it is advisable to use the ratio S1/S2.
Fig. 2. Experimental data for the air pressure drop in in-line finned tube bundles in cross flow for the following tube arrangements in the bundle: 3 S1/S2 = 0.60; 7 S1/S2 = 1.17; 8 S1/S2 = 1.41; 9 S1/S2 = 1.76; 12 S1/S2 = 2.75; 13 S1/S2 = 2.97; 14 S1/S2 = 3.17; 16 S1/S2 = 3.80.
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Fig. 3. Experimental data for the air pressure drop in in-line finned tube bundles in cross flow, for the following tube arrangements in the bundle: 1 S1/S2 = 0.29; 2 S1/S2 = 0.40; 3 S1/S2 = 0.60; 5 S1/S2 = 0.84; 10 S1/S2 = 1.76.
both the slope of the curve ‘gEu0 = f(‘gRee) and the pressure drop in tube bundles depend greatly on their geometric characteristics. With this in mind, it can be concluded that the generalization of the experimental data is possible with an acceptable accuracy over a wide range of geometric parameters only by taking into account the dependence of the slope of the curve ‘gEu0 = f(‘gRee) on the tube bundle geometry. From Figs. 2 and 3 it can also be noted that the behavior of the function Eu0 = f(Ree) is not the same in different regions of the investigated Ree range. In order to determine the conditions, at which the correct comparison and generalization of data are possible, the analysis has been carried out and the obtained results are illustrated in Fig. 4. This figure compares the data [19,10] generalized in terms of the equivalent diameter and the present work measurement results for the air pressure drop in three finned tube bundles with the close values of the parameters H/F and S1/S2. In [10], the data for the air pressure drop at the finned tube heat exchange surfaces are shown in the form of the friction factor U as a function of the Reynolds number U = f (Ree). Based on this work, the Euler numbers for heat exchange surfaces are determined by the expression
Eu0 ¼
2US2 : d
ð11Þ
The experimental data in Fig. 4 are combined satisfactorily and allow one to follow the behavior of the function Eu0 = f(Ree) in the range where Reynolds numbers vary by about three orders of magnitude. The dotted line was made by extrapolating to the region with H/F = 20, the plots of the friction factor are proposed in [10] for rectangular channels with the aspect ratio 1:3, for which the Euler numbers can be calculated using the relation
Eu0 ¼ U
H : 2F
ð12Þ
For tube bundles, in different regions of the Ree number range in Fig. 4, the curve ‘gEu0 = f(‘gRee) has a different slope, which corresponds to different exponents in Eq. (1), generally used to
approximate the experimental data on the air pressure drop in heat exchanger tubes with helical, annular and square finning. For Ree < 5 104, the flow development in a tube bundle is similar to that seen in a smooth channel. At a transition from a laminar to a turbulent flow regime (Ree = 2 103. . .5 103), both for the flow and the channel, the slope of the curve ‘gEu0 = f(‘gRee) is close to zero. Over the Ree range from 5 103 to 5 104, the total pressure drop component associated with friction remains essential in tube bundles. Its contribution to the total air pressure drop determines the slope value. For Ree > 5 104 in tube bundles the air pressure drop is predominant and flow separation phenomena dictate the quadratic behavior of the function Eu0 = f(Ree), i.e., this Reynolds number range is self-similar. From this it follows that when the equivalent diameter of the minimum flow section is taken as the characteristic dimension of tube bundles with the same H/F and S1/S2 values, dimensionless characteristics of the air pressure drop in the tube bundle must be identical and the experimental data generalized in terms of the above dimensions and approximated by the formula of form (1) can be compared and generalized only within the limits of each of the four specific regions, into which the studied Reynolds number range (Ree = 2 102. . .2 105) can be divided:
region 1 5 102 6 Ree < 2 103 ; region 2 2 103 6 Ree < 5 103 ; region 3 5 103 6 Ree < 6 104 ; region 4 6 104 < Ree : Whereas most of the Reynolds number range investigated in this work is occupied by the third region, the below analysis and the generalization of data are valid for the range of Ree = 5 103. . .6 104. 3.1. Staggered tube bundles Figs. 2 and 3 and Tables 2 and 3 illustrate the experimental results for characteristic staggered tube bundles. Some of them are plotted in Fig. 5 in such a manner that it is possible to follow the behavior of the relations n = f (S1/S2) and Cr = f (S1/S2) when H/ F = const. Although the deviation of the parameter H/F from its mean value in each series reaches ±10%, the plots in Fig. 5 demonstrate convincingly the interrelation between the quantities n and Cr, on one hand, and the tube arrangement parameter S1/S2, on the other. The relations n = f (S1/S2) and Cr = f (S1/S2) are self-similar and have a maximum lying in the range of the tube arrangement parameter S1/S2 from 1.5 to1.6. To the left of the maximum, the lower values of the exponent n can be explained by the fact that as the tube arrangement parameter S1/S2 is decreased, the tube bundle gradually transforms into a system of interacting transverse rows of tubes remote from each other downstream and thus not influencing each other (it is known
Table 2 Experimental n and Cr values in Formula (1) for staggered tube bundles. Series 7 and 8. S1 S2
Fig. 4. Comparison of the experimental data for the air pressure drop in staggered finned tube bundles at H/F = const and S1/S2 = const generalized in terms of the same equivalent diameter de: 1 – bundle RRP-3 [10], 2 – bundle at H/F = 11.10; S1/ S2 = 1.76 in series 6 (Table 1), 3 – bundle 12 [19], 4 – bundle for a smooth rectangular channel at H/F = 20 [10].
0.42 0.74 1.16 1.59 1.89 2.42 3.00 4.00
H/F
50.32 35.55 50.32 35.55 23.60 23.60 26.60
Series 7
H/F
n
Cr
0.26 0.27 0.31 0.31 0.28 0.27 0.25
3.05 2.94 4.72 4.42 3.33 3.12 2.35
33.80 24.28 33.80 24.28 16.34 16.34 16.34 16.34
Series 8 N
Cr
0.21 0.24 0.28 0.28 0.25 0.23 0.21 0.17
1.73 2.17 3.44 3.20 2.19 1.92 1.508 0.775
E.N. Pis’mennyi et al. / International Journal of Heat and Mass Transfer 73 (2014) 293–302 Table 3 Experimental n and Cr values in Formula (1) for staggered tube bundles. Series 9 and 11. S1 S2
0.42 1.01 1.15 1.16 1.59 1.89 2.42 2.50 3.00 4.00
Series 9
Series 11
H/F
n
Cr
H/F
n
Cr
21.40 21.40 21.40 15.62 10.66 10.66 10.66
0.24 0.27 0.27 0.24 0.27 0.19 0.16
3.12 4.57 4.57 2.15 2.17 1.093 0.577
19.9 19.9 8.9 8.9 8.9
0.25 0.25 0.25 0.23 0.22
2.380 2.263 2.364 2.142 1.932
297
n and Cr are well combined by the expressions based on the functions
p q S H S1 r 1 y¼a e S2 : F S2
ð13Þ
Using a standard program for finding linear regression coefficients, the processing of the experimental n and Cr values (Tables 2 and 3 and Figs. 2 and 3) depending on the influencing parameters H/F and S1/S2 as applied to the expression of form (13) yielded the following results:
0:25 0:57 H S1 n ¼ 0:17 e0:36S1 =S2 ; F S2
ð14Þ
0:53 1:30 H S1 C r ¼ 1:4 e0:90S1 =S2 : F S2
ð15Þ
The curves obtained with the use of these formulas are plotted in Fig. 5 illustrating that they are well combined with the experimental data. It should be noted that the scatter of the points relative to the calculated curves in Fig. 5 is to some extent connected with the deviations of the parameter H/F values for the selected tube bundles from those, in terms of which the corresponding curves are constructed. The assessment of the accuracy of Eqs. (14) and (15) for calculation of Eu0 numbers was made in Fig. 5 using the complex following from formulas (1) and (15)
"
K Eu
#1 1:3 S1 n 0:90ðS1 =S2 Þ ¼ Eu0 1:4 e Ree ; S2
ð16Þ
where n is defined by formula (14). The values of the complex were determined in terms of Eu0 calculated by Eq. (1) through the experimental values of the exponent n and the coefficient Cr (Figs. 2 and 3, Tables 2 and 3) for tube bundles in series 1–9 (Table 1) at the boundary Reynolds numbers Ree = 5 103 and Ree = 6 104. Fig. 6 plots the results for the higher deviation of the boundary values of the experimental and calculated Eu0 numbers. The dashed lines bound the region, where the deviation is less than ±20%. 94.5% of the points located inside this region (see Tables 4–6). Fig. 7 illustrates the comparison results of relations (1), (14), and (15) with the experimental data published elsewhere in [9,10,19,20,23,27,34,39]. As can be seen, Fig. 6 shows the points for the higher deviation of the compared values of the Eu0 numbers for Ree = 5 103 and Ree = 6 104. If the experimental data do not allow these two Ree values to be compared, then calculations are Fig. 5. Exponent n (a) and the coefficient Cr (b), in formula (1) vs. the tube arrangement parameter S1/S2 and the relative length of the extended surface H/F for staggered tube bundles.
that for the one-row arrangement, as well as for the first row of a standard tube bundle, the value of the exponent n always greatly reduces). To the right of the maximum, the reduction in the exponent values is due to the fact that as the arrangement parameter S1/S2 is increased, the arrangement of tubes in the bundle gradually approaches the arrangement typical for platen surfaces, for which according to [26], the exponent n should be equal to zero. In Fig. 5, the scatter of the curves in the parameter H/F as a whole indicates the growth of the exponent n and the coefficient Cr when the relative length of the extended surface is increased. The above-mentioned similarity of the plots n = f (H/F; S1/S2), H/ F = const and Cr = f (H/F; S1/S2), H/F = const, and also the fact that the correlation coefficient between the n and Cr values is equal to unity, permits one to suggest that the both relations can be described by the same-type functions. The experimental points for
Fig. 6. Assessment of the accuracy of relations (1), (14), (15) for tube bundles in series 1–9 (Table 1).
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Table 4 Experimental n and Cr values in Formula (1) for in-line tube bundles. Series 10, 12, 13. S1 S2
Series 10 H/F
N
Cr
H/F
n
Cr
H/F
n
Cr
0.26 0.52 1.04 1.40 2.94 1.18 1.79 2.36 0.79 0.74 1.37
29.8 29.8 29.8 17.0 6.5 8.6 12.8 8.6 8.6 29.8
0.210 0.180 0.100 0.070 0.000 0.035 0.000 0.155
1.680 1.100 0.360 0.206 0.049 0.114 0.056 0.759
15.1 15.1 15.1 9.0 3.6 4.7 6.9 4.7 4.7 15.1 15.1
0.200 0.180 0.110 0.070 0.000 0.110 0.035 0.000 0.150 0.075
1.440 1.030 0.342 0.183 0.046 0.225 0.093 0.054 0.680 0.200
8.0 8.0 8.0 4.9 2.0 2.7 3.8 2.6 2.6 8.0 8.0
0.195 0.175 0.110 0.065 0.000 0.090 0.030 0.000 0.130 0.145 0.075
1.360 0.972 0.342 0.183 0.045 0.182 0.105 0.052 0.348 0.625 0.206
Series 12
Table 5 Experimental n and Cr values in Formula (1) for in-line tube bundles. Series 14 and 15. S1 S2
2.43 1.46 0.79 1.83 1.20 0.48 0.83
Series 14
Series 15
H/F
N
Cr
H/F
n
Cr
3.9 9.0 9.0 6.1
0.000 0.065 0.135 0.025
0.065 0.197 0.666 0.100
5.4 1.8 5.4 5.4 1.8
0.065 0.130
0.166 0.616 0.298 0.997 0.350
0.090 0.160 0.105
Table 6 Experimental n and Cr values in Formula (1) for in-line tube bundles. Series 8 and 9. S1 S2
0.81 1.00 1.31 0.56 1.16 1.39
Series 8
Series 9
H/F
n
Cr
H/F
n
Cr
23.9 23.9 23.9 23.9 37.6
0.150 0.125 0.090 0.175 0.105
0.690 0.439 0.275 0.957 0.429
15.2 15.2 15.2 15.2 23.3 23.3
0.160 0.120 0.085 0.175 0.110 0.080
0.648 0.394 0.238 0.825 0.369 0.249
Series 13
data from the nine papers reviewed are generalized with an uncertainty of ±20%, and with an uncertainty of ±30% it gets up to 87%. As for all the 167 staggered arrangements of tubes examined in this work with an uncertainty of ±20%, 85.6% of the experimental data are generalized using formulas (1), (14), and (15). As noted above, in the present work the air pressure drop is also investigated by varying the number of transverse rows of tubes z2 in the tube bundle. The results obtained for staggered arrangements are in good agreement with those in [17,22]. So the influence of z2 on the Eu0 value can with a sufficient accuracy be taken into account by introducing into formula (1) the correction factor C 0z to be determined by the proposed relation graphically presented in [22] and included in the normative documents [26]. The processing of the obtained experimental data along with those given in [17,22] yielded the following relations to calculate the correction factor C 0z for staggered finned tubes with the small number of rows: when z2 < 6
6 C 0z ¼ exp 0:1 1 ; z2
ð17Þ
when z2 P 6
C 0z ¼ 1:0: These relations are graphically in [26].
ð18Þ fully
consistent
with those
presented
3.2. In-line tube bundles
Fig. 7. Comparison of relations (1), (14), and (15) with the data of other authors: 1 – [27], 2 – [19], 3 – [34], 4 – [9], 5 – [10], 6 – [39], 7 – [20,23].
performed in terms only of one of them. Such a situation has occurred with the experimental data [10,20,23] where the upper limit of the investigated Reynolds number range is about Ree 104, as well as with the data [39] obtained for the majority of tube bundles at Re > 2 104.With a deviation of less than ±20%, 100% of the data [9,20,23], 77% of the data [27], 93% of the data [10], 71% of the data [19,34] and 57% of the data [39] are generalized. In total, 78% of the
The results for the air pressure drop in forty-nine in-line helically and annularly finned tube bundles are summarized in Tables 5.5–5.7. Based on these tables Fig. 8 shows the exponent n and the coefficient Cr in formula (1) as a function of the relative length of the extended surface H/F for different values of the tube arrangement parameter S1/S2. From this figure it can be seen that as the relative length of the tube bundle is increased, n and Cr values grow monotonically, showing evidence of stabilization in the region of high H/F values. The analysis also showed that over the investigated Reynolds number range, the exponent n value varies from 0 to 0.2, reaching its highest values for the minimum values of the tube arrangement parameter S1/S2 observed in the experiments. The function n = f (S1/ S2) at H/F = const, whose form can be seen in Fig. 9, is characterized by an almost linear decrease in the exponent n with increasing parameter S1/S2. At S1/i2 P 2.1 over the entire H/F range the exponent becomes equal to zero (n = 0), which can be attributed to the transformation of the in-line tube arrangement to the platen one. Owing to this, the functions n = f(H/F; S1/S2) and Cr = f(H/F; S1/S2)
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299
The accuracy was assessed by comparing for Ree = 5 103 and Ree = 6 104 the experimental and calculated by formulas (1), Eqs. (19)–(22) Euler numbers based on the complex
K 0Eu ¼
Fig. 8. Exponent n (a) and the coefficient Cr (b) in formula (1) vs. the relative length of the extended surface H/F and the arrangement parameter S1/S2 for in-line tube bundles: 1 – S1/S2 = 0.26; 2 – S1/S2 = 0.52; 3 – S1/S2 = 0.70. . .0.74; 4 – S1/ S2 = 1.04. . .1.18; 5 – S1/S2 = 1.36. . .1.46; 6 – S1/S2 = 1.78; 7 – S1/S2 = 2.36. . .2.43; 8 – S1/S2 = 2.94.
Eu0 100%; C r Ren 1
ð23Þ
where the n and Cr values are defined by (19)–(22). Fig. 10a shows the results of this comparison for seventy-five in-line finned tubes studied by the author and in works [3,8,12,40]. As in the case of comparing the data for staggered tube bundles, here the points corresponding to the higher deviation of the compared Eu0 number values are plotted. The obtained equations for calculation of the air pressure drop in in-line finned tube bundles in cross flow describe 88% of the experimental data with an error not greater than ±20%. Fig. 10b shows the comparison of the same experimental data with those obtained by the function [26], indicating a significantly higher accuracy of the generalizing relations offered in this work. The influence of the number of transverse rows of tubes z2 on the pressure drop of air flowing across in-line tube bundles was also investigated. The results of generalization of the data for in-line tube bundles with the small number of rows are plotted in Fig. 11 in the form of the correction factor C 0z added to formula (1) as the number of transverse rows of tubes (curve 1). This figure also shows the functions C 0z = f (z2) for in-line (2) and staggered (3) tube bundles available in normative documents [26]. As can be seen from this figure, when the general trend of increasing C 0z with decreasing z2 is kept, one can see a significant deviation of generalizing curve (1) from normative relation (2) obtained only in the graphical form in [22] using a very limited amount of experimental data. Normative curve (2) is significantly higher than normative curve (3), which is not consistent with the in-line tube bundle depth variation of the Eu0 number: in this case, as the number of transverse rows over the range 1 6 z2 < 6 is increased, Eu0 should fall faster
Fig. 9. Functions: 1 – n = f (S1/i2) and 2 – Cr = f (S1/S2) at H/F = const.
cannot be approximated by sufficiently simple and convenient relations, being the same as in the case of the staggered arrangement of tubes, for the entire investigated range of the tube arrangement parameter S1/S2. Equations for data generalization were obtained separately for two ranges of S1/S2 values: When 0.25 6 S1/S2 < 2.1 n and Cr values are described with a reasonable accuracy by the expressions of the form
n ¼ ðH=FÞ0:08 ð0:184 0:088S1 =S2 Þ;
ð19Þ
C r ¼ 1:25ðH=FÞ0:25 expð1:7S1 =S2 Þ;
ð20Þ
when S1/S2 P 2.1, respectively,
n ¼ 0;
ð21Þ
Cr ¼ ðH=FÞ0:1 ð0:066 0:008S1 =S2 Þ:
ð22Þ
Fig. 10. Assessment of the deviation of the experimental data from the calculations by equations (1), Eqs. (19)–(22) (a) and by the method [26] (b): 1 – data of Tables (5.5)–(5.7), 2 – data [12], 3 – data [3], 4 – data [8], 5 – data [40].
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n¼
0:08 H S1 ; 0:184 0:088 F S2
0:25 S H 1:70S1 2; C 0r ¼ 2:5 e F and at S1/S2 > 2.1
n ¼ 0; C 0r ¼
0;1 H S1 0; 132 0; 016 F S2
The relative length of the extended surface H/F is defined by the expression
Fig. 11. Correction factor C0 Z as a function of the small number of rows of tubes in the bundle: 1 – curve for in-line tube bundles obtained according to the present work data and 2, 3 – normative curves for staggered and in-line tube bundles, respectively.
than Eu0 does in the case of staggered tube bundles at other things being equal; in addition, the values of these quantities should coincide when z2 = 1. This fact points to the insufficient accuracy of standard relation (2) for in-line tube bundles. In Fig. 11, relation (1) for determination of the correction factor C 0z as a function of the small number of rows of in-line tubes in the bundle obtained in this work corresponds to the above trend and is described by the following relations: when z2 < 6
C 0z ¼ 1:0 þ
0:65 ðz2 Þ3
ð24Þ
;
when z2 P 6
C 0z ¼ 1:0:
ð25Þ C 0z
H p½dt þ 2hd þ 2hðh þ dÞ ¼ : F S1 t ðdt þ 2hdÞ As a characteristic length in the Reynolds numbers when calculating the pressure drop of air flowing in in-line and staggered tube bundles in cross flow with uconv 6 2 the equivalent diameter of the minimum flow section of the tube bundle is taken as follows
de ¼
2½tðS1 dÞ 2hd ; 2h þ t
and when calculating the pressure drop in staggered tube bundles with uconv > 2, the following expression is used 0
de ¼
2de
uconv
The physical constants in similarity numbers refer to the average temperature of gases in the tube bundle and the velocity – to the narrowest flow (transverse or diagonal) section of the tube bundle. The correction factor for the small number of rows of tubes in the bundle depending on the tube arrangement is calculated using the following relations: for the staggered arrangement and z2 < 6
It should also be noted that the values defined by Eqs. (17) and (24) at z2 = 1 are practically the same.
6 C 0z ¼ exp 0:1 1 ; z2
4. Generalized method for calculation of air pressure drop in transversely finned tube bundles. comparison with the generalizations of other authors
for the in-line arrangement and z2 < 6
As a result of the studies made, the methods for calculation of the air pressure drop in helically and annularly finned tube bundles in cross flow have been developed. The geometric characteristics of bundles are over the range of w = 1.2 . . . 39.0, H/F = 1.5 . . . 70 and S1/ S2 = 0.3...4.0. Following the methodology valid over the Ree number range from 5 103 to104, the pressure drop in a tube bundle based on one transverse row is given by the relation
n0 ¼ C 0z C 0r Ren e :
ð26Þ
For the staggered arrangement of tubes in the bundle the exponent n and the coefficient C 0r in Eq. (26) are determined by the expressions
n ¼ 0:17
C 0r ¼ 2:8
0:25 0:57 S H S1 0:36S1 2; e F S2
0:53 1:30 S H S1 0:90S1 2: e F S2
In the case of the in-line arrangement of tubes with S1/S2 6 2.1, the following equations should be used for determination of n and C 0r
C 0z ¼ 1 þ
0:65 ðz2 Þ3
;
for any tube arrangement in the bundle and z2 P 6
C 0z ¼ 1:0: The proposed method provides a more accurate calculation of the pressure drop of air flowing across helically and annularly finned tubes compared with others that are most widely used [25,28,34,39], including those entering the normative method [26]. Our results for comparison of the calculation methods are presented in Tables 7 and 8 and Figs. 10–12. The comparison covers the experimental data for the pressure drop in 167 staggered finned tube bundles and in 75 in-line finned tube bundles in cross flow.
Table 7 Comparison of the methods for calculation of the air pressure drop in helically and annularly staggered finned tube bundles.
[28] [26] Present work
D 6 ±20%
D 6 ±30%
D 6 ±40%
D 6 ±50%
D 6 ±60%
55.3 65.2 87.0
70.2 79.5 91.9
82.0 90.7 96.9
86.3 93.2 98.8
90.1 97.5 98.8
E.N. Pis’mennyi et al. / International Journal of Heat and Mass Transfer 73 (2014) 293–302 Table 8 Comparison of the methods for calculation of the air pressure drop in helically and annularly in-line finned tube bundles.
[26] Present work
D 6 ±20%
D 6 ±30%
D 6 ±40%
D 6 ±50%
D 6 ±60%
53.2 88.0
67.5 92.2
77.0 98.7
93.5 100.0
93.5 100.0
301
structure of the equation proposed for calculation of Euler numbers does not allow its use for tubes with high fins (for h/d values close to unity, the equation gives too high values and for h/d > 1 negative values of Eu numbers). Because of this, the comparison with other calculating systems makes no sense. The comparison results with the methods [25,32], obtained over the narrow ranges of geometric characteristics and therefore yielding the low calculation accuracy, are not included into the summary tables.
References
Fig. 12. Comparison of the accuracy of methods for calculation of the drag of staggered finned tubes in the bundle: a – normative relation [26]; b – relation [28] c – relations (1), (14), (15); 1 – Ree = 5000; 2 – Ree = 60000.
In Tables 7 and 8, for each of the above calculation methods, the results are given as the percentage of tube bundles, in which the pressure drop is described by the corresponding equations with an error not exceeding a specified value of D. Table 7 contains no results compared with the method [39], because there the
[1] S.L. Jameson, Tube spacing in finned-tube banks, ASME Trans. 67 (8) (1945) 633–642. [2] A.Y. Gunter, W.A. Shaw, A general correlation of friction factors for various types of surfaces in cross flow, ASME Trans. 67 (1945) 643–660. [3] V.M. Antuf’ev, G.C. Beletskiy, Heat Transfer and Aerodynamic Resistance of Tube Surfaces in Cross Flow, Moscow, Mashgiz, 1948. pp. 119. [4] Th.E. Schmidt, Heat transmission and pressure drop in banks of finned tubes and in laminated coolers, Gen. Discuss. Heat Transfer (Inst. Mech. Eng.) (1951) 186–188. [5] A.L. London, W.M. Kays, Heat-transfer and flow-friction characteristics of some compact heat-exchanger surfaces, Trans. ASME 74 (7) (1952) 1167–1178. [6] D.J. Ward, E.H. Young, Heat transfer and pressure drop of air in forced convection across triangular pitch banks of finned tubes, Chem. Eng. Symp. Ser. 55 (29) (1959) 37–44. [7] V.G. Fastovskiy, Yu.V. Petrovskiy, Heat transfer and pressure drop of staggered bundles of tubes with continuous spiral finning, Teploenergetika 6 (1960) 69–72. [8] H. Brauer, Wärme und Strömungstechnische Untersuchungen an quer angeströmten Rippenrohr bündeln, Chemie Ing. Technik 33 (5) (1961) 327–335. [9] V.M. Antufiev, Efficiency of Different Shapes of Convective Heating Surfaces, Energia, Moscow-Leningrad, 1966. p. 184. [10] V.M. Kays, A.L. London, Compact Heat Exchangers, Moscow, Energia, 1962. p. 160. [11] D.E. Briggs, E.H. Young, Convection heat transfer and pressure drop of air flowing across triangular pitch banks of finned tubes, Chem. Eng. Prog. Symp. Ser. 89 (41) (1963) 1–10. [12] V.F. Yudin, L.S. Tokhtarova, Heat transfer and pressure drop of staggered and in-line bundles of finned tubes, vol. 1, Energomashinostroyenye, Moscow, 1964, pp. 11–13. [13] V.F. Yudin, L.S. Tokhtarova, Investigations of heat transfer and pressure drop of staggered bundles of tubes with different forms of fins, Heat Mass Transfer Sci. Technol. 1 (1965) 249–259. [14] I. Vampola, Generalization of relationships related to heat transfer and pressure drop in cross flow bundles of finned tubes, Heat Mass Transfer Sci. Technol. 1 (1965) 260–269. [15] K.K. Robinson, D.E. Briggs, Pressure drop of air flowing across triangular pitch banks of finned tubes, Chem. Eng. Prog. Symp. Ser. 62 (64) (1965) 177–184. [16] V.F. Yudin, L.S. Tokhtarova, Heat transfer and pressure drop of staggered bundles of finned tubes with different fin heights and pitches, Proc. I. I. Polzunov TsKTI 73 (1966) 249–259. [17] V.F. Yudin, L.S. Tokhtarova, Influence of the number of rows of staggered and in-line bundles of finned tubes on heat transfer and pressure drop, vol. 4, Energomashinostroyenye, Moscow, 1971, pp. 41–42. [18] V.F. Yudin, L.S. Tokhtarova, Pressure drop of bundles of finned tubes in cross flow, vol. 9, Energomashinostroyenye, Moscow, 1972, pp. 44–45. [19] V.F. Yudin, L.S. Tokhtarova, Heat transfer and pressure drop of finned tube bundles with different fin heights and pitches for high Reynolds numbers, vol. 12, Energomashinostroyenye, Moscow, 1972, pp. 21–23. [20] I.G. Taranyan, F.M. Yokhvedov, V.B. Kuntysh, Investigation of fin parameters influence on heat transfer and pressure drop of staggered bundles of tubes with annular smooth and integral fins, High Temp. Thermophys. 10 (5) (1972) 1049–1054. [21] V.F. Yudin, L.S. Tokhtarova, Heat transfer and pressure drop of staggered bundles of finned tubes in cross flow, Teploenergetika 2 (1973) 49–52. [22] V.F. Yudin, L.S. Tokhtarova, Pressure drop of finned tube bundles in cross flow, vol. 6, Energomashinostroyenye, Mocow, 1974, pp. 30–32. [23] V.B. Kuntish, F.M. Yokhvedov, I.G. Taranyan, Heat transfer, pressure drop and comparative analysis of bundles of finned tubes, Izv. VUZov, Energetika, vol. 6, 1974, pp. 132–137. [24] Z. Mircovic, Heat transfer and flow resistance correlation for helically finned and staggered tube banks in cross flow, in: N.H. Afgan, E.U. Schlünder (Eds.), Heat Exchangers: Design and Theory Sourcebook, Hemisphere, Washington, 1974, pp. 559–584. [25] C. Weierman, Correlations ease the selection of finned tubes, Oil Gas J. 74 (36) (1976) 94–100. [26] Standards of aerodynamic resistance calculation of steam generators, in: S.I. Mochan (Ed.), 3rd ed., Energia, Moscow, 1977, p. 256. [27] V.M. Legkiy, Yu.K. Tupitsyn, E.N. Pysmennyy, Influence of tube-pitch relationships on heat transfer and aerodynamic resistance of staggered bundles of finned tubes, in: Heat Transfer in Power Units, Naukova Dumka, Kiev, 1978, pp. 78–82.
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[28] V.A. Lokshin, B.N. Fomina, Generalization of experimental investigation data on pressure drop of finned tube bundles, Teploenergetika 6 (1978) 36–39. [29] K. Schack, Berechnung des Druckverlustes im Querstrom von Rippenrohrbündeln, Chem. Ing. Technol. 51 (3) (1979) 986–994. [30] V.B. Kuntish, A.E. Piir, L.F. Kolobova, N.I. Mirmov, L.M. Fedotova, Correlation equation for aerodynamic resistance of tube bundles in air-cooled heat exchangers, Chem. Technol. Fuels Oils 15 (5–6) (1979) 341–344. [31] R.L. Webb, Air-side heat transfer in finned tube heat exchanger, Heat Transfer Eng. 1 (3) (1980) 33–49. [32] T.J. Rabas, P.W. Eckels, R.A. Sabatino, The effect of fin density on the heat transfer and pressure drop performance of low-finned tube banks, Chem. Eng. Commun. 10 (1–3) (1981) 127–147. [33] V.B. Kuntish, L.M. Fedotova, N.M. Kuznetsov, Influence of Finned Tube Bundle Geometry on Heat Transfer and Pressure Drop, vol. 8, Kholodilnaya Tekhnika, Moscow, 1981. pp. 25–28. [34] V.F. Yudin, Heat Transfer of Tubes with Transversal Fins, Mashinostroyenye, Leningrad, 1982. p. 189. [35] A.A. Zhukauskas, R.V. Ulinskas, F.V. Zinyavichyus, Heat transfer and pressure drop in staggered bundles of finned tubes, Energetika i Transp. 4 (1983) 117– 124. [36] E.M. Sparrow, F. Samie, Heat transfer and pressure drop results for one-andtwo row arrays of finned tubes, Int. J. Heat Mass Transfer 28 (12) (1985) 2247– 2258. [37] T.J. Rabas, J. Taborek, Survey of turbulent forced-convection heat transfer and pressure drop characteristics of low-finned tube banks in cross flow, Heat Transfer Eng. 8 (2) (1987) 49–62.
[38] S. Haaf, Warmeubertragung in Luftkuhlern, Handbuch der Kaltechnik, Bd. 6, Teil B, Warmeaustauscher, Berlin u.a.: Springer Verlag, 1988, pp. 435–491. [39] J. Stasiulevicius, V.J. Survila, Heat Transfer of Finned Tube Bundles in Cross Flow, Hemisphere Publishing, 1988. [40] A. Khavin, A study of heat transfer and drag of spiral-ribbon finned tubes in the bundle and the implementation results, Kiev, 1975, p. 24 (Abstract of the Thesis for a degree of Candidate of Technical Sciences). [41] A. Nir, Heat transfer and friction factor correlations for cross flow over staggered finned tube banks, Heat Transfer Eng. 12 (1) (1991) 43–58. [42] B.G. Srbislav, M.J. Branislav, R.L. Boris, Research on air pressure drop in helically-finned tube heat exchangers, Appl. Therm. Eng. 26 (5–6) (2006) 478– 485. [43] P. Vassallo, P. Symolon, Friction factor measurements in an equally spaced triangular tube array, Report LM-07K012, US Department of Energy, 2007. [44] E.N. Pis’mennyi, V.M. Legkii, Calculating heat transfer of multi-row staggered bundles of tubes with transverse finning, Therm. Eng. (English Translation Teploenergetika) 31 (6) (1984) 349–352. [45] N. Pis’mennyy Ye, V.M. Legkiy, Correlation of experimental data on the drag of bundles of staggered tubes with external circular fins, Fluid Mech. Sov. Res. 16 (2) (1987) 42–47. [46] V.M. Legkij, E.N. Pis’mennyj, A.M. Terekh, Aerodynamic resistance of corridor bundles of finned tubes being laterally flown about, Teploenergetika 5 (1994) 58–60. [47] E.N. Pis’mennyi, An asymptotic approach to generalizing the experimental data on convective heat transfer of tube bundles in cross flow, Int. J. Heat Mass Transfer 54 (19–20) (2011) 4235–4246.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Universal relations for calculation of the drag of transversely finned tube bundles E.N. Pis’mennyi a,⇑, A.M. Terekh a, G.P. Polupan b, I. Carvajal-Mariscal b, F. Sanchez-Silva b a Heat Power Engineering Faculty, Nuclear Power Plants and Engineering Thermophysics Department, National Technical University of Ukraine ‘‘Kiev Polytechnic Institute’’ (NTUU ‘‘KPI’’), 37, Peremogy Ave., Kiev 03056, Ukraine b National Polytechnic Institute of Mexico, Av. IPN s/n, Edificio 5, U.P.A.L.M., Col. Lindavista, Del. G.A.M., Mexico D.F. CP 07738, Mexico
a r t i c l e
i n f o
Article history: Received 13 August 2013 Received in revised form 23 December 2013 Accepted 5 February 2014 Available online 6 March 2014 Keywords: Finned tube Tube bundle Aerodynamic drag Calculation method
a b s t r a c t The drag of cross flow staggered and in-line helically and annularly finned tube bundles is investigated experimentally. The widest range of the reported geometric characteristics of finned tubes and tube bundles is considered. It is shown that when generalizing the experimental data obtained the best result is achieved by taking into account the Reynolds number exponent in the similarity equation as a function of geometric characteristics. Here as dimensionless parameters the geometric similarity of the reduced extended surface and the transverse-to-longitudinal tube pitch ratio are used, and also as the determining dimension in the Reynolds number – the equivalent diameter of the minimum flow section of the tube bundle. The results obtained by this approach describe with a sufficient accuracy 90% of all reported experimental data. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction The aerodynamic drag of transversely finned tube heat exchangers (Fig. 1), which find wide use in power engineering and industry, to a considerable extent determines expenses for their maintenance. Therefore, to design heat exchangers with required construction and performance characteristics, in addition to the thermal design methods, it is necessary to have generalized relations for calculation of their aerodynamic drag. The review of the works dealing with such relations [1–43], as well as of the existing normative documents [26] has shown that the generalized relations outlined their reliably describe the experimental data over the relatively narrow ranges of the geometric characteristics S1/S2 = 1.0 . . . 1.5; w = 5.0 . . . 12.0. At the same time, the range of geometric characteristics of tube bundles being of practical and scientific interest is much wider both in finning parameters and tube arrangement characteristics. In particular, this also relates to the so-called contracted tube bundles (S1/S2 > 2.0). Of late, for the reasons of technology, operation, etc. they are widespread in constructions of generating units. To our opinion, the mentioned drawbacks of the existing generalized relations are associated with the following basic reasons.
⇑ Corresponding author. Tel.: +380 503583720; fax: +380 444068087. E-mail addresses: [email protected], [email protected] (E.N. Pis’mennyi). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.02.013 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
The generalized relations were obtained from the small number of experiments. Most of the experiments were made over the range of the finning coefficient w = 5.0 . . . 10.0, although in individual cases, tube bundles with w values attaining 21 were used. The largest number of staggered tube bundles (up to 60%) data relates to a very narrow range of tube arrangement parameters S1/ S2 = 1.0 . . . 1.5. Approximately three quarters of experiments over the given range were performed on the so-called equilateral arrangements (S1/S2 = 1.15) or on those close to them. The range S1/S2 = 1.5 . . . 2.0 was studied greatly little (about 20% of experiments). The range of tube arrangement characteristics S1/S2 > 2.0 was not practically studied. Data on in-line tube bundles are very scanty: a relatively small number of experiments is mainly associated with ‘‘square’’ tube arrangements or with those close to them (S1/S2 = 0.8 . . . 1.25). The structure of the proposed calculation expressions based on the similarity equation of the form
Eu0 ¼ C r Ren e ;
ð1Þ
does not take into account fully the influence of the finning and pitch of tube bundles on their drag. This is first of all expressed in the fact that the Reynolds number exponent in the formula for the Euler number is taken as to be constant. The search for the possibilities to generalize experimental data in many cases is directed to derive complex expressions for the determining dimension in the Reynolds number, new forms
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Nomenclature a b Cr Cz
D D dconv de Eu Euo F Fmid Fmin H H H/F
cross section width of the channel where the tube bundle is placed cross section height of the channel where the tube bundle is placed coefficient in the similarity equation for the aerodynamic drag coefficient allowing for the influence of the number of transverse rows in the tube bundle on the aerodynamic drag tube diameter in the bundle outer diameter of finning conventional diameter of the transversely finned tube equivalent diameter of the minimum flow section of the bundle Euler number Euler number based on one transverse row of tubes in the bundle area of the minimum transverse flow section of the bundle midsection of the finned tube area of the minimum flow section of the tube bundle area of the heat exchange surface of one transverse row of tubes in the bundle fin height reduced length of the extended surface
of generalized tube arrangement parameters, as well as complex dependences of the quantity Cr in equation (1). Owing to the above-mentioned, at the NTUU KPI, along with heat transfer experiments [47], study was made of the drag of staggered and in-line helically and annularly finned tube bundles over ultimately wide ranges of their geometric characteristics covering all practical and theoretical needs.
2. Investigation methods Experiments were carried out in an open-type wind tunnel under isothermal conditions at air flow temperatures ranging from 293 to 300 K with the use of the experimental methodology
DP R Re S1 S2 S02 T U z1 z2
pressure drop Euler number exponent Reynolds number transverse pitch of tubes longitudinal pitch of tubes diagonal pitch of tubes fin pitch flow velocity number of tubes in the transverse row number of transverse rows in the bundle
Greek symbols d fin thickness D deviation of the experimental values of the Euo number from those calculated by the generalized relations no = 2Euo drag based on one transverse row of tubes in the bundle q heat carrier density r1 = S1/d relative transverse pitch of tubes in the bundle r2 = S2/d relative longitudinal pitch of tubes in the bundle U friction factor uconv = (S1 dconv)/(S02 dconv) arrangement parameter of finned tubes in the staggered tube bundle W finning coefficient
presented elsewhere in [45,46]. Fifteen series of finned tubes (Table 1) were analyzed. In total, there were 77 staggered arrangements of tubes, with 33 variants of tube location in accordance with the relative tube pitches r1 from 1.69 to 6.57, r2 from 0.63 to 9.48, and S1/S2 from 0.29 to 5.28. Furthermore, 54 in-line arrangements of tubes, with 26 variants of tube location in accordance with the relative tube pitches r1 from 2.38 to 6.57 and r2 from 1.28 to 9.52 were studied. For each series, 5 to 10 different locations of tubes were investigated. They were selected in a manner to cover the widest range of tube location characteristics and to observe their influence on the thermal and hydraulic characteristics of one of the tube pitches when fixing another. Several tube locations were made in a way to get at or close values of the parameter S1/S2 for relative transverse and longitudinal tube pitch values significantly different, which at a certain level allowed assessing the possibility of using the parameter S1/S2 as the parameter for the staggered arrangement of tubes in the bundle while generalizing the data obtained. The air pressure drop DP was determined in terms of the static pressure difference in front of and behind the heat exchange surface with one, two, three, four, etc. rows of tubes. Frictional pressure loss in the wind tunnel walls was also considered. With DP values, one obtains the Euler numbers for the entire heat exchange surface
Eu ¼ DP=qU 2 ;
ð2Þ
and
Eu0 ¼ Eu=z2 :
Fig. 1. Geometric characteristics of the finned tube bundle.
ð3Þ
for a row of tubes. The experimental results are presented as the relation Eu0 = f(Ree). While generalizing the data, the flow velocity over the narrowest cross section of the tube bundle was taken as the characteristic velocity. In the staggered arrangement of tubes the minimum flow section area, Fmin, of the tube bundle can be found
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E.N. Pis’mennyi et al. / International Journal of Heat and Mass Transfer 73 (2014) 293–302 Table 1 Geometric characteristics of studied finned tubes. Series
Tube diameter d, mm
Fin height h, mm
Fin pitch t, mm
Fin thickness d, mm
Finning coefficient w
D/d
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
16.0 16.0 16.0 16.0 16.0 16.0 21.0 21.0 21.0 21.0 42.0 21.0 21.0 16.0 16.0
6.50 4.60 2.85 1.40 0.70 6.50 30.0 24.0 18.0 12.0 14.5 7.25 7.25 3.50 1.00
7.10 7.10 7.10 7.10 7.10 3.55 4.0 4.0 4.0 4.0 8.0 4.0 8.0 3.55 3.55
1.10 1.10 1.10 1.10 1.10 1.10 1.25 1.25 1.25 1.25 1.20 1.20 1.20 1.10 1.10
3.70 2.76 2.00 1.46 1.22 6.40 38.32 27.27 18.25 10.79 5.98 6.08 3.54 3.54 1.64
1.81 1.58 1.36 1.18 1.09 1.81 3.86 3.29 2.71 2.14 1.69 1.69 1.69 1.44 1.13
both in the transverse and diagonal planes. Which of them is correct can be determined using the parameter uconv that was proposed for the first time in [44]
uconv ¼ ðS1 dconv Þ=ðS02 dconv Þ;
ð4Þ
where the quantity dy corresponds to the diameter of the reference smooth tube with the midsection equal to that of the investigated finned tube
dconv ¼ d þ 2hd=t:
ð5Þ
When uy 6 2 the minimum flow section of the tube bundle is in the transverse plane and its area is determined by the formula
F min ¼ a b z1 F mid :
ð6Þ
In [10], it was shown that the relative length of the extended surface H/F is determined as the ratio of this surface area to the area of the minimum flow section. Since the air pressure drop in staggered tube bundles in cross flow is directly proportional to the number of transverse rows of tubes z2, it is found that for one row
H p½dt þ 2hd þ 2hðh þ dÞ ¼ : F S1 t ðdt þ 2hdÞ
ð10Þ
In the experiments of this work, the Ree number range was from 2 103 to 7 104. The results for most of the characteristic tube bundles of various series are shown in Figs. 2 and 3, from which it can be seen that
When uy > 2 the minimum flow section of the tube bundle is in the diagonal plane and its area is determined by the formula
F min ¼ ða b z1 F mid Þ:
ð7Þ
The Reynolds number for both in-line and staggered arrangements of tubes when uconv 6 2 was calculated in terms of the equivalent diameter of the contracted cross section
de ¼
2½tðS1 dÞ 2hd : 2h þ t
ð8Þ
For the staggered arrangement of tubes in the bundle, when
uconv > 2, the quantity 0
de ¼
2de
uconv
;
ð9Þ
was taken as the equivalent diameter. The evaluation of the experiments accuracy showed that the error in defining the Euler number was in the range of ±3.6 to 15.1%. 3. Analysis and generalization of experimental data The preliminary analysis of our experimental data and other authors’ results on the pressure drop of air flowing across tubes with helical, annular, and square finning [10,22,39] showed that for the experimental data to be generalized considering the geometry of finned tubes as a principal parameter, it must be recommended to take the relative length of the extended surface H/F [10] instead of the finning coefficient w. Moreover, the Reynolds number is defined in terms of the equivalent diameter, de , of the minimum (perpendicular or diagonal) flow section where the characteristic flow velocity is calculated. As a parameter which considers the geometry of tube arrangement in the bundle, it is advisable to use the ratio S1/S2.
Fig. 2. Experimental data for the air pressure drop in in-line finned tube bundles in cross flow for the following tube arrangements in the bundle: 3 S1/S2 = 0.60; 7 S1/S2 = 1.17; 8 S1/S2 = 1.41; 9 S1/S2 = 1.76; 12 S1/S2 = 2.75; 13 S1/S2 = 2.97; 14 S1/S2 = 3.17; 16 S1/S2 = 3.80.
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Fig. 3. Experimental data for the air pressure drop in in-line finned tube bundles in cross flow, for the following tube arrangements in the bundle: 1 S1/S2 = 0.29; 2 S1/S2 = 0.40; 3 S1/S2 = 0.60; 5 S1/S2 = 0.84; 10 S1/S2 = 1.76.
both the slope of the curve ‘gEu0 = f(‘gRee) and the pressure drop in tube bundles depend greatly on their geometric characteristics. With this in mind, it can be concluded that the generalization of the experimental data is possible with an acceptable accuracy over a wide range of geometric parameters only by taking into account the dependence of the slope of the curve ‘gEu0 = f(‘gRee) on the tube bundle geometry. From Figs. 2 and 3 it can also be noted that the behavior of the function Eu0 = f(Ree) is not the same in different regions of the investigated Ree range. In order to determine the conditions, at which the correct comparison and generalization of data are possible, the analysis has been carried out and the obtained results are illustrated in Fig. 4. This figure compares the data [19,10] generalized in terms of the equivalent diameter and the present work measurement results for the air pressure drop in three finned tube bundles with the close values of the parameters H/F and S1/S2. In [10], the data for the air pressure drop at the finned tube heat exchange surfaces are shown in the form of the friction factor U as a function of the Reynolds number U = f (Ree). Based on this work, the Euler numbers for heat exchange surfaces are determined by the expression
Eu0 ¼
2US2 : d
ð11Þ
The experimental data in Fig. 4 are combined satisfactorily and allow one to follow the behavior of the function Eu0 = f(Ree) in the range where Reynolds numbers vary by about three orders of magnitude. The dotted line was made by extrapolating to the region with H/F = 20, the plots of the friction factor are proposed in [10] for rectangular channels with the aspect ratio 1:3, for which the Euler numbers can be calculated using the relation
Eu0 ¼ U
H : 2F
ð12Þ
For tube bundles, in different regions of the Ree number range in Fig. 4, the curve ‘gEu0 = f(‘gRee) has a different slope, which corresponds to different exponents in Eq. (1), generally used to
approximate the experimental data on the air pressure drop in heat exchanger tubes with helical, annular and square finning. For Ree < 5 104, the flow development in a tube bundle is similar to that seen in a smooth channel. At a transition from a laminar to a turbulent flow regime (Ree = 2 103. . .5 103), both for the flow and the channel, the slope of the curve ‘gEu0 = f(‘gRee) is close to zero. Over the Ree range from 5 103 to 5 104, the total pressure drop component associated with friction remains essential in tube bundles. Its contribution to the total air pressure drop determines the slope value. For Ree > 5 104 in tube bundles the air pressure drop is predominant and flow separation phenomena dictate the quadratic behavior of the function Eu0 = f(Ree), i.e., this Reynolds number range is self-similar. From this it follows that when the equivalent diameter of the minimum flow section is taken as the characteristic dimension of tube bundles with the same H/F and S1/S2 values, dimensionless characteristics of the air pressure drop in the tube bundle must be identical and the experimental data generalized in terms of the above dimensions and approximated by the formula of form (1) can be compared and generalized only within the limits of each of the four specific regions, into which the studied Reynolds number range (Ree = 2 102. . .2 105) can be divided:
region 1 5 102 6 Ree < 2 103 ; region 2 2 103 6 Ree < 5 103 ; region 3 5 103 6 Ree < 6 104 ; region 4 6 104 < Ree : Whereas most of the Reynolds number range investigated in this work is occupied by the third region, the below analysis and the generalization of data are valid for the range of Ree = 5 103. . .6 104. 3.1. Staggered tube bundles Figs. 2 and 3 and Tables 2 and 3 illustrate the experimental results for characteristic staggered tube bundles. Some of them are plotted in Fig. 5 in such a manner that it is possible to follow the behavior of the relations n = f (S1/S2) and Cr = f (S1/S2) when H/ F = const. Although the deviation of the parameter H/F from its mean value in each series reaches ±10%, the plots in Fig. 5 demonstrate convincingly the interrelation between the quantities n and Cr, on one hand, and the tube arrangement parameter S1/S2, on the other. The relations n = f (S1/S2) and Cr = f (S1/S2) are self-similar and have a maximum lying in the range of the tube arrangement parameter S1/S2 from 1.5 to1.6. To the left of the maximum, the lower values of the exponent n can be explained by the fact that as the tube arrangement parameter S1/S2 is decreased, the tube bundle gradually transforms into a system of interacting transverse rows of tubes remote from each other downstream and thus not influencing each other (it is known
Table 2 Experimental n and Cr values in Formula (1) for staggered tube bundles. Series 7 and 8. S1 S2
Fig. 4. Comparison of the experimental data for the air pressure drop in staggered finned tube bundles at H/F = const and S1/S2 = const generalized in terms of the same equivalent diameter de: 1 – bundle RRP-3 [10], 2 – bundle at H/F = 11.10; S1/ S2 = 1.76 in series 6 (Table 1), 3 – bundle 12 [19], 4 – bundle for a smooth rectangular channel at H/F = 20 [10].
0.42 0.74 1.16 1.59 1.89 2.42 3.00 4.00
H/F
50.32 35.55 50.32 35.55 23.60 23.60 26.60
Series 7
H/F
n
Cr
0.26 0.27 0.31 0.31 0.28 0.27 0.25
3.05 2.94 4.72 4.42 3.33 3.12 2.35
33.80 24.28 33.80 24.28 16.34 16.34 16.34 16.34
Series 8 N
Cr
0.21 0.24 0.28 0.28 0.25 0.23 0.21 0.17
1.73 2.17 3.44 3.20 2.19 1.92 1.508 0.775
E.N. Pis’mennyi et al. / International Journal of Heat and Mass Transfer 73 (2014) 293–302 Table 3 Experimental n and Cr values in Formula (1) for staggered tube bundles. Series 9 and 11. S1 S2
0.42 1.01 1.15 1.16 1.59 1.89 2.42 2.50 3.00 4.00
Series 9
Series 11
H/F
n
Cr
H/F
n
Cr
21.40 21.40 21.40 15.62 10.66 10.66 10.66
0.24 0.27 0.27 0.24 0.27 0.19 0.16
3.12 4.57 4.57 2.15 2.17 1.093 0.577
19.9 19.9 8.9 8.9 8.9
0.25 0.25 0.25 0.23 0.22
2.380 2.263 2.364 2.142 1.932
297
n and Cr are well combined by the expressions based on the functions
p q S H S1 r 1 y¼a e S2 : F S2
ð13Þ
Using a standard program for finding linear regression coefficients, the processing of the experimental n and Cr values (Tables 2 and 3 and Figs. 2 and 3) depending on the influencing parameters H/F and S1/S2 as applied to the expression of form (13) yielded the following results:
0:25 0:57 H S1 n ¼ 0:17 e0:36S1 =S2 ; F S2
ð14Þ
0:53 1:30 H S1 C r ¼ 1:4 e0:90S1 =S2 : F S2
ð15Þ
The curves obtained with the use of these formulas are plotted in Fig. 5 illustrating that they are well combined with the experimental data. It should be noted that the scatter of the points relative to the calculated curves in Fig. 5 is to some extent connected with the deviations of the parameter H/F values for the selected tube bundles from those, in terms of which the corresponding curves are constructed. The assessment of the accuracy of Eqs. (14) and (15) for calculation of Eu0 numbers was made in Fig. 5 using the complex following from formulas (1) and (15)
"
K Eu
#1 1:3 S1 n 0:90ðS1 =S2 Þ ¼ Eu0 1:4 e Ree ; S2
ð16Þ
where n is defined by formula (14). The values of the complex were determined in terms of Eu0 calculated by Eq. (1) through the experimental values of the exponent n and the coefficient Cr (Figs. 2 and 3, Tables 2 and 3) for tube bundles in series 1–9 (Table 1) at the boundary Reynolds numbers Ree = 5 103 and Ree = 6 104. Fig. 6 plots the results for the higher deviation of the boundary values of the experimental and calculated Eu0 numbers. The dashed lines bound the region, where the deviation is less than ±20%. 94.5% of the points located inside this region (see Tables 4–6). Fig. 7 illustrates the comparison results of relations (1), (14), and (15) with the experimental data published elsewhere in [9,10,19,20,23,27,34,39]. As can be seen, Fig. 6 shows the points for the higher deviation of the compared values of the Eu0 numbers for Ree = 5 103 and Ree = 6 104. If the experimental data do not allow these two Ree values to be compared, then calculations are Fig. 5. Exponent n (a) and the coefficient Cr (b), in formula (1) vs. the tube arrangement parameter S1/S2 and the relative length of the extended surface H/F for staggered tube bundles.
that for the one-row arrangement, as well as for the first row of a standard tube bundle, the value of the exponent n always greatly reduces). To the right of the maximum, the reduction in the exponent values is due to the fact that as the arrangement parameter S1/S2 is increased, the arrangement of tubes in the bundle gradually approaches the arrangement typical for platen surfaces, for which according to [26], the exponent n should be equal to zero. In Fig. 5, the scatter of the curves in the parameter H/F as a whole indicates the growth of the exponent n and the coefficient Cr when the relative length of the extended surface is increased. The above-mentioned similarity of the plots n = f (H/F; S1/S2), H/ F = const and Cr = f (H/F; S1/S2), H/F = const, and also the fact that the correlation coefficient between the n and Cr values is equal to unity, permits one to suggest that the both relations can be described by the same-type functions. The experimental points for
Fig. 6. Assessment of the accuracy of relations (1), (14), (15) for tube bundles in series 1–9 (Table 1).
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Table 4 Experimental n and Cr values in Formula (1) for in-line tube bundles. Series 10, 12, 13. S1 S2
Series 10 H/F
N
Cr
H/F
n
Cr
H/F
n
Cr
0.26 0.52 1.04 1.40 2.94 1.18 1.79 2.36 0.79 0.74 1.37
29.8 29.8 29.8 17.0 6.5 8.6 12.8 8.6 8.6 29.8
0.210 0.180 0.100 0.070 0.000 0.035 0.000 0.155
1.680 1.100 0.360 0.206 0.049 0.114 0.056 0.759
15.1 15.1 15.1 9.0 3.6 4.7 6.9 4.7 4.7 15.1 15.1
0.200 0.180 0.110 0.070 0.000 0.110 0.035 0.000 0.150 0.075
1.440 1.030 0.342 0.183 0.046 0.225 0.093 0.054 0.680 0.200
8.0 8.0 8.0 4.9 2.0 2.7 3.8 2.6 2.6 8.0 8.0
0.195 0.175 0.110 0.065 0.000 0.090 0.030 0.000 0.130 0.145 0.075
1.360 0.972 0.342 0.183 0.045 0.182 0.105 0.052 0.348 0.625 0.206
Series 12
Table 5 Experimental n and Cr values in Formula (1) for in-line tube bundles. Series 14 and 15. S1 S2
2.43 1.46 0.79 1.83 1.20 0.48 0.83
Series 14
Series 15
H/F
N
Cr
H/F
n
Cr
3.9 9.0 9.0 6.1
0.000 0.065 0.135 0.025
0.065 0.197 0.666 0.100
5.4 1.8 5.4 5.4 1.8
0.065 0.130
0.166 0.616 0.298 0.997 0.350
0.090 0.160 0.105
Table 6 Experimental n and Cr values in Formula (1) for in-line tube bundles. Series 8 and 9. S1 S2
0.81 1.00 1.31 0.56 1.16 1.39
Series 8
Series 9
H/F
n
Cr
H/F
n
Cr
23.9 23.9 23.9 23.9 37.6
0.150 0.125 0.090 0.175 0.105
0.690 0.439 0.275 0.957 0.429
15.2 15.2 15.2 15.2 23.3 23.3
0.160 0.120 0.085 0.175 0.110 0.080
0.648 0.394 0.238 0.825 0.369 0.249
Series 13
data from the nine papers reviewed are generalized with an uncertainty of ±20%, and with an uncertainty of ±30% it gets up to 87%. As for all the 167 staggered arrangements of tubes examined in this work with an uncertainty of ±20%, 85.6% of the experimental data are generalized using formulas (1), (14), and (15). As noted above, in the present work the air pressure drop is also investigated by varying the number of transverse rows of tubes z2 in the tube bundle. The results obtained for staggered arrangements are in good agreement with those in [17,22]. So the influence of z2 on the Eu0 value can with a sufficient accuracy be taken into account by introducing into formula (1) the correction factor C 0z to be determined by the proposed relation graphically presented in [22] and included in the normative documents [26]. The processing of the obtained experimental data along with those given in [17,22] yielded the following relations to calculate the correction factor C 0z for staggered finned tubes with the small number of rows: when z2 < 6
6 C 0z ¼ exp 0:1 1 ; z2
ð17Þ
when z2 P 6
C 0z ¼ 1:0: These relations are graphically in [26].
ð18Þ fully
consistent
with those
presented
3.2. In-line tube bundles
Fig. 7. Comparison of relations (1), (14), and (15) with the data of other authors: 1 – [27], 2 – [19], 3 – [34], 4 – [9], 5 – [10], 6 – [39], 7 – [20,23].
performed in terms only of one of them. Such a situation has occurred with the experimental data [10,20,23] where the upper limit of the investigated Reynolds number range is about Ree 104, as well as with the data [39] obtained for the majority of tube bundles at Re > 2 104.With a deviation of less than ±20%, 100% of the data [9,20,23], 77% of the data [27], 93% of the data [10], 71% of the data [19,34] and 57% of the data [39] are generalized. In total, 78% of the
The results for the air pressure drop in forty-nine in-line helically and annularly finned tube bundles are summarized in Tables 5.5–5.7. Based on these tables Fig. 8 shows the exponent n and the coefficient Cr in formula (1) as a function of the relative length of the extended surface H/F for different values of the tube arrangement parameter S1/S2. From this figure it can be seen that as the relative length of the tube bundle is increased, n and Cr values grow monotonically, showing evidence of stabilization in the region of high H/F values. The analysis also showed that over the investigated Reynolds number range, the exponent n value varies from 0 to 0.2, reaching its highest values for the minimum values of the tube arrangement parameter S1/S2 observed in the experiments. The function n = f (S1/ S2) at H/F = const, whose form can be seen in Fig. 9, is characterized by an almost linear decrease in the exponent n with increasing parameter S1/S2. At S1/i2 P 2.1 over the entire H/F range the exponent becomes equal to zero (n = 0), which can be attributed to the transformation of the in-line tube arrangement to the platen one. Owing to this, the functions n = f(H/F; S1/S2) and Cr = f(H/F; S1/S2)
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299
The accuracy was assessed by comparing for Ree = 5 103 and Ree = 6 104 the experimental and calculated by formulas (1), Eqs. (19)–(22) Euler numbers based on the complex
K 0Eu ¼
Fig. 8. Exponent n (a) and the coefficient Cr (b) in formula (1) vs. the relative length of the extended surface H/F and the arrangement parameter S1/S2 for in-line tube bundles: 1 – S1/S2 = 0.26; 2 – S1/S2 = 0.52; 3 – S1/S2 = 0.70. . .0.74; 4 – S1/ S2 = 1.04. . .1.18; 5 – S1/S2 = 1.36. . .1.46; 6 – S1/S2 = 1.78; 7 – S1/S2 = 2.36. . .2.43; 8 – S1/S2 = 2.94.
Eu0 100%; C r Ren 1
ð23Þ
where the n and Cr values are defined by (19)–(22). Fig. 10a shows the results of this comparison for seventy-five in-line finned tubes studied by the author and in works [3,8,12,40]. As in the case of comparing the data for staggered tube bundles, here the points corresponding to the higher deviation of the compared Eu0 number values are plotted. The obtained equations for calculation of the air pressure drop in in-line finned tube bundles in cross flow describe 88% of the experimental data with an error not greater than ±20%. Fig. 10b shows the comparison of the same experimental data with those obtained by the function [26], indicating a significantly higher accuracy of the generalizing relations offered in this work. The influence of the number of transverse rows of tubes z2 on the pressure drop of air flowing across in-line tube bundles was also investigated. The results of generalization of the data for in-line tube bundles with the small number of rows are plotted in Fig. 11 in the form of the correction factor C 0z added to formula (1) as the number of transverse rows of tubes (curve 1). This figure also shows the functions C 0z = f (z2) for in-line (2) and staggered (3) tube bundles available in normative documents [26]. As can be seen from this figure, when the general trend of increasing C 0z with decreasing z2 is kept, one can see a significant deviation of generalizing curve (1) from normative relation (2) obtained only in the graphical form in [22] using a very limited amount of experimental data. Normative curve (2) is significantly higher than normative curve (3), which is not consistent with the in-line tube bundle depth variation of the Eu0 number: in this case, as the number of transverse rows over the range 1 6 z2 < 6 is increased, Eu0 should fall faster
Fig. 9. Functions: 1 – n = f (S1/i2) and 2 – Cr = f (S1/S2) at H/F = const.
cannot be approximated by sufficiently simple and convenient relations, being the same as in the case of the staggered arrangement of tubes, for the entire investigated range of the tube arrangement parameter S1/S2. Equations for data generalization were obtained separately for two ranges of S1/S2 values: When 0.25 6 S1/S2 < 2.1 n and Cr values are described with a reasonable accuracy by the expressions of the form
n ¼ ðH=FÞ0:08 ð0:184 0:088S1 =S2 Þ;
ð19Þ
C r ¼ 1:25ðH=FÞ0:25 expð1:7S1 =S2 Þ;
ð20Þ
when S1/S2 P 2.1, respectively,
n ¼ 0;
ð21Þ
Cr ¼ ðH=FÞ0:1 ð0:066 0:008S1 =S2 Þ:
ð22Þ
Fig. 10. Assessment of the deviation of the experimental data from the calculations by equations (1), Eqs. (19)–(22) (a) and by the method [26] (b): 1 – data of Tables (5.5)–(5.7), 2 – data [12], 3 – data [3], 4 – data [8], 5 – data [40].
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n¼
0:08 H S1 ; 0:184 0:088 F S2
0:25 S H 1:70S1 2; C 0r ¼ 2:5 e F and at S1/S2 > 2.1
n ¼ 0; C 0r ¼
0;1 H S1 0; 132 0; 016 F S2
The relative length of the extended surface H/F is defined by the expression
Fig. 11. Correction factor C0 Z as a function of the small number of rows of tubes in the bundle: 1 – curve for in-line tube bundles obtained according to the present work data and 2, 3 – normative curves for staggered and in-line tube bundles, respectively.
than Eu0 does in the case of staggered tube bundles at other things being equal; in addition, the values of these quantities should coincide when z2 = 1. This fact points to the insufficient accuracy of standard relation (2) for in-line tube bundles. In Fig. 11, relation (1) for determination of the correction factor C 0z as a function of the small number of rows of in-line tubes in the bundle obtained in this work corresponds to the above trend and is described by the following relations: when z2 < 6
C 0z ¼ 1:0 þ
0:65 ðz2 Þ3
ð24Þ
;
when z2 P 6
C 0z ¼ 1:0:
ð25Þ C 0z
H p½dt þ 2hd þ 2hðh þ dÞ ¼ : F S1 t ðdt þ 2hdÞ As a characteristic length in the Reynolds numbers when calculating the pressure drop of air flowing in in-line and staggered tube bundles in cross flow with uconv 6 2 the equivalent diameter of the minimum flow section of the tube bundle is taken as follows
de ¼
2½tðS1 dÞ 2hd ; 2h þ t
and when calculating the pressure drop in staggered tube bundles with uconv > 2, the following expression is used 0
de ¼
2de
uconv
The physical constants in similarity numbers refer to the average temperature of gases in the tube bundle and the velocity – to the narrowest flow (transverse or diagonal) section of the tube bundle. The correction factor for the small number of rows of tubes in the bundle depending on the tube arrangement is calculated using the following relations: for the staggered arrangement and z2 < 6
It should also be noted that the values defined by Eqs. (17) and (24) at z2 = 1 are practically the same.
6 C 0z ¼ exp 0:1 1 ; z2
4. Generalized method for calculation of air pressure drop in transversely finned tube bundles. comparison with the generalizations of other authors
for the in-line arrangement and z2 < 6
As a result of the studies made, the methods for calculation of the air pressure drop in helically and annularly finned tube bundles in cross flow have been developed. The geometric characteristics of bundles are over the range of w = 1.2 . . . 39.0, H/F = 1.5 . . . 70 and S1/ S2 = 0.3...4.0. Following the methodology valid over the Ree number range from 5 103 to104, the pressure drop in a tube bundle based on one transverse row is given by the relation
n0 ¼ C 0z C 0r Ren e :
ð26Þ
For the staggered arrangement of tubes in the bundle the exponent n and the coefficient C 0r in Eq. (26) are determined by the expressions
n ¼ 0:17
C 0r ¼ 2:8
0:25 0:57 S H S1 0:36S1 2; e F S2
0:53 1:30 S H S1 0:90S1 2: e F S2
In the case of the in-line arrangement of tubes with S1/S2 6 2.1, the following equations should be used for determination of n and C 0r
C 0z ¼ 1 þ
0:65 ðz2 Þ3
;
for any tube arrangement in the bundle and z2 P 6
C 0z ¼ 1:0: The proposed method provides a more accurate calculation of the pressure drop of air flowing across helically and annularly finned tubes compared with others that are most widely used [25,28,34,39], including those entering the normative method [26]. Our results for comparison of the calculation methods are presented in Tables 7 and 8 and Figs. 10–12. The comparison covers the experimental data for the pressure drop in 167 staggered finned tube bundles and in 75 in-line finned tube bundles in cross flow.
Table 7 Comparison of the methods for calculation of the air pressure drop in helically and annularly staggered finned tube bundles.
[28] [26] Present work
D 6 ±20%
D 6 ±30%
D 6 ±40%
D 6 ±50%
D 6 ±60%
55.3 65.2 87.0
70.2 79.5 91.9
82.0 90.7 96.9
86.3 93.2 98.8
90.1 97.5 98.8
E.N. Pis’mennyi et al. / International Journal of Heat and Mass Transfer 73 (2014) 293–302 Table 8 Comparison of the methods for calculation of the air pressure drop in helically and annularly in-line finned tube bundles.
[26] Present work
D 6 ±20%
D 6 ±30%
D 6 ±40%
D 6 ±50%
D 6 ±60%
53.2 88.0
67.5 92.2
77.0 98.7
93.5 100.0
93.5 100.0
301
structure of the equation proposed for calculation of Euler numbers does not allow its use for tubes with high fins (for h/d values close to unity, the equation gives too high values and for h/d > 1 negative values of Eu numbers). Because of this, the comparison with other calculating systems makes no sense. The comparison results with the methods [25,32], obtained over the narrow ranges of geometric characteristics and therefore yielding the low calculation accuracy, are not included into the summary tables.
References
Fig. 12. Comparison of the accuracy of methods for calculation of the drag of staggered finned tubes in the bundle: a – normative relation [26]; b – relation [28] c – relations (1), (14), (15); 1 – Ree = 5000; 2 – Ree = 60000.
In Tables 7 and 8, for each of the above calculation methods, the results are given as the percentage of tube bundles, in which the pressure drop is described by the corresponding equations with an error not exceeding a specified value of D. Table 7 contains no results compared with the method [39], because there the
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