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Calculation of Heat Transfer from Pressure Drop in Tube Bundles

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Calculation of Heat Transfer from Pressure Drop in Tube Bundles 1155 3rd European Thermal Sciences Conference 2000 E.W.P. Hahne, W. Heidemann and K. Spindler (Editors) © 2000 Edizioni ETS, Pisa. All rights reserved. CALCULATION OF HEAT TRANSFER FROM PRESSURE DROP IN TUBE BUNDLES Holger Martin and Volker Gnielinski Universität Karlsruhe (TH), Thermische Verfahrenstechnik, D-76128 Karlsruhe ABSTRACT A new type of analogy between pressure drop and heat transfer has been discovered, that may be used in chevron-type plate heat exchangers, in packed beds, and in tube bundles. It is based on the „generalized Lévêque equation“. Available experimental data on heat transfer in tube bundles in crossflow, both inline, and staggered arrangements, have been tested in greater detail. Using an empirical correlation for pressure drop in these arrangements from the literature that has been successfully tested against a large number of experimental pressure drop data, we found that the heat transfer data collected earlier could be very well represented from the pressure drop correlation and the generalized Lévêque equation. The fraction xf of the total pressure drop that is due to fluid friction only, and therefore related to the average shear rate at the surface, turned out to be a constant over the whole range of Reynolds numbers. The new method has the advantage, that apart from the empirical pressure drop correlation, the heat or mass transfer coefficients are obtained from an equation based on theory. The method results in a heat transfer prediction for bundles with more than two rows of tubes, which is at least as good as the existing empirical heat transfer correlations that do not make use of an analogy. 1. STATE OF THE ART [1, 2, 3] fits the data for inline bundles with an acceptable degree of approximation in the whole range of the experimental Heat transfer in tube bundles in cross flow is a classical variables. The data for staggered bundles are also well problem of process heat transfer which has been investigated represented for gases (air mainly), while for the liquids (see by many researchers already in the first half of the 20th Fig. 8 in [3]) the data systematically tend to give higher values century. The present state of the art can be found in the with increasing Prandtl and decreasing Reynolds numbers. handbooks, like the Heat Exchanger Design Handbook [1] and the VDI-Heat Atlas [2]. In these handbooks, the heat transfer coefficient between the outer cylindrical surfaces of a tube 2. THE LÉVÊQUE-ANALOGY bundle and a fluid flowing through the bundle in cross flow is calculated from the corresponding equations for the Nusselt A new type of analogy between pressure drop and heat number of a single (row of ) cylinder(s) in cross flow, transfer has been discovered, that may be used in chevron- multiplied by an empirically correlated arrangement factor fa(a, type plate heat exchangers [5], in packed beds, and in tube b, type), which was found to be a function of the type of bundles [6]. It is based on the „generalized Lévêque bundle, i.e. inline or staggered arrangements, and of the equation“. transverse and longitudinal pitch ratios a and b respectively. 1/3 1/3 2 1/3 The pitches ad and bd are defined as the distances of two Nu/Pr = Sh/Sc = 0.404 (xf Re dh/L) (1) adjacent tube centerlines of the bundle in a direction perpendicular to the main flow direction (lateral, ad) and in Available experimental data on heat transfer in tube bundles in flow direction (longitudinal, bd), where d is the outer diameter crossflow, both inline, and staggered arrangements, have of the tubes. The equations to calculate Nubundle(Re, Pr, a, b, been tested in greater detail. Using the empirical correlation type) as recommended in the Heat Exchanger Design for pressure drop in these arrangements by Gaddis and Handbook [1] and in the VDI-Heat Atlas [1] have been Gnielinski [4] that has been successfully tested against a developed by the second author in 1978 (reference [3] is the large number of experimental pressure drop data, we found English translation of an original paper in German). In this that the heat transfer data for bundles with more than 2 tube paper [3], the experimental data then available from the rows, collected by Gnielinski [3] could be very well literature had been used to find out the appropriate represented from the pressure drop correlation and the arrangement factors fa. The method was tested against a large generalized Lévêque equation. number of experimental data from the 20 sources as given here When using the pressure drop correlation by Gaddis and in Table 1. The comparison between the correlation (curve) Gnielinski [4], it has to be taken into account, that they and the experimental data (symbols) was shown graphically in defined their coefficient x (= x total) as the pressure drop per reference [3] in eight figures containing up to fourteen single number of main resistances (=number of rows of tubes N) curves of a (Nubundle/fa) vs. Re plot with Pr as a parameter. From 2 divided by the dynamic pressure (ru0 /2) using the velocity in these figures one can find out that this state of the art method the narrowest cross section (see APPENDIX, 9.1). 1156 4 4 10 10 1/3 1/3 1.5 X 1.5 1.25 X 1.25 Nu/Pr Nu/Pr 3 3 10 10 1 / 3 1 /3 Nu æ d ö Nu æ dh ö = 0.404 Hg h = 0.404ç Hg ÷ 1/ 3 ç ÷ Pr1 / 3 è L ø Pr è L ø 2 + 30% 2 10 10 + 30% 101 102 105 - 30% 110 - 30% 10 10 101 112 114 108 115 113 118 114 1 1 2 3 4 5 7 2 3 4 5 7 1 10 10 10 10 10 Re0 10 1 10 10 10 10 10 Re0 10 Fig. 1 Heat Transfer Group Nu/Pr1/3 vs. Reynolds-number, Fig. 2 Heat Transfer Group Nu/Pr1/3 vs. Reynolds-number, inline bundles a=1.25, b=1.25, for the number of data sources inline bundles a=1.5, b=1.5, for the number of data sources (at 2 2 (at the symbols) see Table 1. Hg=(x/2)Re0 . the symbols) see Table 1. Hg=(x/2)Re0 . As the pressure gradient (Dp/Dz) at the heated surface is 3 needed, the values can be used directly for b>1, while they 10 have to be divided by b for b<1. 1/3 1.25 X 1.08 The hydraulic diameter for the tube bundles was calculated as Nu/Pr dh=((4a/p)-1)d (for b >1) (2) 1 / 3 and 2 Nu d 10 æ h ö 1 / 3 = 0.404ç Hg ÷ dh=((4ab/p)-1)d (for b <1) (3) Pr è L ø + 30 % where a and b are the lateral and the longitudinal pitch ratios respectively, and d is the outer diameter of the tubes. - 30 % Longitudinal pitches of b<1 are only possible for staggered 10 bundles. The length L in the generalized Lévêque equation 114 115 has been taken as the longitudinal and diagonal pitch, 118 respectively 117 1 L = bd (for inline bundles) (4) 2 3 4 6 and 1 10 10 10 10 Re0 10 L = cd (for staggered bundles) (5) 1/3 Fig. 3 Heat Transfer Group Nu/Pr vs. Reynolds-number, staggered bundles a=1.25, b=1.08, for the number of data where c is the diagonal pitch ratio 2 sources (at the symbols) see Table 1. Hg=(x/2)Re0 . c =((a/2)2+b2)0.5. (6) 3 It was found that using 10 1/3 1.5 X 1.3 . xf = xf x (7) Nu/Pr 1 / 3 Nu æ dh ö 2 = 0.404ç Hg ÷ leads to a very reasonable agreement between the analogy 10 Pr1 / 3 è L ø predictions and the experimental results. The fraction xf of the total pressure drop coefficient x, that is due to fluid friction + 30 % only, and therefore related to the average shear rate at the surface, turned out to be a constant over the whole range of 10 - 30 % Reynolds numbers. The heat transfer data had been collected by Gnielinski [3] 114 from about twenty different sources. Fortunately, the 115 collection of data had been conserved in a usable format, so it 1 was possible to re-evaluate this experimental information. 2 3 4 6 Re0 1/3 1 10 10 10 10 10 The Figs. 1 through 4 show the dimensionless group Nu/Pr plotted vs. the Reynolds number Re0, for typical inline and Fig. 4 Heat Transfer Group Nu/Pr1/3 vs. Reynolds-number, staggered bundles. The curves in these figures are calculated staggered bundles a=1.5, b=1.3, for the number of data from the generalized Lévêque-equation (1), with dh/L from 2 sources (at the symbols) see Table 1. Hg=(x/2)Re0 . Eqs. (2 –5), and the friction factor xf taken as x/2, with the total pressure drop coefficient x calculated from the correlation of Gaddis and Gnielinski [4], which is also recommended in the The total amount of 1606 data points for the inline tube handbooks [1, 2]. bundles, when compared with the prediction of the generalized Lévêque equation, using the empirical correlation for x in these arrangements by Gaddis and Gnielinski [4] by 1157 minimizing the root mean square (RMS)-deviation, leads to an The figures therefore may be also seen as a plot of optimum of xf,,inline=0.54 with RMSinline=19.5%.
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