The CRC Handbook of Mechanical Engineering. Chapter 4. Heat and Mass Transfer (776127), страница 33
Текст из файла (страница 33)
Fora sharp square inlet configuration, Recrit is about 10 to 15% lower than that for a rounded inlet configuration. For most exchangers, the entrance configuration would be sharp. Some information on Recrit isprovided by Ghajar and Tam (1994).© 1999 by CRC Press LLC4-144Section 4Transition flow and fully developed turbulent flow Fanning friction factors (within ±2% accuracy) aregiven by Bhatti and Shah (1987) asf = A + BRe -1 m(4.5.37)whereA = 0.0054,B = 2.3 ´ 10 -8 ,m = -2 3,for 2100 £ Re £ 4000A = 0.00128,B = 0.1143,m = 3.2154,for 4000 £ Re £ 10 7The transition flow and fully developed turbulent flow Nusselt number correlation for a circular tube isgiven by Gnielinski as reported in Bhatti and Shah (1987) asNu =( f 2) (Re - 1000) Pr121 + 12.7( f 2) (Pr 2 3 - 1)(4.5.38)which is accurate within about ±10% with experimental data for 2300 £ Re £ 5 ´ 106 and 0.5 £ Pr £ 2000.A careful observation of accurate experimental friction factors for all noncircular smooth ducts revealsthat ducts with laminar fRe < 16 have turbulent f factors lower than those for the circular tube; whereasducts with laminar fRe > 16 have turbulent f factors higher than those for the circular tube (Shah andBhatti, 1988).
Similar trends are observed for the Nusselt numbers. Within ±15% accuracy, Equations(4.5.37) and (4.5.38) for f and Nu can be used for noncircular passages with the hydraulic diameter asthe characteristic length in f, Nu, and Re; otherwise, refer to Bhatti and Shah (1987) for more accurateresults for turbulent flow.For hydrodynamically and thermally developing flows, the analytical solutions are boundary conditiondependent (for laminar flow heat transfer only) and geometry dependent.
The hydrodynamic entrancelengths for developing laminar and turbulent flows are given by Shah and Bhatti (1987) and Bhatti andShah (1987) asì0.0565Reï=íDh ï1.359Re1 4îLhyfor laminar flow (Re £ 2100)(for turbulent flow Re ³ 10 4)( 4.5.39)( 4.5.40)Analytical results are useful for well-defined constant-cross-sectional surfaces with essentially unidirectional flows. The flows encountered in heat exchangers are generally very complex having flowseparation, reattachment, recirculation, and vortices. Such flows significantly affect Nu and f for thespecific exchanger surfaces.
Since no analytical or accurate numerical solutions are available, theinformation is derived experimentally. Kays and London (1984) and Webb (1994) present most of theexperimental results reported in the open literature. In the following, empirical correlations for onlysome important surfaces are summarized due to space limitations. Refer to section 4.2, subsectionExternal Flow Forced Convection for bare tube banks.Plate-Fin Extended Surfaces.Offset strip fins.
This is one of the most widely used enhanced fin geometries (Figure 4.5.15) inaircraft, cryogenics, and many other industries that do not require mass production. This surface hasone of the highest heat transfer performances relative to the friction factor.
The most comprehensivecorrelations for j and f factors for the offset strip fin geometry is provided by Manglik and Bergles (1995)as follows.© 1999 by CRC Press LLC4-145Heat and Mass TransferFIGURE 4.5.15 An offset strip fin geometry.j = 0.6522 Re-0.5403æ söè h¢ ø-0.1541ædf öçl ÷è f ø0.1499ædf öç s ÷è ø-0.06780.456-1.055 ùé0.504 ædf öædf ö1.340 æ s ö-5êú´ 1 + 5.269 ´ 10 Reçl ÷ç s ÷è h¢ øêúè øèf øëûsf = 9.6243Re -0.7422 æ öè h¢ ø-0.1856ædf öçl ÷è f ø0.3053ædf öç s ÷è ø0.1(4.5.41)-0.26593.7670.236 ùé0.920 ædf öædf ösú´ ê1 + 7.669 ´ 10 -8 Re 4.429 æ öçl ÷ç s ÷è h¢ øêúè øèf øëû0.1(4.5.42)whereDh = 4 Ao( A l ) = 4sh¢l [2(slfff)+ h ¢l f + d f h ¢ + d f s](4.5.43)Geometric symbols in Equation (4.5.43) are shown in Figure 4.5.15.These correlations predict the experimental data of 18 test cores within ±20% for 120 £ Re £ 104.Although all the experimental data for these correlations are obtained for air, the j factor takes intoconsideration minor variations in the Prandtl number, and the above correlations should be valid for0.5 < Pr < 15.Louver fins.
Louver or multilouver fins are extensively used in the auto industry because of their massproduction manufacturability and hence lower cost. The louver fin has generally higher j and f factorsthan those for the offset strip fin geometry, and also the increase in the friction factors is in generalhigher than the increase in the j factors. However, the exchanger can be designed for higher heat transferand the same pressure drop compared to that with the offset strip fins by a proper selection of exchangerfrontal area, core depth, and fin density. Published literature and correlations on the louver fins aresummarized by Webb (1994) and Cowell et al. (1995), and the understanding of flow and heat transferphenomena is summarized by Cowell et al. (1995). Because of the lack of systematic studies reported© 1999 by CRC Press LLC4-146Section 4in the open literature on modern louver fin geometries, no correlation can be recommended for the designpurpose.Tube-Fin Extended Surfaces.Two major types of tube-fin extended surfaces as shown in Figure 4.5.5 are (1) individually finned tubesand (2) flat fins (also sometimes referred to as plate fins) with or without enhancements/interruptionson an array of tubes.
An extensive coverage of the published literature and correlations for these extendedsurfaces are provided by Webb (1994), Kays and London (1984), and Rozenman (1976). Empiricalcorrelations for some important geometries are summarized below.Individually finned tubes. This fin geometry, helically wrapped (or extruded) circular fins on a circulartube as shown in Figure 4.5.5a, is commonly used in process and waste heat recovery industries. Thefollowing correlation for j factors is recommended by Briggs and Young (see Webb, 1994) for individuallyfinned tubes on staggered tube banks.0.2( ) (s d )j = 0.134 Re -d0.319 s l f0.11(4.5.44)fwhere lf is the radial height of the fin, df the fin thickness, s = pf – df is the distance between adjacentfins and pf is the fin pitch.
Equation (4.5.44) is valid for the following ranges: 1100 £ Red £ 18,000,0.13 £ s/lf £ 0.63, 1.01 £ s/df £ 6.62, 0.09 £ lf /do £ 0.69, 0.011 £ df /do £ 0.15, 1.54 £ Xt /do £ 8.23, finroot diameter do between 11.1 and 40.9 mm, and fin density Nf (= 1/pf) between 246 and 768 fin/m. Thestandard deviation of Equation (4.5.44) with experimental results was 5.1%.For friction factors, Robinson and Briggs (see Webb, 1994) recommended the following correlation:ftb = 9.465Re -d0.316 ( Xt do )-0.927(XtXd )0.515(4.5.45)Here Xd = ( Xt2 + Xl2 )1 / 2 is the diagonal pitch, and Xt and Xl are the transverse and longitudinal tubepitches, respectively.
The correlation is valid for the following ranges: 2000 £ Red £ 50,000, 0.15 £ s/lf£ 0.19, 3.75 £ s/df £ 6.03, 0.35 £ lf /do £ 0.56, 0.011 £ df /do £ 0.025, 1.86 £ Xt /do £ 4.60, 18.6 £ do £40.9 mm, and 311 £ Nf £ 431 fin/m. The standard deviation of Equation (4.5.45) with correlated datawas 7.8%For crossflow over low-height finned tubes, a simple but accurate correlation for heat transfer is givenby Ganguli and Yilmaz (1987) asj = 0.255Re -d0.3 (de s)-0.3(4.5.46)A more accurate correlation for heat transfer is given by Rabas and Taborek (1987). Chai (1988) providesthe best correlation for friction factors:æ lf öftb = 1.748Re d-0.233 ç ÷è sø0.552æ do öçX ÷è tø0.599æ do öçX ÷è 1ø0.1738(4.5.47)This correlation is valid for 895 < Red < 713,000, 20 < q < 40°, Xt /do < 4, N ³ 4, and q is the tubelayout angle.
It predicts 89 literature data points within a mean absolute error of 6%; the range of actualerror is from –16.7 to 19.9%.Flat plain fins on a staggered tubebank. This geometry, as shown in Figure 4.5.5b, is used in airconditioning/refrigeration industry as well as where the pressure drop on the fin side prohibits the useof enhanced/interrupted flat fins. An inline tubebank is generally not used unless very low fin sidepressure drop is the essential requirement.
Heat transfer correlation for Figure 4.5.5b flat plain fins on© 1999 by CRC Press LLC4-147Heat and Mass Transferstaggered tubebanks is provided by Gray and Webb (see Webb, 1994) as follows for four or more tuberows.j4 = 0.14 Re -d0.328 ( Xt Xl )-0.502(s d )0.031o(4.5.48)For the number of tube rows N from 1 to 3, the j factor is lower and is given byjN-0.031= 0.991 2.24 Re -d0.092 ( N 4)j4[]0.607( 4 - N )(4.5.49)Gray and Webb (see Webb, 1994) hypothesized the friction factor consisting of two components: oneassociated with the fins and the other associated with the tubes as follows.f = ffdf öAf ö ææ+ ft ç 1 ÷ ç1 AA øèp f ÷øèAf(4.5.50)where1.318f f = 0.508Re d-0.521 ( Xt do )(4.5.51)and ft (defined the same way as f) is the Fanning friction factor associated with the tube and can bedetermined from Eu of Figure 19 of Zukaukas (1987) as ft = EuN(Xt – do)/pdo.
Equation (4.5.50) correlated90% of the data for 19 heat exchangers within ±20%. The range of dimensionless variables of Equations(4.5.50) and (4.5.51) are 500 £ Re £ 24,700, 1.97 £ Xt /do £ 2.55, 1.7 £ Xl /do £ 2.58, and 0.08 £ s/do £ 0.64.Exchanger Design MethodologyThe problem of heat exchanger design is complex and multidisciplinary (Shah, 1991). The major designconsiderations for a new heat exchanger include process/design specifications, thermal and hydraulicdesign, mechanical design, manufacturing and cost considerations, and trade-offs and system-basedoptimization, as shown in Figure 4.5.16 with possible strong interactions among these considerations asindicated by double-sided arrows. The thermal and hydraulic design methods are mainly analytical, andthe structural design is analytical to some extent.
Most of the other major design considerations involvequalitative and experience-based judgments, trade-offs, and compromises. Therefore, there is no uniquesolution to designing a heat exchanger for given process specifications. Further details on this designmethodology is given by Shah (1991).Two important heat exchanger design problems are the rating and sizing problems. Determination ofheat transfer and pressure drop performance of either an existing exchanger or an already sized exchangeris referred to as the rating problem. The objective here is to verify vendor’s specifications or to determinethe performance at off-design conditions. The rating problem is also sometimes referred to as theperformance problem.
In contrast, the design of a new or existing type of exchanger is referred to asthe sizing problem. In a broad sense, it means the determination of the exchanger construction type,flow arrangement, heat transfer surface geometries and materials, and the physical size of an exchangerto meet the specified heat transfer and pressure drops. However, from the viewpoint of quantitativethermal-hydraulic analysis, we will consider that the selection of the exchanger construction type, flowarrangement, and materials has already been made.
Thus, in the sizing problem, we will determine thephysical size (length, width, height) and surface areas on each side of the exchanger. The sizing problemis also sometimes referred to as the design problem.© 1999 by CRC Press LLC4-148FIGURE 4.5.16 Heat exchanger design methodology.© 1999 by CRC Press LLCSection 44-149Heat and Mass TransferThe step-by-step solution procedures for the rating and sizing problems for counterflow and crossflow single-pass plate-fin heat exchangers have been presented with a detailed illustrative example byShah (1981).
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
