Heat Transfer and Pressure Drop Correlations

ELSEVIER Heat Transfer and Pressure Drop Correlations for the Rectangular Offset Strip Fin Compact Heat Exchanger Raj M. Manglik Department of Mechanical, Industrial and Nuclear Engineering, University of Cincinnati, Cincinnati, Ohio Arthur E. Bergles Department of Mechanical Engineering, Aeronautical Engineering and Mechanics, Rensselaer Polytechnic Institute, Troy, New York • The development of thermal-hydraulic design tools for rectangular offset strip fin compact heat exchangers and the associated convection process are delineated. On the basis of current understanding of the physical phenomena and enhancement mechanisms, existing empirical f and j data for actual cores are reanalyzed. The asymptotic behavior of the data in the deep laminar and fully turbulent flow regimes is identified. The respective asymptotes for f and j are shown to be correlated by power law expressions in terms of Re and the dimensionless geometric parameters a, 6, and y. Finally, rational design equations for f and j are presented in the form of single continuous expressions covering the laminar, transition, and turbu- lent flow regimes. Keywords: enhanced heat transfer, compact heat exchangers, offset strip fins, thermal-hydraulic performance, heat exchanger design INTRODUCTION Compact heat exchangers are used in a wide variety of applications. Typical among these are automobile radia- tors, air-conditioning evaporators and condensers, elec- tronic cooling devices, recuperators and regenerators, and cryogenic exchangers. The need for lightweight, space- saving, and economical heat exchangers has driven the development of compact surfaces. The modem automo- bile radiator perhaps best exemplifies the advancement and technological development in compact heat exchang- ers since its vintage predecessor of the early 1900s. In recent times the thrust for energy conservation and the use of alternative energy resources has extended their application to ocean thermal energy conversion, solar, and geothermal systems [1]. Compact heat exchangers are generally characterized by extended surfaces with large surface area/volume ra- tios that are often configured in either plate-fin or tube-fin arrangements [2]. In a plate-fin exchanger, which finds diverse applications, a variety of augmented surfaces are used: plain fins, wavy fins, offset strip fins, perforated fins, pin fins, and louvered fins [2]. For these complex geome- tries, which are usually set up in a cross-flow arrangement, few predictive models or generalized correlations are available [2, 3], and actual databases are often employed for design. Though containing relatively old data, the monograph by Kays and London [4] is perhaps the most comprehensive design sourcebook. Design issues have also been addressed in several reviews by Dubrovsky [5], Shah et al. [6], Shah and Webb [2], and Shah [7, 8]. Of the many enhanced fin geometries described earlier, offset strip fins are very widely used. They have a high degree of surface compactness, and substantial heat trans- fer enhancement is obtained as a result of the periodic starting and development of laminar boundary layers over uninterrupted channels formed by the fins and their dissi- pation in the fin wakes. There is, of course, an associated increase in pressure drop due to increased friction and a form-drag contribution from the finite thickness of the fins. Typically, many offset strip fins are arrayed in the flow direction, as schematically shown in Fig. 1. Their surface geometry is described by the fin length l, height h, transverse spacing s, and thickness t. The fin offset is usually uniform and equal to a half-fin spacing; a nonuni- form offset will introduce an additional geometrical vari- able. Furthermore, manufacturing irregularities such as burred edges, bonding imperfections, and separating plate roughness also influence the flow and heat transfer char- acteristics in actual heat exchanger cores. There has been considerable effort to understand the convection mechanisms and to predict the thermal- hydraulic behavior in offset strip fin cores, and it contin- ues to attract much research attention [9-12]. The many reported studies include experimental data for actual cores or scaled-up models, empirical correlations, flow visualiza- tion, mass transfer data, analytical models, and numerical Address correspondence to Dr. Raj M. Manglik, Department of Mechanical, Industrial and Nuclear Engineering, University of Cincinnati, Cincinnati, OH 45221-0072. Experimental Thermal and Fluid Science 1995; 10:171-180 © Elsevier Science Inc., 1995 655 Avenue of the Americas, New York, NY 10010 0894-1777/95/$9.50 SSDI 0894-1777(94)00096-Q 172 R.M. Manglik and A. E. Bergles r.-8~ a = s lh t h 6 = t / t ,y = t ls Figure 1. Geometrical description of a typical offset strip fin core. solutions. Over eight decades of work has been associated with offset strip fin heat exchangers, and Manglik and Bergles [12] have compiled an extensive bibliography. Though experimental investigations predominate in the literature [5, 12], analytical modeling and numerical solu- tions have also been carried out. However, most theoreti- cal solutions suffer from an oversimplification of the flow-channel geometry. Except for Patankar and Prakash [13], Suzuki et al. [14], and Xi et al. [10], all others have considered zero fin thickness. Furthermore, stable laminar wakes have been assumed, which is contrary to the results of flow visualization studies; a transition from steady lami- nar flow to an oscillating or vortex-shedding flow occurs at higher flow rates [15-17]. The flow is generally character- ized by a progression of laminar, second laminar (transi- tional, or vortex-shedding, or oscillating flow), and turbu- lent flow regimes [17]. Joshi and Webb [16] attempted to incorporate these observations and geometrical parame- ters of the offset strip fin flow channel into an analytical model. Although a fully analytical model describes the laminar flow region, a semiempirical approach is adopted for the turbulent flow predictions. Despite this broad investigative effort, reliable predic- tion of heat transfer and pressure drop in offset strip fin heat exchangers remains a difficult, restrictive, and often uncertain process. The analytical models are either very cumbersome or oversimplified. Most of the available em- pirical correlations tend to inadequately describe the trends in a larger database; the deficiencies of prevailing correlations are discussed at length in the ensuing section. Given the importance and wide range of applications of offset strip fin heat exchangers, reliable prediction of heat transfer coefficients and friction factors is necessary. This is addressed in the present paper, and improved, more generalized correlations for f and j are presented. These are based on a reexamination of the available data in the literature, identification of the effects of geometrical fea- tures of offset strip fins and the associated enhancement mechanism, and correlation of the appropriate asymptotic behavior of the data. Finally, predictive equations are devised that represent the data continuously from laminar to turbulent flow. THERMAL-HYDRAULIC PERFORMANCE Many different correlations for heat transfer and pressure drop in offset strip fin heat exchangers have been reported in the literature. These are chronologically listed in Table 1, and typical comparisons between some of them are graphed in Figs. 2 and 3. Manson [18] appears to have made the first attempt at developing predictive equations. However, the database he employed consists of dissimilar geometries: scaled-up and actual offset strip fins, louvered fins, and finned flat tubes. Kays [22] made one of the first attempts at analytical modeling of the heat transfer and friction loss in offset strip fins and proposed a modified laminar boundary layer solution that includes the form- drag contribution of the blunt fin edges. The often cited Wieting [23] correlations are power law curve fits through data for 22 geometries [Eqs. (6)-(9), Table 1] for laminar or turbulent flows. For predicting f and j in the transition region, extrapolating the equations up to their respective intersection point is suggested. This, however, tends to misrepresent the transition region as seen from Figs. 2 and 3. Also, there appears to be some discrepancy in the hydraulic diameter definition, and it is not clear whether all the data were referenced to one consistent definition. On the basis of previously reported flow visualization studies [15, 30, 31] and their own experiments, Joshi and Webb [16] attempted to identify the transition from lami- nar flow. As the flow rate increases, oscillating velocities develop in the wakes, leading to vortex shedding with further increase in Re; this acts as free-stream turbulence for the downstream fins, thereby increasing the heat and momentum transfer. The onset of oscillating flow and the consequent change in wake structure were found to corre- spond approximately to the departure from the laminar region log-linear behavior of f and j. It was observed that the velocity profile in the wake was affected by the fin spacing s, thickness t, and length I. From the data on 21 offset strip fin cores given by Ix)ndon and Shah [25], Walters [26], and Kays and London [4], it was determined that the transition point can be predicted by the correla- tion for Re* given in Eq. (14) (see Table 1). Furthermore, Joshi and Webb [16] developed elaborate analytical models to predict f and j. They incorporate the heat transfer from the fin ends, the form drag due to the finite fin thickness, and heat transfer and friction loss from the parting plates. An attempt was also made to model the effects of fin burrs and roughness with a semiempirical approach. However, these models are quite cumbersome, and in cognizance of the need for an easy- to-use correlation, Joshi and Webb [16] reevaluated the empirical equations of Wieting [23]. Adjusting the database of 21 geometries [4, 25, 26] to their hydraulic diameter definition (Table 1) and choosing the laminar and turbu- lent flow limits as Re < Re* and Re > (Re*+ 1000), respectively, they presented Eqs. (10) and (12) to predict j and Eqs. (11) and (13) to predict f (see Table 1). Their predictions are compared with typical data in Figs. 2 and 3. Correlations for Offset Strip Fin Surface 173 Table 1. Chronological Listing of Heat Transfer and Friction Factor Correlations for Offset Strip Fin Cores Investigator(s) Correlation Database / Remarks 1. Manson [18] (0.6( l /Dh)°SRe °'5, l /D h < 3.5 ( la) Norris and Spofford [19], three J = ~ 0.321 Re °5, l /D a > 3.5 ( lb) four°ffSetscaled-upStrip fin cores;C°res; LondonJ°yner [20],and Ferguson [21], one louvered fin For Re < 3500: core and one finned fiat tube ( l l .8( l /D h) Re °'67, I /D h < 3.5 (2a) core. f = ~ 3.371 Re °67, l /D h > 3.5 (2b) For Re > 3500: 0.38(1/D h) Re 0"24, l /D h < 3.5 (3a) f = 0.1086Re °24, I /D h > 3.5 (3b) where the hydraulic diameter is defined by D h = 2sh/(s + h). 2. Kays [22] j = 0.665 Ret -°'5 (4) Analytical model for purely laminar f = 0.44(t/l) + 1.328 Re/- 0.5 (5) boundary layer flow over interrupted plate surface. 3. Wieting [23] 4. Joshi and Webb [16] 5. Mochizuki et al. [27] 6. Dubrovsky and Vasiliev [28] Re _< 1000: j = 0.483(l/Dh)-°162ot -0"184 Re 0.536 (6) f = 7.661(l/Dh)-°3840~ -°'°92 Re 0.712 (7) Re >_ 2000: j = 0.242(l/Dh)-°322(t/Dh) °'°89 Re- 0.368 (8) f= 1.136(l/Dh ) o.781(t/Dh) 0.534 Re-0.198 (9) where D h = 2sh/(s q- h). Re _< Re*: j = 0.53 Re-°'5(l/Dh)-°~Sc~ -°'14 (10) f = 8.12 Re- 0.74(l/Dh )- 0.41Og- 0.02 (1 1) Re >_ Re* + 1000: j = 0.21 Re-°4°(l/Oh)- °24( t /Oh )0"02 (12) f = 1.12 Re- °36( l /Oh) - °65(t//Oh )0"17 (13) where - 1 Re* =257(~) l23(~)°58Dh[ t+ 1.328(/~-h ) - °s ] (14) and D h = 2(s - t)h/[(s + h) + th/l]. Re < 2000: j = 1.37(1/D h) °25a-°184 Re -°'67 (15) f = 5.55(l/Oh)-O.32ot 0.092 Re-0.67 (16) Re > 2000: j = 1.17(l/D h + 3.75)-l(t/Dh) °'°89 Re 0.36 (17) f = 0.83(l/D h + 0.33)-°5(t/Dh) °'534 Re -°'2° (18) where D h = 2sh/(s + h). Re < Relim: Nu = O.O00437(t/Dh)- 2"6(l/Oh)- 0.15 Re x (19) where x = 2.2(t/Dh)°55(l/Dh) -°°2 Re > Reli m Nu = O.O0723(t/D h) l6(l/Oh) °9REX (20) where x = 1.2(t/Dh)°34(l/Oh) 0"15 and Reli m = 3960(t/Dh)°25(l/Dh)°42; (21) Re ___ Relim: = 1.05(t/Dh) 1.OS(l/Dh) 0.217 Re x (22) where x = - 0.277(t/D h) °285(1/Dh )°'°64. Re > Relim: = O.13l(t/Oh)-°'4a(l/Dh )-0"234 Re x (23) where x = -O.O042(t /Dh)- l25( l /Oh) 0"39 and Reli m = 448(t/D h) °653(l/Dh )0"09. (24) Kays and London [24], 10 cores; London and Shah [25], nine cores; Waiters [26], two cores; London and Ferguson [21], one louvered fin core. Kays and London [4], 18 cores; London and Shah [25], one core; Walters [26], two cores. Experimental data for five scaled-up offset strip fin cores [27]. Eleven cores in a double sandwich arrangement but without the splitter plate, which leaves a " leakage" path between top and bottom rows of strip fins. The definition of D h is not explicitly given. Also, the same Nu equations are cited by Kalinin et al. [29], and have been used to correlate data for non rectangular offset strip fin geometries. 174 R.M. Manglik and A. E. Bergles j ,f 10 010 001 Surface: 1 /8 - 16.O0(D) 0 f. } ~ = 0.477, 6 = 0.1148, 3' = 0.106 • I 0 002 i , J ~ L L = ~ [ ~ , , ~ , 150 103 104 Re Figure 2. Comparison of f and j correlations with experi- mental data of Kays and London [4] for the 1/8-15.61 offset strip fin core. ( - - - ) Joshi and Webb [16]; ( ) Wieting [23]; ( - - ) Mochizuki et al. [27]. Subsequently, two other sets of correlations appeared in the literature. As seen in Table 1, the Mochizuki et al. [27] correlations [Eqs. (15)-(18)] are once again a reworking of the Wieting [23] equations, with the coefficients and expo- nents modified to fit their own experimental data for five scaled-up rectangular offset strip fin surfaces. Only fully developed laminar or turbulent flow is considered, with an abrupt change of flow regime at Re = 2000. Dubrovsky and Vasiliev [28] presented experimental data for 11 dif- ferent rectangular offset strip fin cores with a double- sandwich construction. These are similar to surface 1/8- 20.06(D), for example, of Kays and London [4], except that they do not have a splitter plate between the two tiers of offset strip fins; instead, a "leakage" flow passage of three to four times the fin thickness is provided. No specific reasons were given for such a construction, and the effect of the "leakage" path on the performance has not been established. To predict Nusselt numbers and Darcy fric- tion factors, Eqs. (19)-(24) were presented (Table 1) [28]. As in the case of Mochizuki et al. [27], only laminar and turbulent flow conditions are considered. The limiting Reynolds number for the two flow regimes are given by Eqs. (21) and (24) for the Nusselt number and friction loss predictions, respectively. Furthermore, the form of the correlations is rather awkward, with the same geometrical parameters appearing as coefficients and exponents of Re. The Nu equations given here were also reported by Kalinin j,f 10 010 001 0002 150 . . . . . i Sur[ace: 1/8 - 15.61 • ,1, = 0.244, ~ = 0.032, 7 = 0.067 • J - . % - - ' ' . . . . . J i i i t , i ~ i 10 3 10 ' Re Figure 3. Comparison of f and j correlations with experi- mental data of Kays and London [4] for the 1/8-16.00(D) offset strip fin core. ( - - - ) Joshi and Webb [16]; ( - - ) Wiet- ing [23]; (- -) Mochizuki et al. [27]. et al. [29] as being applicable to triangular offset strip fins, and the origin of the equations is attributed to Voronin and Dubrovsky [32]. As seen in Figs. 2 and 3, the existing correlations tend to present a wide performance envelope, and they have several shortcomings. In all cases, only established lami- nar or turbulent flow is considered and the transition region is ignored. This extends over a fairly large Re range of 1000 in the case of Wieting [23] and Joshi and Webb [16]. Mochizuki et al. [27] and Dubrovsky and Vasiliev [28] completely ignore the transition region and consider an abrupt change from laminar to turbulent flow; the evi- dence in the data for actual cores [4, 25] is contrary to this. Furthermore, most of the equations are a reworking of the Wieting [23] expressions, with constants and expo- nents changed to fit different data sets. Little attempt has been made to consider a large database that would extend their general validity. Also, the equations do not necessar- ily relate the performance to all the pertinent geometrical attributes of the offset strip fin surface. NEW DESIGN CORRELATIONS Analysis of Experimental Data To understand the effects of various geometrical at- tributes of offset strip fins, experimental data for airflows Correlations for Offset Strip Fin Surface 175 with heat transfer for 18 different cores I given by Kays and London [4], Waiters [26], and London and Shah [25] have been examined. These surfaces are listed in Table 2 along with their pertinent geometrical parameters. As Joshi and Webb [16] have established, for the case of uniform offset of half-fin spacing, the dimensionless pa- rameters a = s /h , 6 = t / l , and 3" = t / s describe the off- set strip fin geometry. Their influence has also been documented in other experiments [33, 34] and numerical studies [13, 14, 35]. Furthermore, the listed hydraulic diameter in Table 2 is given by the following definition: 4A~ 4shl D h (27) Al l 2(sl + hl + th) + ts As discussed previously [12], there appears to be no con- sensus in the literature over definition of the hydraulic (or equivalent) diameter in terms of the geometrical dimen- sions of offset strip fins. The usual definition employed is either 4Ac/P or 4Ac/ (A / I ) , where Ac, P, and A have been evaluated differently by various investigators; at least three different expressions for D h can be identified in the literature [12]. In the present case, on a unit fin channel basis, the free flow or channel flow area is taken as Ac = sh, and the mass flux per channel is given by G = rh/Nsh. In evaluating the heat transfer area A, the blunt fin edges, both vertical and lateral, have been included as well as the channel surface area. This is consistent with the conventions of London and Shah [25] and Joshi and Webb [16]. In Fig. 4, the f and j data for two pairs of surfaces, each with the same value of 6 but different aspect ratio a, are presented. The influence of a is clearly discernible; the effect is almost the same in both laminar and turbu- lent flows, with higher f and j for smaller values of a. The fin thickness introduces a form drag and affects the heat transfer. Also, as the boundary layer grows over the fin surface, it is abruptly disrupted at the end of the fin offset length I. Essentially, for the flow over short lengths of fins of finite thickness, there is an outward displace- ment near the leading edge, .followed by a local accelera- tion near the trailing edge and the eventual dissipation of the boundary layer in the fin wakes. As documented by Xi et al. [10] and Lee and Kwon [9], these effects are shown schematically in Fig. 5; the fin thickness and offset length tend to have a competing influence on the flow field. Moreover, thicker fins have larger form drag and heat transfer contributions from blunt fin edges, whereas with slender and longer fins, f and j are