Heat transfer and pressure drop correlations for the rectangular offset strip fin compact heat exchanger

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 turbulent flow regimes.

Keywords: enhanced heat transfer, compact heat exchangers, offset strip fins, thermal-hydraulic performance, heat exchanger design

I N T R O D U C T I O N

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 automobile radiator perhaps best exemplifies the advancement and technological development in compact heat exchangers 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 geometries, 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 transfer 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 dissipation 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 nonuniform offset will introduce an additional geometrical variable. Furthermore, manufacturing irregularities such as burred edges, bonding imperfections, and separating plate roughness also influence the flow and heat transfer characteristics 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 visualization, 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 l h

t h 6 = t / t ,y = t l s

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 solutions 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 laminar flow to an oscillating or vortex-shedding flow occurs at higher flow rates [15-17]. The flow is generally characterized by a progression of laminar, second laminar (transi- tional, or vortex-shedding, or oscillating flow), and turbulent flow regimes [17]. Joshi and Webb [16] attempted to incorporate these observations and geometrical parameters 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 prediction 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 empirical 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 t h e 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 laminar 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 correlation 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 turbulent flow limits as Re < Re* and Re > ( R e * + 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.

Correlat ions for Offset Strip Fin Surface 173

Table 1. Chronological Listing of Hea t Transfer and Frict ion Factor Correlat ions for Offset Strip Fin Cores

Investigator(s) Correlation Database / Remarks

1. Manson [18] (0 .6 ( l /Dh)°SRe °'5, l / D h < 3.5 ( l a ) Norris and Spofford [19], three

J = ~ 0.321 Re °5, l / D a > 3.5 ( l b ) 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 d iameter 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. Wiet ing [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 R e - 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 R e - 0.74(l/Dh ) - 0.41Og- 0.02 (1 1)

Re >_ Re* + 1000: j = 0.21 Re-°4°(l/Oh)- ° 2 4 ( t / O h )0"02 (12) f = 1.12 R e - ° 3 6 ( l / O h ) - °65( t / /Oh )0"17 (13)

where - 1

Re* = 2 5 7 ( ~ ) l 2 3 ( ~ ) ° 5 8 D h [ 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 .O0 4 2 ( t /Dh ) - l 25 ( 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; Wal ters [26], two cores.

Experimental data for five scaled-up offset strip fin cores [27].

Eleven cores in a double sandwich arrangement but without the split ter plate, which leaves a " leakage" path between top and bot tom 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.


j , f

10

010

001

Surface: 1 / 8 - 16 .O0(D)

0 f. } ~ = 0.477, 6 = 0.1148, 3' = 0.106 • I

0 0 0 2 i , J ~ L L = ~ [ ~ , , ~ ,

1 5 0 1 0 3 1 0 4

Re

Figure 2. Comparison of f and j correlations with experimental 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 exponents 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 different 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 friction 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

1 0 3 1 0 '

R e

Figure 3. Comparison of f and j correlations with experimental 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 laminar 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 exponents 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 attributes 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 parameters a = s / h , 6 = t / l , and 3" = t / s describe the offset 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)

A l 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 4 A c / P or 4 A c / ( 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 = r h / N s h . 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 turbulent 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 displacement near the leading edge, .followed by a local acceleration 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 influenced only by the momentum and energy transfer from the fin sides. These effects can be characterized by the parameter 6 = t / l [10, 16] and are evident in Fig. 6, where data for a pair of surfaces with the same a but different 6 are presented. Furthermore, thicker fins lead to smaller passages and a smaller fin density. There is a consequent reduction in the free-flow area and, as shown by Patankar and Prakash [13], Suzuki et al. [14], and Joshi and Webb [16], this effect is best described by 3" = t / s .

1 Data for three other cores contained in Kays and London [4] have been excluded due to the uncertainty in reevaluating their hydraulic diameters.

0 2

0 .1

0 . 0 2

. . . . . . . I . . . . . .

~ _ S U R F A C E a = 8/7~ di = t f l o 1 / 1 0 - 2 7 . 0 3 0 . 1 3 4 - ' ~ 0 0 4 0

• 1 /10o19 .35 0 .670 J - ' -

x,.'-,, '-%. ~ v 8 - 1 5 6 1 0 2 4 4 ~ 0 0 3 2

i i i i i i i

004 . . . . . . .

0.01

0 . 0 0 5 I I i = = i i I I I i I I 200 1000 10000

Re

Figure 4. Effect of aspect ratio ( a = s / h ) on the experimental data for f and j from Kays and London [4].

Correlat ions for f a n d j

From the foregoing it is evident that f and j are function- ally related to Re, a , 6, and 3, and can be represented as

f = A Real( a )a2( 6 )a3(3' )a4 (28) and

j = B Re 61(a)bE(6 )ba(3')b4. (29)

The use .of these power law expressions is justified because variations in f and j with Re, a , 6, and 3' follow constant-slope log-linear lines in both deep laminar and fully turbulent flow regions. The data for airflows (Pr = 0.7) for the 18 geometries described in Table 2 were analyzed separately in the laminar and turbulent flow regions. The limit on the laminar flow range was taken as Re _< Re*, and the limit on the turbulent flow range as Re >_ ( R e * + 1000); Re* was evaluated from Eq. (14) (given by Joshi and Webb [16]) and then adjusted to conform with the Dh definition in the present case [Eq. (27)]. (Also, note that Nu was correlated instead of j [ = N u / ( R e prl/3)] to avoid a compounding effect of Re.) A multivariable regression analysis [36] yielded the following results. 2

For Re < Re* (laminar flow): f = 9.6243 R e -0"7422 a-0"185660"30533" -0"2659, (30)

j = 0.6522 Re-°"54°3 Of- 0"15416 0.14993'- 0.0678, (31)

2 The calculations were carried to four decimal places (four or five significant digits) only to reduce the roundoff errors in the arith- metic. This does not reflect the accuracy of the original data but minimizes the influence of roundoff errors in calculating the coefficients and exponents of the correlations.


local flow acceleration

displacement \ wake recovery

(a)

Figure 5. Schematic of the flow behavior in a typical offset strip fin array (adapted from [9] and [10]). (a) Growth and disruption of boundary layer; (b) isoveiocity contours [9].

I __..1

(b)

02

0

f

0 0:

0 025

. . . . . . I . . . . . . .

S U R F A C E 6 = ~fl e = s/h o ,,lO:27o3 o o,o

, , , , , , i , , , , , , ,

0 005

. . . . . . l ' '

i l i i , i I , ,

150 1000 IOOOO

R e

Figure 6. Effect of fin thickness/offset length ratio (6 = t/l) on the experimental data for f and j from Kays and London [4].

and for Re > (Re* + 1000) (turbulent flow):

f = 1.8699 Re - 0.2993 O ~ - 0 . 0 9 3 6 t ~ 0 . 6 8 2 0 , ~ - 0 . 2 4 2 3 , (32)

j = 0.2435 Re -0"4063 OL-0"1037t~0"1955"y 0.1733, (33)

where the regression fits for f and j, respectively, had multivariable correlation coefficients of 0.987 and 0.942 in the laminar flow range and 0.923 and 0.972 in the turbulent flow range, all at the 0.99 confidence level. Equations (30)-(33) establish an inverse relationship for f and j with Re, a, and y, and a direct relationship with 6. This clearly substantiates the statement in the preceding section that smaller values of a (small aspect ratio) and 3' (thin fins) and a large value of 6 (shorter offset length) increase both f and j. Furthermore, these equations reflect the appropriate asymptotic behavior in both the laminar (0 < Re _< Re*) and turbulent (Re* + 1000 < Re < ~) flow regimes, and they describe the data very well; from the statistical analysis, the maximum standard error in the predicted value of any data point was 0.054.

It is seen from Figs. 2-4 and 6 that the experimental data do not exhibit the characteristic discontinuity, or "jump," normally associated with the transition from laminar to turbulent flow. Instead, there is a continuous varia- tion of f and j with Re. This suggests that the data can be described by a single equation for the laminar, transition, and turbulent regions. To devise such expressions, the asymptote-matching technique of Churchill [37] was em-


Table 2. Geometrical Parameters for the Database of Offset Strip Fin Cores

Reference Surface Dh(mm) a ~ 7

Kays and London [4]

London and Shah [25]

Walters [26]

1/8-15.61 2.383 0.244 0.032 0.067 1/8-19.86 1.542 0.494 0.032 0.086 1/9-22.68 1.735 0.135 0.036 0.100 1/9-25.01 1.495 0.184 0.036 0.111 1/9-24.12 1.209 0.528 0.036 0.107 1/10-27.03 1.423 0.134 0.040 0.121 1/10-19.35 1.403 0.672 0.040 0.084 1/10-19.74 1.219 0.997 0.020 0.041 3/32-12.22 3.414 0.162 0.043 0.051 1/2-11.94(D) 2.266 0.711 0.012 0.077 1/6-12.18(D) 2.636 0.461 0.022 0.051 1/7-15.75(D) 2.070 0.410 0.028 0.067 1/8-16.00(D) 1 .863 0.477 0.048 0.106 1/8-19.82(D) 1 .539 0.484 0.032 0.086 1/8-20.06(D) 1 .491 0.491 0.032 0.087

#501 0.646 1.024 0.020 0.038 20R/19.43 1.133 1.034 0.060 0.132 28R/27.20 0.917 0.700 0.060 0.195

18 0

, ,,-,,

b . I } - -

U

~ 1 0 -1 W

O.

10 -2 -2

0

I I I I I f I I I I I I I I I

1 0 - 1 1 0 m

f EXPERIMENTRL

Figure 7. Comparison of predictions for f given by Eq. (34) with experimental data for offset strip fin cores listed in Table 2.

ployed. Equations (30) and (31) describe the laminar region asymptotes for f and j, respectively, as Re ~ 0; likewise, Eqs. (32) and (33) are the asymptotes as Re ~ ~. Consequently, the following equations are obtained:

f = 9.6243 Re -0.7422 Ot-0"1856~ 0.3053~-0.2659

× [1 + 7.669 x 10 -8 Re4429o~°92°~3767"y°236] 0"1 (34)

and

j = 0.6522 R e -0"5403 ~¢ 0.1541~0.1499,y 0.0678

×[1 + 5.269 × 10 -5 Rel'34°ot°5°4~°456T-l°55]°l.

(35)

These equations correlate the experimental data for the 18 cores of Table 2 within + 20%, as is seen in the scatter plots of Figs. 7 and 8. The good agreement of Eqs. (34) and (35) with typical data is also evident in Figs. 9 and 10. More important, they describe the right trend in the heat transfer and friction loss behavior of offset strip fins in laminar, transition, and turbulent flow regimes and obvi- ate the need for determining the specific flow regime for any given operating condition. This is particularly useful in predicting off-design operating performance in most practical applications. It may be noted, however, that there is inherent scatter in the data due to manufacturing variations in the offset strip fin cores. Consequently, there is an implicit limit on the accuracy of the empirical correlation. As seen from Figs. 7 and 8, there is a some- what larger scatter in the f data than in the j data; this was also the case in the statistical correlations developed by Ravigururajan and Bergles [38] for enhanced tubes.

Furthermore, the data for the cores listed in Table 2 were obtained usin~ air (Pr = 0.7) as a working fluid in the 120 < Re _< 10" range. Though Prandtl number was not a specific test variable, its effect is included in a

classical manner by employing the Colburn factor, j [St Pr 2/3 or Nu / (Re prl/3)]. This is applicable over a moderate range of Prandtl numbers, that is, for fluids with Pr ranging from 0.5 to 15 [4], and Eq. (35) is expected to be valid for all gases and most liquids with moderate Pr values. The experiments of Tinaut et al. [11] with lubricat- ing oil further support this contention; Pr 1/3 is found to suitably correlate the results. However, it is noted that the Pr exponent is often 0.4 instead of 1/3 in many enhanced channel flow situations. Thus, caution should be exercised in using the correlations for high Prandtl number viscous liquids and high-temperature waste-heat recovery applica-

1 0 - t

W

U

Q I 0 - e W r r I Z

"-3

-3 1 0

1 0

I l I I I I I I I I I I I I 1,1

3 ~ 1 / I I I I I I I I 2 I I I I I I I

I O

j EXPERIMENTRL

~ 0 - t

Figure 8. Comparison of predictions for j given by Eq. (35) with experimental data for offset strip fin cores listed in Table 2.


10 I i , , , , i , i i r , T i r , t - -

r S U R F A C E : 1 / 8 - 1 5 . 6 1 a = 0 . 2 4 4 , 6 = 0 . 0 3 2 , y = 0 . 0 6 7

• [ ) K a y s a n d L o n d o n [4 ]

o,o 413 j,f

o 01

0 0 0 2 I I I I I I I [ [ 1 I I i I i I 150 103 10"

R e

Figure 9. Comparison of predictions from Eqs. (34) and (35) with experimental data for the 1/8-15.61 core of Kays and London [41.

10 ' ' ~ . . . . I

j f

0 1 0

001

S U R F A C E : 1 / 8 - 1 6 . 0 0 ( D ) o, = 0 4 7 7 , 6 = 0 0 4 8 , g = 0 . 1 0 6

oQ [ ) K a y s a n d L o n d o n [4 ]

0002 i I , i , i , I I l i i i I i I r 150 103 104

Re

Figure 10. Comparison of predictions from Eqs. (34) and (35) with experimental data for the 1/8-16.00(D) core of Kays and London [4].

tions. Here, the influence of temperature-dependent fluid properties could be significant, and the Prandtl number effects may not be adequately described by its 1/3 power in j. Because no experimental data are currently available, these effects and the applicability of the correlations presented here cannot be verified for such flow conditions.

In order to understand the performance of different compact heat exchanger surfaces relative to their special geometrical attributes, the flow area goodness factor (j/f) is often used [2, 7], among other figures of merit [3, 4]. The significance of j / f is that it is inversely proportional to the core free-flow area A C for fixed operating conditions. Thus, larger values for j / f suggest a lower frontal area requirement for the heat exchanger. The area goodness factors of four geometries given in Table 2 are graphed in Fig. 11. As is evident, the influence of the geometrical parameters a, ~, and y on the core's performance is dependent upon the flow Re. There seems to be a relatively small effect due to changes in a, 6, and 7 in the deep laminar region compared to that at higher flow rates (Re >_ 1000). In general, for a given a, offset strip fin geometries with small 6 and 7 tend to give superior performance. Of the four geometries in Fig. 11, the core with 6 = 0.02 and 7 = 0.038 appears to perform the best, particularly for flows with Re > 1000.

Two additional issues that are of considerable significance for the design of offset strip fin heat exchangers are the "whistling" noise emitted from the cores within a certain range of operation and the staggering of fin offsets. Mullisen and Loehrke [39] reported a direct correlation between the onset of oscillating flows and the genera- tion of "audible tones." Other studies have found that the development of oscillating flows and the consequent vortex shedding are influenced by fin separation (or length) and fin offset [13, 15, 16]. One could speculate that the core acts like a "flute" for certain fin arrangements and flow rates, and a tentative solution appears to be the rearrangement of the fin offset. In the recent study by Kurosaki et al. [40], it was shown by laser holographic interferometric measurements that nonuniform staggering influences the heat transfer performance. A suggested arrangement to improve heat transfer rates envisages a

0 . 3 5

0 . 3 0

0 . 2 5

0.20

0.15 o 0 ~ o a = 0 . 1 3 5 , 6 = 0 . 0 3 6 , 7 - 0 . 1 0 0 . a a a o a = 0 . 1 3 4 , 6 = 0 . 0 4 0 , 7 = 0 .121 ~ A ~ a - 1 . 0 2 4 , 6 = 0 . 0 2 0 , 7 - 0 . 0 3 8 A A A A A a = 1 .034 , 6 = 0 . 0 6 0 , 7 = 0 . 1 3 2

i i i i ~ J 0.1C~0L ~ . . . . . i000

Re

Figure 11. Influence of geometrical features of offset strip fins on their flow area goodness factor.


large enough offset to put successive fins out of the wakes of their immedia te ups t ream neighbors. This appears to have considerable promise, and more research is needed to make any definite design recommendat ions .

C O N C L U S I O N S

The heat transfer and friction factor da ta for 18 offset strip fin surfaces have been analyzed and are shown to be affected by the fin geometr ic parameters a , 6, and 3' along with the Reynolds number. Equat ions that describe the asymptotic behavior of the data in the deep laminar and fully turbulent regions have been developed. These asymptotes have been combined to form single predictive correlat ions for f and j , as given by Eqs. (34) and (35), respectively. These equations represent the da ta continuously in the laminar, transition, and turbulent flow regions. However, to extend the validity of these correlations and to lend support for the methodology adopted in the present study, more da ta should be included, part icu- larly for liquids. Also, the issues of staggering, or rearrang- ing, the fin offsets and the emission of whistling sounds at certain operat ing condit ions have been briefly discussed. Both influence to some extent the enhancement of the thermal-hydraul ic per formance of offset strip fin compact heat exchangers. In the absence of quanti tat ive data, no definite design recommendat ions can be made, and more investigation is warranted.

NOMENCLATURE

A heat transfer area of offset strip fin surface, m 2 Z c free-flow area, m 2 D h hydraulic diameter , m

f average Fanning friction factor in the offset strip fin array, dimensionless

G mass velocity ( = r h / A c = r h / N s h ) , kg/(mZs) h height of the offset strip fin channel, m j Colburn factor [ = St Pr 2/3 or N u / ( R e prl /3)] ,

dimensionless l length of fin, m

rh mass flow rate, k g / s N number of fin channels, dimensionless

Nu average overall Nusselt number based on hydraulic diameter , dimensionless

P wet ted per imeter , m Pr Prandtl number, dimensionless

Re Reynolds number based on hydraulic diameter , dimensionless

Re* transit ion point Reynolds number, Eq. (14), dimensionless

Re t Reynolds number based on fin length, dimensionless St Stanton number based on hydraulic diameter ,

dimensionless s lateral fin spacing, m t fin thickness, m

Greek Symbols a aspect rat io s / h , dimensionless 3' rat io t / s , dimensionless 6 rat io t / l , dimensionless s c Darcy friction factor, dimensionless

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Received December 21, 1993; revised May 16, 1994

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Heat transfer and pressure drop correlations for the rectangular offset strip fin compact heat exchanger