Abstract

The development of rationalized correlations for Fanning friction factor f and Colburn j factor for forced air-flow convection in rectangular offset-strip-fin cores is presented in this study with 100 ≤ Re ≤ 8000. New experimental data and three-dimensional conjugate heat transfer computational simulations were acquired to understand the flow physics and heat transfer phenomena. The offset arrangement of the fins disrupts the fully-developed condition prevalent in plain fins to promote secondary flow and enhanced heat transfer, and this effect is found to be fundamentally scaled by offset length ratio λ(=/dh). Furthermore, because of the blunt surface edge or finite thickness of the offset fin, the flow stagnation and wake effects are integral parts to the secondary flow. The influence is found to be characterized by the thickness ratio ζ(=tRe/dh) as well as the rectangular flow cross section aspect ratio α(=s/hf). New models are proposed for f and j in both laminar and turbulent regimes based on the enhanced convection effects, which are represented by these scaling parameters, and are superimposed on the fully-developed condition in a rectangular channel. The correlations are thereby devised from the new sets of experimental data as well as that given in the open literature and thus cover a wide range of λ, ζ, and α. Because the transition from laminar to turbulent regimes is smooth and continuous, the general correlations of f and j are developed by asymptotic matching as single expressions and are shown to predict the extended dataset to within ±20%.

1 Introduction

Compact and enhanced plate-fin heat exchangers are adopted in a broad range of industrial applications. Besides effecting high surface area density, geometrically modified extended or finned surfaces significantly improve the thermal effectiveness [1], especially for air and gas flows that tend to have high convective thermal resistance and dominate the overall performance of heat exchangers [24]. While there are several different examples of augmented plate fins, e.g., perforated fins, wavy or ruffled fins, slotted fins, and more [1,3,512], the offset-strip fin is of particular interest in many different air/gas flow applications [5,1316]. Compared to the plain rectangular plate-fin, the heat transfer coefficient can be increased substantially due to periodic boundary layer disruption and early triggers of transitional flow. The convective enhancement is primarily engendered by the different geometric attributes of offset-strip fins, and, as depicted schematically along with a photographic view in Fig. 1, these are described by the fin height hf, interfin spacing s, fin thickness t, and offset or lanced length . Because of this performance advantage, offset-strip fins have been the focus of both experimentations to acquire data and devise predictive correlations as well as computational simulation for exploring the convective flow field in numerous studies over the past eight decades [1422].

Fig. 1
Schematic and photographic depiction of a typical offset-strip fin core with markers for geometric descriptors
Fig. 1
Schematic and photographic depiction of a typical offset-strip fin core with markers for geometric descriptors
Close modal

As delineated in a detailed prior review [14], one of the earliest attempts to provide predictive correlations for friction factors and heat transfer coefficients in gas or air flows through offset strip-fin (OSF) cores were reported in 1950 by Manson [23]. A two-equation model was devised, one for Re ≤ 3500 and the other for Re > 3500, wherein, besides Re, each regime has an additional two-equation model with one for (/dh)3.5 and another for (/dh)>3.5 to describe the performance variations. Since then, considerable effort has been directed into developing correlations for f and j, and they reflect the evolving understanding of the enhanced convective behavior as updated with new observations and data [15,16,1822,24]. Wieting [15] developed correlations that are power-law fits through the data for 21 different OSF geometries compiled in Refs. [2,2527]. Separate expressions for the laminar and turbulent regimes are given, and besides Re and (/dh) used in Ref. [23], the fin-channel cross section aspect ratio (s/hf), and dimensionless fin thickness (t/dh) were included to describe the parametric variation in convection. While (s/hf) was suggested to influence laminar flow (Re ≤ 1000) but not turbulent flow, (t/dh) was a significant parameter only in the latter case (Re ≥ 1000).

Subsequently, Joshi and Webb [16] readdressed this work [15] by adhering to the same scaling parameters, and devised another set of two-equation model correlations for the same 21 OSF cores dataset [2,14]. Their two expressions for f and j are applicable for Re ≤ Re* and Re ≥ (Re* + 1000), thereby leaving a significant flowrate gap. The demarcating Re*, a function of Re, t, s, , and dh, is based on parametric visualization experimental data analysis. It marks the onset of what was termed as flow oscillation that approximately corresponds with the log-linear departure in the f and j curves of the laminar regime. Furthermore, working with experimental data from 5 scaled-up (>10× or larger and with 1.6 mm-thick “fins” that are not viable in practice) OSF cores, Mochizuki et al. [24] reworked the Wieting [15] two-equation model with new f and j correlations for Re < 2000 and Re ≥ 2000 flows. There were a few other similar attempts as been summarized and commented upon elsewhere [14,18]. Besides differences in scaling with differing treatment of OSF geometry, there is no consensus in the demarcation of flow regime transition as well as in the definition for dh [14,18].

Perhaps the most widely used and recommended [6,28,29] correlations for f and j in OSF cores are the ones devised by Manglik and Bergles [18]. By reevaluating the available data for 18 different offset-strip fin cores [2,25,26], the scaling of enhanced convection was shown to be described by the cross section aspect ratio α(s/hf), thickness to interfin spacing ratio γ(t/s), and thickness to offset length ratio δ(t/). The air flow and convective heat transfer behaviors in laminar and turbulent regimes were decomposed and demarcated using the Re* introduced by Joshi and Webb [16], and regime-restricted power-law expressions were devised. These were then combined by asymptotic matching [18,30] as single set of equations, respectively, for f and j as follows:
f=9.624Re0.7422×δ0.3053α0.1856γ0.2659[1+7.669×108Re4.429α0.92δ3.767γ0.236]0.1
(1a)
j=0.6522Re0.5403×δ0.1499α0.1541γ0.0678[1+5.269×105Re1.34α0.504δ0.456γ1.055]0.1
(1b)

A subsequent study by Dong et al. [19] added another power-law scaling factor (Lf/), or the ratio of total core flow length to offset length, so as to account for the developing flow effect. While this modification to Eqs. (1a) and (1b) predicted their own experimental data to within ±10%, the applicable range and influence of entrance effect have not been specified. Moreover, this is an unusual parameter as this would perhaps reflect on whether periodicity is established or not, and flow entrance effects would tend to be a function of offset length and channel hydraulic diameter dh.

A few computational studies have also explored the convective behavior in offset-strip-fin core channels [2022,31,32]. Either a 2D approach is taken [31,32], or instead of extracting local convective flow behavior and enhancement mechanisms, they have focused on developing correlations based on simulated results and are thus not instructive. For instance, one study [20] obtained results using standard k-ε model and devised correlations with identical scaling as in Manglik and Bergles [18]. It also suggests that the recirculation zone behind and in the wake of the thick fins considered are significant in the turbulent regime. This restates what was already reported in a much earlier and different computational study [31] with no new insights. Kim et al. [21] indicate that their CFD results for f and j are underestimated by the correlations in [18] when the blockage ratio, or t/(s+t), is higher than 20%. Much of this work [2022,31] borders on impracticality as in most applications, even with very high fin densities, the so-called frontal blockage due to fin thickness is no more than ∼ 10% [33].

Given the lack of consensus and variance in the existing correlations, including the definition for dh, it is instructive to compare their predictions against sample experimental data. Because the importance of offset length has been flagged in the newer modification [19] of the prevailing widely-adopted correlation [6,18,28] as well as in some earlier cases [15,16,24], comparisons of f and j predictions, with 100 ≤ Re ≤ 8000, for an OSF core with λ=(/dh)=0.33 and (Lf/)19.2 are presented in Fig. 2. Considerable disagreements between predictions from various correlations are clearly evident. The only broad consensus in both laminar and turbulent regimes is between those given by Wieting [15] and Manglik and Bergles [18]. The latter tends to have the best agreement, as shown in Ref. [18], with the older dataset in Kays and London [2]. To clarify these observed discrepancies and the parametric assertions in some of the newer studies, albiet not substantiated with data from different sources, warrants a reexamination of the enhancement mechanisms and their scaling in forced convection of gas flows inside OSF cores is necessary. This has been rigorously and carefully addressed in the present study, where new experimental data for a judiciously selected set of OSF cores with broad geometric variations are presented. A wide range of flows spanning the laminar-to-turbulent regimes found in most applications [33] are considered. The consequent parametric influences of the different OSF geometric features are further supported with computational simulations, which provide the needed insights for appropriately resolving the scaling of the flow and convective behavior. Based on these results, a new refined set of rationalized correlations are presented for predicting f and j in OSF cores for design of compact heat exchangers for gas-flow applications.

Fig. 2
Comparison of predictions from different available correlations for forced convection in an offset-strip fin core geometry with isothermal partition walls and α = 0.32, λ = 0.33, and ξ = 0.038: (a) Fanning friction factor f and (b) Colburn j factor
Fig. 2
Comparison of predictions from different available correlations for forced convection in an offset-strip fin core geometry with isothermal partition walls and α = 0.32, λ = 0.33, and ξ = 0.038: (a) Fanning friction factor f and (b) Colburn j factor
Close modal

2 Experimentation and Analysis

To characterize the forced convection behavior in OSF cores and the associated thermal-hydrodynamic performance, both experimental measurements and computational simulations were carried out. Air (Pr ∼ 0.71) flows have been considered with flow rates that span laminar to turbulent regimes with 100 ≤ Re ≤ 8,000. Six different offset-strip-fin cores, all fabricated with Al 3033 (kAl = 190 W/m·K), were tested, and their different dimensionless features, as described in previous studies as well as those considered in the present work, are listed in Table 1. It may be noted from this listing that the blockage ratio (ξ=t/dh) is < 0.1; large fin thickness with high blockage ratios yield impractical low fin density and compactness, which negates the use of high-effectiveness plate-fin heat exchangers [6,28,33].

Table 1

Listing of of the offset-strip-fin cores considered in this study and their respective dimensions as well as dimensionless representations

N (fpi)s (mm)hf (mm)t (mm) (mm)α(s/hf)δ(t/)γ(t/s)dh (mm)λ(/dh)ξ(t/dh)
OSF-1121.969.370.1523.180.210.0480.0783.250.980.047
OSF-2121.899.300.2296.350.200.0360.1213.142.020.073
OSF-3121.8618.800.2541.980.100.1280.1363.390.580.075
OSF-482.9512.470.2291.980.240.1150.0784.770.420.048
OSF-582.9218.800.2543.180.160.0800.0875.060.630.050
OSF-664.0012.470.2291.980.320.1150.0576.060.330.038
N (fpi)s (mm)hf (mm)t (mm) (mm)α(s/hf)δ(t/)γ(t/s)dh (mm)λ(/dh)ξ(t/dh)
OSF-1121.969.370.1523.180.210.0480.0783.250.980.047
OSF-2121.899.300.2296.350.200.0360.1213.142.020.073
OSF-3121.8618.800.2541.980.100.1280.1363.390.580.075
OSF-482.9512.470.2291.980.240.1150.0784.770.420.048
OSF-582.9218.800.2543.180.160.0800.0875.060.630.050
OSF-664.0012.470.2291.980.320.1150.0576.060.330.038

2.1 Experimental Setup and Measurements.

A low-pressure wind tunnel, schematically depicted in Fig. 3, was employed, where a continuous and steady supply of filtered air is provided to the test section. An intake end honeycomb flow straightener ensures uniform and freestream turbulence-free steady air flow rates, which are recorded via a mass and/or volumetric gas-flowmeter (depending upon the flowrate range). Precision (±0.5 °C) precalibrated T-type thermocouples, located at four different points in the flow channel at the inlet of the test section and at 10 different points at the outlet, provide the respective Ti and To temperature measurements. The pressure drop across the test section is measured by precision differential pressure transducers and/or manometer from inlet-outlet pressure taps located very close to the test section.

Fig. 3
Schematic description of the experimental setup and apparatus for measuring the plate-fin thermal-hydrodynamics performance
Fig. 3
Schematic description of the experimental setup and apparatus for measuring the plate-fin thermal-hydrodynamics performance
Close modal

An electrically heated test section that can accommodate different fin heights and two different flow lengths (76.2 mm and 38.1 mm) is fitted with multiple T-type thermocouples placed inside predrilled holes in the encasing copper plates and extending close to the fin surface. Flexible surface heaters, affixed over thick (15.88 mm) copper plates that enclose the fin core, provide the heating, and their power is controlled by a variable transformer. The fin cores, fabricated with Al 3033 (kAl = 190 W/m·K) sheet metal, are sandwiched between the heated copper plates (kCu = 391 W/m·K) with a thin layer (∼ 50 μm) of thermal paste (kth = 2.31 W/m·K) so as to ensure good thermal contact with heated surface. The thick copper plates proved effective heat spreading in order to maintain a uniform surface temperature (≤2 °C) at the top and bottom surfaces of the fin cores. The entire test section is well insulated (thick fiberglass insulation) so as to minimize heat loss. All measurements were recorded and stored in real-time with a computerized data acquisition system, and the process, as well as more details of the apparatus, are the same as for the ongoing fin-performance research reported in Refs. [3436].

To determine the fin-core heat transfer coefficient, the heat transfer rate q was measured by the enthalpy change of the air flowrate through it, after carefully controlling the steady-state energy balance, with < 10% difference between the convective heat gain by the fluid flow and the electrical heat input, which is given by
[q=m˙cp(ToTi)][qe=(VI)e]
(2)
The heat transfer coefficient h, for uniform wall temperature conditions and based on the log-mean temperature difference ΔTlm [37], is thus obtained as
h=qAΔTlm;ΔTlm=(TwTo)(TwTi)ln[(TwTo)/(TwTi)]
(3)
The average wall or fin-base temperature was determined from the multiple wall-embedded thermocouple measurements as follows:
Tw=1n(T1+T2+...+Tn)q/2Ab(tcukcu+tthkth)
(4)
The corrected heat transfer coefficient ho, after accounting for the fin effects as per [38], was then obtained iterativelyfrom
ho=hηo;ηo=1AfA(1ηf)
(5a)
ηf=tanh(mho/2)(mho/2);m=2h0kAlt
(5b)
The Nusselt number and/or the Colburn factor are thus calculated as
Nu=(hodh/k);andj=(Nu/RePr1/3)
(6)
For evaluating the pressure drop penalty to flow through the enhanced fin core, the pressure transducer measurement (Δpme=pipo) are corrected to account for entrance and exit losses, and the actual core pressure drop obtained as
Δp=Δpme1ρ(Kc+Ke)ρu22
(7)
The entrance and exit loss coefficients, Kc and Ke, are obtained as outlined in Refs. [2,39]. Thus, the hydraulic-diameter-base Fanning friction factor is determined by its definition as
f=(ΔpL)(dh2ρu2)
(8)

All primitive measurements (flowrate, pressure drop, temperature, and electrical voltage and current) were made with high-precision instruments. The uncertainties in the derived variables Re, f, and j, were determined using a single-sample error-propagation analysis [40], and these were within ±3%, ±9%, and ±7%, respectively. Moreover, the apparatus, experimental measurements, and data reduction method were vetted with multiple measurements for air flows in plain rectangular cores and f and j data were in excellent agreement with established results as presented elsewhere [34,41].

2.2 Computational Methods.

Three-dimensional conjugate steady-state simulations for forced convection heat transfer with air (Pr ∼ 0.71) flows in OSF channels were carried over the computational domain depicted in Fig. 4. The latter spans two periods of the offset-strip fins with periodicity conditions at the inlet and outlet (streamwise direction), as addressed by Patankar et al. [42], and symmetry conditions at the two side surfaces. Also, with top and bottom partition plates at constant and uniform surface temperature and no-slip conditions, three-dimensional conduction through the solid fins was included in the simulations so as to account for the appropriate fin effects.

Fig. 4
Simulation model for the offset-strip fin core: (a) computational domain and (b) mesh or grid structure with nonuniform near-wall and inlet-outlet refinement
Fig. 4
Simulation model for the offset-strip fin core: (a) computational domain and (b) mesh or grid structure with nonuniform near-wall and inlet-outlet refinement
Close modal
The simulated flow range (100 ≤ Re ≤ 8000) covers laminar, transition, and turbulent regimes, and hence both laminar and turbulent models were appropirate. Thus, the general incompressible constant-property steady-state governing equations are as follows:
V=0
(9a)
ρ(V)V=p+μ2Vρvv¯
(9b)
ρ(V)T=kaircp2TρvT¯
(9c)
Note that v=(V+v) is the velocity vector where V is the mean velocity vector and v the fluctuating part for turbulent flows; likewise pressure is decomposed as p+p, and temperature as T+T. These fluctuating parts, or last terms in Eqs. (9b)  and (9c), vanish for laminar flows. The Reynolds-averaged Navier–Stokes (RANS) model with two additional transport equations, or the RNG k-ε model [43] was implemented for turbulent flows. With its additional strain-dependent correction term, it was found to resolve swirl or rapidly strained flows better (as compared to an alternative SST k-ω model [41,44]). The two additional equations required for k and ε are as follows:
xj(ρuik)=xj[αkμeff(kxj)]+τijSijρε
(10a)
xj(ρuiε)=xj[αεμeff(εxj)]+C1ετijSijC2ε*ρε2k
(10b)
where the effective viscosity is expressed as
μeff=μ+μt;μt=ρCμ(k2/ε)
(10c)
and the strain-dependent constant C1ε* is given by
C2ε*=C2ε+η(1η/η0)1+βη3;η=kε2SijSij
(10d)

The other analytically derived RNG constants are: Cμ=0.0845, αk=αε=1.39, C1ε=1.42, C2ε=1.68, η0=4.377, and β=0.012.

To ensure mesh quality with lower number of grid points, hexahedron cells were adopted for both fluid and solid domains (Fig. 4(b)). The air-side grids close to the wall were further refined to capture effectively the high wall-gradients, and the enhanced wall treatment (EWT) option was employed more precise resolution of gradients in the viscous sublayer; the farthest distance of the center of the first grid set adjacent to the wall was kept at y+=(y/v)τw/ρ1. Second-order upwind scheme was utilized to discretize all differential equations, and pressure–velocity coupling was solved by the coupled algorithm. Convergence for the velocity field, turbulent kinetic energy, and turbulent dissipation rate was set with relative residuals < 10−6 in the local field values of each variable, and 10−8 for the local temperature field. Furthermore, relative grid independence was established with a mesh count of ∼1,000,000 based successive mesh refinements, and when f and j results yielded less than 1% difference, in both laminar and turbulent regimes, with a finer grid.

3 Results and Discussions

The experimental f and j data for a typical fin core (OSF-1; Table 1) are presented in Fig. 5, along with the computationally simulated results as well as the predictions of Manglik and Bergles [18] correlations. That the former agrees rather well with the data, especially in the laminar regime is selfevident. The disrupted flow behavior engendered in OSF cores, triggered by periodic boundary layer inter-ruption and high stagnation pressure in flow channels, is seen to effect early transition (at approximately Re ∼ 800) compared to plain plate-fin. This is indicated by the change in slopes of the f and j curves, and the typical channel-flow discontinuity in the transition region is not manifest in the experimental data. The correlations [18] are seen to predict f data very well but considerably underpredict the j data in the laminar regime. Similar trends, with substantial deviations between predictions and data across regimes in different cases, were also observed in comparisons with most other OSF core data [41]; in some cases, there are considerable differences in both f and j. This reiterates and underscores the need for revisiting the development of a more generalized set of correlations for OSF cores.

Fig. 5
Comparison of experimental data for Fanning friction factor f and Colburn j factor with predictions from computational simulations and correlations for OSF-1 (Table 1)
Fig. 5
Comparison of experimental data for Fanning friction factor f and Colburn j factor with predictions from computational simulations and correlations for OSF-1 (Table 1)
Close modal

3.1 Offset-Fin Geometry Effects on Flow Convection.

Local flow streamlines in the offset-strip-fin core at midplane along the fin height (at 0.5hf) over the offset length graphed in Fig. 6 illustrate typical boundary layer disruption and its effects. The flow stagnation of the inflow at the leading blunt edge of a fin causes flow displacement around the fin corners, which pushes the adjoining flow stream into the unfinned core. There is a consequent thinning of the boundary layer at the trailing edge of the adjacent offset fin. This is manifest in the uptick or relative increase in the local friction factor along the fin surface when (x/)> 0.8 (see Fig. S1(a) available in the Supplemental Materials on the ASME Digital Collection). As the flow becomes turbulent, the higher stagnation pressure at the blunt edge causes even larger flow displacement thereby inducing flow separation at the leading edge and a recirculation zone. The latter flow reversal yields negative local friction loss results in the (x/)< 0.1 part of the fin surface (Fig. S1(b) available in the Supplemental Materials on the ASME Digital Collection). In both laminar and turbulent flows, the sudden change in fin geometry and boundary-layer interruption produces a wake region at the rear-edge of the fin. The subsequent flow recovery downstream is over a shorter length in the turbulent regime, due to higher momentum exchange in the transverse direction. These results suggest that besides the wall shear, additional form drag contribution could be significant. The concomitant influence on heat transfer, or the local variations in Nux along (x/), is not much different from that in (fxRe) in the laminar regime (Fig. S2(a) available in the Supplemental Materials). The leading edge recirculation zone in turbulent flow, however, remains favorable to heat transfer due to the inherent and induced chaotic mixing (Fig. S2(b) available in the Supplemental Materials).

Fig. 6
Characteristic streamline distributions in the flow through an offset-strip-fin core (OSF-1; Table 1): (a) in the laminar regime at Re = 600 and (b) in the turbulent regime at Re = 4000
Fig. 6
Characteristic streamline distributions in the flow through an offset-strip-fin core (OSF-1; Table 1): (a) in the laminar regime at Re = 600 and (b) in the turbulent regime at Re = 4000
Close modal

Very distinct from the flow-loss that is effected in plain rectangular plate-fin cores, the offset-fin arrangement, with its interrupted and redeveloping the boundary layer, not only increases the shear drag but also induces additional form drag. This is primarily due to the blunt edges of the finite thickness and the consequent flow blockage. It is thus instructive to delineate the specific contributions of wall shear fshear and form drag fform to the total flow friction loss, orf=(fshear+fform). It is clear from the computed results graphed in Fig. 7(a) for a typical offset-fin-core geometry (OSF-1; Table 1) that the wall shear stress is dominant in the laminar flow (Re < 600 – 800) regime. It constitutes more than 75% of the total flow penalty, with over 80% in the lower flow rates (Re ∼ 100). In contrast, as seen in Fig. 7(b), form drag contribution is substantially larger in the turbulent regime and is the dominant pressure-loss force when Re > 2000. This goes from ∼60% to >80% with increasing flowrate, whereas the influence of the wall shear stress declines commensurately. This essentially points to the fin-thickness effects and a different scaling argument, as enunciated subsequently, for correlating the pressure loss in offset-strip-fin cores.

Fig. 7
Relative contributions of shear stress and form drag to the total flow friction loss in an offset-strip-fin core: (a) laminar flow regime and (b) turbulent flow regime
Fig. 7
Relative contributions of shear stress and form drag to the total flow friction loss in an offset-strip-fin core: (a) laminar flow regime and (b) turbulent flow regime
Close modal
Contrasting with the flow behavior in plain rectangular plate-fin channels, the transverse direction pressure distribution in the offset-strip-fin channel is not uniform. This can primarily be attributed to the blunt-edge stagnation and wake regions engendered by the fin thickness. This produces form drag, which can be described by the form friction factor fp based on the periodic pressure in the flow channel as
(fpRe)=(2p*/ρu2)Re
(11)
The secondary flow induced by the offset-strip fin of finite thickness results in a nonuniform pressure distribution in the flow cross section, unlike that in fully developed flow in a rectangular channel. This effect has previously been correlated by a blockage ratio ξ, defined as (=t/dh) [15,16,24,45], and is found to be less effective in normalizing the pressure distribution or form friction factor [41]. However, drawing from the case of cross-flow in staggered-tube banks [37] and to capture the wake and drag effects of blunt or bluff surfaces, a more appropriate scaling parameter can be defined as
ζ=(ρut/μ)=(t/dh)Re
(12)

The computed results for the transverse distribution of (fpRe)with constant ζ in both laminar and turbulent flow regimes, as seen in Figs. S3(a) and S3(b) available in the Supplemental Materials, respectively, bear out this scaling. The variations or profiles are more-or-less same, especially in the turbulent regime where form drag is more dominant.

Another important fin-geometry attribute is the offset-strip length , which can be scaled with the hydraulic diameter dh and represented in dimensionless form as λ=(/dh). As deduced in some previous computational analyses [32,46], with increasing fin-offset length the velocity and temperature distributions tend to approach that in a plain rectangular uninterrupted plate-fin channel. This is illustrated in Fig. 8 with the computed results for dimensionless velocity u* and temperature T*, which are respectively defined as follows:
u*=(u/ub),andT*=[(TwT)/(TwTb)]
(13)
Fig. 8
Fluid flow and thermal fields at midway locations along the fin height and offset-fin length (at hf/2 and ℓ/2) in an offset-strip-fin core and their variations with altered offset-fin length or λ(=ℓ/dh)in laminar (Re = 600) and turbulent (Re = 4000) regimes: (a) dimensionless velocity u*, and (b) dimensionless temperature T*
Fig. 8
Fluid flow and thermal fields at midway locations along the fin height and offset-fin length (at hf/2 and ℓ/2) in an offset-strip-fin core and their variations with altered offset-fin length or λ(=ℓ/dh)in laminar (Re = 600) and turbulent (Re = 4000) regimes: (a) dimensionless velocity u*, and (b) dimensionless temperature T*
Close modal

Two different offset-fin lengths (λ = 0.49 and 3.91) in otherwise the same core are considered, and u* and T* profiles at midfin-height (z=0.5hf) and halfway along the offset-fin flow length are graphed. With shorter λ ( = 0.49) and more frequent boundary layer disruption, a significant dip or cleavage at the centerline is seen in the velocity and temperature profiles along with sharper wall gradients in both laminar (Re = 600) and turbulent (Re = 4000) regimes. In the latter case, reflecting a shorter flow recovery length (see Fig. 6), the magnitude of this dip is substantially smaller. As the offset-fin length increases, (λ = 3.91), the centerline cleavage reduces in magnitude and has a gentler and relatively more uniform profile in both regimes. The flow recovery and augmented fluid momentum exchange with the growth of boundary layers over a longer fin surface effectively renders this profile redistribution, which tends to be similar to that in a plain uninterrupted (orλ) rectangular plate-fin channel.

Finally, the role of the plate-fin core flow cross section aspect ration, or α(=s/hf) in scaling the forced convection behavior can be rationalized from the pressure loss. This is evident from the variation in (fRe), and its two components (fRe)shear and (fRe)form, with α given in Fig. 9. In laminar fully-developed flow inside plain rectangular plate-fin cores, (fRe) and Nu are a function of flow cross section aspect ratio α [34,47]. They are respectively correlated and predicted by the following polynomial expressions [47]
(fRe)fd,lam=24.0(11.355α+1.947α21.701α3+0.956α40.254α5)
(14)
Nufd,lam=7.541(12.61α+4.97α25.119α3+2.702α40.548α5)
(15)
Fig. 9
Contributions of viscous drag and form drag in flows through offset-strip-fin cores with different cross section aspect ratio α: (a) in laminar regime and (b) turbulent regime
Fig. 9
Contributions of viscous drag and form drag in flows through offset-strip-fin cores with different cross section aspect ratio α: (a) in laminar regime and (b) turbulent regime
Close modal
For fully developed turbulent flow in straight rectangular channels these are respectively predicted by the following correlations [34,37,41], which are not explicit functions of α but its effect is implied in the hydraulic diameter length scale
(fRe)fd,trurb=0.1268Re0.7
(16)
Nufd,turb=0.023Re4/5Pr1/3
(17)

The significant variation of (fRe)form in both regimes and the much more dominant contribution in turbulent flow is evident from Fig. 9. This stems from the fin thickness, or the interfin spacing ratio (t/s); when dh remains the same and as α increases, (t/s) decreases and so does the form drag.

3.2 General Correlations for f and j.

That the secondary flow effects, and the periodic boundary-layer disruption, induced by the offset-fin arrangement are perhaps the primary enhancement mechanism that can be scaled with the offset length ratio λ(=/dh)is amply supported in the foregoing discussion. The flow recirculation is essentially induced by the high stagnation pressure at the blunt edge of the fin wall. The fin thickness, which can be scaled by ζ(=tRe/dh) as in Eq. (12), also contributes to the form drag addition to the otherwise viscous shear pressure loss. The latter is further induced by the relative blockage due to the flow cross section aspect ratio α(=s/hf). To express these scaling predictors in a correlation for predicting (fRe), and Nu in offset-strip-fin cores, the following new models are proposed
fRe=(fRe)fd+A(λ)a1(ζ)a2(α)a3
(18)
Nu=Nufd+A(λ)b1(ζ)b2(α)b3
(19)

In the first term on the RHS of Eqs. (18) and (19), the expressions for (fRe)fd and Nufd are appropriately given by Eqs. (14)(17) for fully developed convection in plain (uninterrupted) rectangular plate-fin cores. This effectively represents the limiting condition where the fin offset length is very large or λ → ∞; correspondingly, λa1 and λb1 → 0. The second terms on the RHS represent the respective effects of the induced secondary flows.

To develop the predictive equations and ascertain the model parameters for Eqs. (18) and (19), experimental performance data for 15 sets of offset-strip-fin cores were considered. They included the nine sets listed in Kays and London [2] along with the six sets of geometries experimentally evaluated in this study. The friction factor results for all of these fin cores are presented in the composite graph of Fig. 10. The demarcation of laminar and turbulent regimes, effected by the respective change in the log-linear slope of f-curves, are indicated in this plot. It is evident that transition Re for each of the two regimes is a function of the offset-strip-fin geometry. These two transitional demarcations were determined to be correlated λ, ξ, and α as per the following respective power-law expressions
Relam*=560.2λ0.08ξ0.13α0.08
(20)
Returb*=1823.1λ0.18ξ0.13α0.13
(21)

where Relam* and Eq. (20) marks the transition from laminar flow, and Returb* and Eq. (21) marks the onset of fully developed turbulent flow; the intervening region is a continuous transition between the two limits.

Fig. 10
Experimental data for friction factor f and its variation with Re in the offset-strip-fin cores of the present study as well as those given in Ref. [2]
Fig. 10
Experimental data for friction factor f and its variation with Re in the offset-strip-fin cores of the present study as well as those given in Ref. [2]
Close modal

Based on the identification of the flow regimes in Fig. 10 and their limits given by Eqs. (20) and (21), the asymptotic equations to represent the dataset for the laminar and turbulent regimes were determined. The consequent expression for (f Re) and Nu, obtained from a multivariable regression analysis, are as follows:

Laminar regime (ReRelam*)
(fRe)lam=(fRe)fd,lam+5.73λ0.65ζ0.38α0.39
(22)
Nulam=Nufd,lam+1.21λ0.63ζ0.49α0.02
(23)
Turbulent regime (ReReturb*)
(fRe)turb=(fRe)fd,turb+0.75λ0.74ζ0.86α0.43
(24)
Nuturb=Nufd,turb+1.55λ0.56ζ0.45α0.01
(25)
The correlations in Eqs. (22) and (23) for laminar flows and Eqs. (24) and (25) for turbulent flows exclude the intervening transition region. However, the absence of any discontinuity or jump in the transition regime (as is commonly encountered in duct flows) in the experimental data in Fig. 10 implies that the data can be described by a single continuous equation (Fig. 11). To achieve this, the asymptotic matching technique proposed by Churchill and Usagi [30] was employed, wherein the laminar and turbulent correlations provide the respective flow-regime asymptotes. Moreover, in order to adhere to the more common practice in heat exchanger design applications and its literature [2,3,6,47], besides expressing friction loss as f instead of (fRe), the heat transfer coefficient is expressed in terms of j(=Nu/RePr1/3) instead of Nu. The general equations so obtained are as follows:
f=(flam15+fturb15)1/15
(26a)
Fig. 11
Asymptotic match of laminar and turbulent region correlations for f and j in offset-strip-fin cores for developing a single continuous expression in each case
Fig. 11
Asymptotic match of laminar and turbulent region correlations for f and j in offset-strip-fin cores for developing a single continuous expression in each case
Close modal
and
j=(jlam15+jturb15)1/15
(26b)

where the exponents were determined by the least-square method in the asymptotic matching process. As seen from Fig. 12, there is very good agreement between the experimental data of f and j with the predictions from their respective correlation expressed in the form of Eq. (26). More than 96% of all the experimental data for the 15 different core geometries in each case are within ±20% of the predicted values. Some of the uncertainties, it may be noted, stem from manufacturing irregularities, such as burr edges and surface roughness; their effects, however, tend to be more in turbulent flows. Nevertheless, Eqs. (26a) and (26b), and their respective components in Eqs. (22)(24) and (23)(25), provide much more generalized and significantly better correlations for the design of compact heat exchangers with offset-strip-fin cores. Their formulations based on a clearer understanding of the enhancement mechanisms and their appropriate phenomenological scaling ensure better predictive accuracy.

Fig. 12
Comparison of predictions of the respective correlations with the experimental data sets: (a) Fanning friction factor f, Eq. (26a) and (b) Colburn j factor, Eq.(26b)
Fig. 12
Comparison of predictions of the respective correlations with the experimental data sets: (a) Fanning friction factor f, Eq. (26a) and (b) Colburn j factor, Eq.(26b)
Close modal

4 Conclusions

The thermal-hydrodynamic performances of offset-strip fins with air (Pr ≈ 0.7) flows have been experimentally and computationally investigated. The flowrate range covers both laminar and turbulent regimes (100 ≤ Re ≤ 8000) so as to revisit the mechanistic and phenomenological enhancement behavior induced by the fin-core geometry. This was necessitated by the relative inadequacy of previously reported correlations [15,16,18,24], most notably in predicting the performance of configurations with low λ (or longer offset-fin length ). The influence of the offset length ratio λ(/dh)cannot be understated for its dominant effect on the disruption and redevelopment of hydraulic and thermal boundary layers in the core flow. With very large λ the flow behavior tends to that in a plain uninterrupted rectangular plate-fin core; smaller λ and the offset-fin arrangement induces secondary flows. Furthermore, these effects are compounded by the wake and drag effects attributed to the blunt edges of the fin, and they are shown to be effectively scaled by ζ(tRe/dh). Relative thickness of the fin edge augments from drag and wake-induced receirculation. Finally, the flow blockage because of the interfin cross section aspect ratio α adds to the secondary flow effects besides the channel-shape effects in the laminar regime.

The flow-mechanism effects of the offset-strip-fin geometry, described scaled by λ, ζ, and α, were correlated by a new form of predictive model proposed in this study for determining their thermal-hydrodynamic performance. In doing so, fifteen sets of experimental data, including those for the six (6) new cores experimentally evaluated in this study and the nine (9) other cores in Ref. [2] were considered. The variation in their f and j with Re on a logarithmic scale revealed the demarcations for the laminar, transition, and turbulent regimes. The corresponding critical of transitional Re* are also seen to be functions of the fin-core geometrical features λ, ζ, and α. Based on the log-linear asymptotic behavior of the frictional loss and heat transfer coefficient in the laminar and turbulent regimes, separate respective predictive equations were devised based on the new proposed correlating model. They were combined by asymptotic matching to develop general correlations, Eqs. (26a) and (26b)f and j, respectively, which cover all the flow regimes and over 96% of the data to within ±20% and provide a valuable design tool for compact heat exchanger applications.

Acknowledgment

This study was carried out with support from the ARID program of ARPA-e (# DOE DE-AR0000577). The technical support from Robinson Fin Machines, Inc., as well as for providing the needed offset-strip plate-fin coupons are also gratefully acknowledged.

Funding Data

  • Advanced Research Projects Agency (Funder ID: 10.13039/100009224).

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

Nomenclature

A =

heat transfer surface area (m)

cp =

specific heat at constant pressure (J/kg·K)

dh =

hydraulic diameter (m)

e =

uncertainty from measurements

f =

fanning friction factor, Eq. (8)

h =

heat transfer coefficient (W/m2K)

h0 =

heat transfer coefficient based overall fin surface efficiency, Eq. (5a) (W/m2K)

hf =

fin height (m)

j =

Colburn factor, Eq. (6)

k =

thermal conductivity (W/mK) or turbulent kinetic energy (m2/s2)

Kc =

coefficient of entrance pressure loss

Ke =

coefficient of exit pressure loss

Lf =

flow length or fin length (m)

l =

offset length (m)

m˙ =

mass flow rate (kg/s)

Nu =

Nusselt number, Eq. (6)

p =

pressure (Pa)

p* =

periodic pressure (Pa)

Pr =

Prandtl number

q =

heat transfer rate (W)

Re =

Reynolds number

s =

fin spacing (m)

T =

temperature (K)

t =

fin thickness (m)

ub =

bulk velocity (m/s)

Greek Symbols
α =

cross section aspect ratio (s/hf)

ε =

rate of dissipation of turbulent kinetic energy (m2/s3)

ζ =

thickness ratio (= ξ Re)

λ =

offset length ratio

μ =

dynamic viscosity (Pa·s)

ξ =

blockage ratio

ρ =

density (kg/m3)

η =

overall efficiency

ω =

turbulent dissipation frequency (1/s)

Subscripts
fd =

fully developed flow

form =

form drag

in =

at flow inlet

lam =

laminar flow

lm =

logarithmic-mean

o =

at flow outlet

PF =

plain fin

shear =

shear drag

turb =

turbulent flow

w =

at wall conditions

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Supplementary data