## Abstract

The characterization and scaling of the thermal-hydraulic performance in wavy plate-fin compact heat exchanger cores, based on the understanding of the physical phenomena and heat transfer enhancement mechanism is delineated. Experimental data are presented for forced convection in air (Pr = 0.71) with flow rates in the range 50 ≤ Re ≤ 4000. A variety of wavy-fin cores that span viable applications, with geometrical attributes described by the cross section aspect ratio α (ratio of fin spacing to height), fin corrugation aspect ratio γ (ratio of 2× corrugation amplitude to wave pitch), and fin spacing ratio ζ (ratio of fin spacing to wave pitch), are considered. To characterize and correlate the vortex-flow mixing in interfin spaces, a Swirl number Sw is introduced from the balance of viscous, inertial and centrifugal forces. It is shown that the laminar, transitional and turbulent flow regimes can be explicitly identified by this Swirl number. Based on the experimental results and extended analysis, new correlations for Fanning friction factor f and Colburn factor j are developed with Sw, α, γ, and ζ as scaling parameters. Requisite expressions are devised by a superposition of both enhancement components due to the corrugated surface area enlargement and induced swirl flow field, and they are combined to cover the laminar, transitional and turbulent regimes by the method of asymptotic matching. The resulting generalized correlations are further shown to predict all available experimental data for f and j factors to within ±20% and ±15%, respectively.

## 1 Introduction

Plate-fin heat exchangers are used in a wide variety of applications to meet the demands of energy efficient and cost-effective heat transfer equipment [16]. They typically include power plant condensers, waste-heat recovery recuperators, automobile radiators, air-conditioning condensers and evaporators, and electronic cooling devices, among others. Their primary attributes are high surface area to volume ratio, which renders high compactness, and enhanced heat transfer. Moreover, by altering the fin surface geometry the flow field can be modulated to promote high heat transfer coefficients albeit with additional pressure loss [13,712]. Lanced or offset-strip, louvered, perforated, corrugated and wavy or ruffled plate-fin surface modifications are some examples. Of these, wavy plate-fins are considered to be effective in not only providing additional surface-area enlargement, compared with plain plate-fins, but also enhancing the heat transfer coefficient by their surface-curvature induced transformation of the convective flow field.

As shown in Fig. 1, the sinusoidal wavy-fin geometry can be described by the fin height H, interfin spacing S, fin thickness t, wave amplitude A, and its wavelength or pitch λ. These geometric features can be scaled by a group of dimensionless variables, namely, the cross section aspect ratio α$(=S/H)$, corrugation aspect ratio γ$(=2A/λ)$, and fin spacing ratio ε$(=S/2A)$ or ζ$(=S/λ)$. The sinusoidal waveform of the ruffled surface, referenced to the x–y axes depicted in Fig. 1(a), is given by $y=A sin(2πx/λ)$. The consequent fin surface-area enlargement, for a fixed linear flow-length footprint, is represented by the dimensionless ratio
$κ=AS,wAS,plain=Lwλ=1λ∫0λ1+y′2dx=1λ∫0λ1+(2πAλ)2 cos2(2πxλ)dx$
(1)
Fig. 1
Fig. 1
Close modal

Note that $y′$ is the first-derivative, or $(dy/dx)$, in the above expression. The larger effective flow length intrinsic in the ruffled or wavy-fin passages thus also allows for larger residence time for heat transfer.

Early studies on wavy plate-fins had a limited focus on attempting to understand the associated flow patterns by either computational simulation or visualizing flow behavior in scaled-up wavy channels [10,1324]. It has generally been observed that the flow behavior in wavy plate-fin channels is characterized by flow separation and generation of vortices. In both experimental and computational simulations [1317,2124] of fluid flow in two-dimensional representations of wavy-fin channels ($H≫S$, and α → 0), lateral vortices in the concavities have been identified. This swirl is induced by the flow separation downstream of the crests of the wavy-fin surface, and the recirculation is encapsulated in the trough region by the core flow that reattaches at an upstream surface of the next crest [14,24]. This lateral vortex structure, its magnitude and spatial coverage, is further seen to be altered by changes in γ and/or ε [21]. In three-dimensional interfin channels (H ∼ S and $α≫0$), the swirl structure is found to consist of periodic longitudinal vortices [10,1820]. The associated helical flow progression and fluid mixing, along with a larger flow length lead to substantial heat transfer enhancement.

Furthermore, it has been observed that in wavy channels with $α≫0$, flow instability sets in and the transition to turbulent regime typically occurs at relatively lower Reynolds number compared with the plain plate-fin channel. Rush et al. [20], for instance, found in their flow visualization experiments that the flow instability are dependent upon both flow rate or Re, and number of corrugation waves in the flow length. Similar observations have been made in two earlier studies [18,19] as well. In fact Nishimura et al. [18] found that instabilities tend to also appear in two-dimensional (α → 0) wavy-plate channels and the flow transitions to the turbulent regime. The transient and transitional flow behavior in wavy-plate channels have also been modeled computationally by using the large Eddy simulation method [25], and a three-dimensional direct numerical simulation [26]. In the latter, for periodically fully developed water-flow in wavy microchannels, it was observed that with increasing Re, the flow transitions from steady-state to temporally periodic with one fundamental frequency, and subsequently to quasi-periodicity with two fundamental frequencies. Because of the complex and chaotic nature of flow transitions in wavy channels, and substantial differences in experimental and computational results, quantitative determination of the transitional Reynolds number has been elusive. A constant critical Re is often employed to distinguish laminar and turbulent flows [2730], which incorrectly disregards the effects of the wavy-fin corrugation severity or γ.

Very few experimental studies, and that too sporadically, have reported measurements of pressure loss and heat transfer in wavy-plate-fin cores. The oldest record is, of course, the Kays and London [1] monograph that gives data for three sinusoidal wavy fin cores, but all with the same corrugation aspect ratio γ of 0.2. A rather limited set of mass transfer results for a nearly two-dimensional channel (α = 0.065), and with γ = 0.25 and ε = 1.857, are likewise presented by Nishimura et al. [14]. A later study by Rush et al. [20] has only measured local Nusselt number variation with nondimensional distance for scaled-up wavy channels. Dong et al. [28] conducted experiments with wavy-fin cores with rather small waviness (γ = 0.07, 0.10, 0.12, and 0.15). Moreover, their two-fluid (water–air) heat-exchanger measurements are indicative of potentially large uncertainties in the energy balance and heat transfer coefficients. The verification and/or comparisons with other data, or correlations, or simulation results from some previous work in the literature is also not provided. A more recent study [31] has provided experimental measurements for forced convection of air in stainless steel wavy fin cores manufactured by selective laser melting (SLM), which is one among the different metal additive manufacturing techniques. This study's data from an air–water heat-exchange experiment also have an unverified water-side heat transfer estimation and are limited to a fixed α (= 0.23) albeit with 0.08 ≤ γ ≤ 0.27 and 1.35 ≤ ε ≤ 2.7. The latter large values tend to diminish swirl effects due to the fin waviness. However, the additional uncertainties in this reported data, brought about by the manufacturing process and associated high surface roughness, need further resolution.

Despite the lack of substantive and cross-verified experimental data sets, several studies have attempted to correlate the thermal-hydraulic behavior of wavy-plate-fin cores based on either the sparse data or computational modeling results. These correlations for Fanning friction factor f and Colburn j factor in forced convection of air are listed in Table 1. Two of the listed are based on computational modeling [27,29] or numerical results, two others on own limited and uncertain data [28,31], and another [32] on the very few data listed in Kays and London [1]. Of these, Awad and Muzychka [32] have provided a single respective expression for f and j, for the entire laminar-to-turbulent regime flow rates by employing an asymptotic matching technique [33]. All others are two different set of equations for the two flow regimes, and are primarily regression fits to the respective data as numerical results. The choice of dimensionless variables to account for the various convective mechanisms at play is also not based on explanatory scaling analyses.

Table 1

Correlations reported in the literature for Fanning friction factor f and Colburn j-factor in sinusoidal wavy-plate-fin cores.

YearStudy/SourceCorrelations
2009Sheik Ismail and Velraj [27]; based on computational simulation resultsFor laminar regime (100 ≤ Re ≤ 800), $f=9.827Re−0.705(HS+t)0.322(2AS+t)−0.394γ0.603j=2.348Re−0.786(HS+t)0.312(2AS+t)−0.192γ0.432$
For turbulent regime (1000 ≤ Re ≤ 15,000), $f=10.628Re−0.359(HS+t)0.264(2AS+t)−0.848γ1.931j=0.242Re−0.375(HS+t)0.235(2AS+t)−0.288γ0.553$
2011Awad and Muzychka [32]; based on Kays and London [1] data$f=(fwavy2+fapp2)1/2fwavy=LwL24Re(1−1.3553α+1.9467α2−1.7012α3+0.9564α4−0.2537α5)fapp=3.44ReLw/2dhRe, Lw=2L1+γ2π2πE(Lπ1+γ2π2)$
$j=(jwavy,T5+jLBL5)1/5jwavy,T=7.541RePr1/3(1−2.610α+4.970α2−5.119α3+2.702α4−0.548α5)jLBL=0.664Re1/22dhLw$
2013Dong et al. [28]; based on own dataa$f=3.865Re−0.416(2(S+t)H)−0.138γ1.098(2A2(S+t))−0.506(Lλ)−0.45j=0.0864RePr1/3Re0.914(2(S+t)H)−0.301γ0.7875(2A2(S+t))−0.226(Lλ)−0.254$
2014Aliabadi et al. [29]; based on computational simulation results.For laminar regime (Re < 1900), $f=38.7488Re−0.3840(Sdh)−1.4790(Hdh)−0.3696(λdh)−1.4542(tdh)0.1016(2Adh)1.0903(Ldh)−0.1549j=0.2951Re−0.1908(Sdh)0.7356(Hdh)0.1378(λdh)−0.3171(tdh)0.0485(2Adh)0.2467(Ldh)−0.4976$
For turbulent regime (Re > 1900), $f=52.2375Re−0.3524(Sdh)−1.6277(Hdh)−0.3529(λdh)−1.7484(tdh)0.1034(2Adh)1.2294(Ldh)−0.2371j=0.7293Re−0.3637(Sdh)0.7966(Hdh)0.2398(λdh)−0.4979(tdh)0.0402(2Adh)0.2012(Ldh)−0.3026$
2018Kuehndel et al. [31]; based on own dataa$f=0.526Re−0.221(2AS+t)0.725γ1.226(λL)−1.083j=0.210RePr1/3Re0.638(2AS+t)0.133γ0.641(λL)−0.413$
YearStudy/SourceCorrelations
2009Sheik Ismail and Velraj [27]; based on computational simulation resultsFor laminar regime (100 ≤ Re ≤ 800), $f=9.827Re−0.705(HS+t)0.322(2AS+t)−0.394γ0.603j=2.348Re−0.786(HS+t)0.312(2AS+t)−0.192γ0.432$
For turbulent regime (1000 ≤ Re ≤ 15,000), $f=10.628Re−0.359(HS+t)0.264(2AS+t)−0.848γ1.931j=0.242Re−0.375(HS+t)0.235(2AS+t)−0.288γ0.553$
2011Awad and Muzychka [32]; based on Kays and London [1] data$f=(fwavy2+fapp2)1/2fwavy=LwL24Re(1−1.3553α+1.9467α2−1.7012α3+0.9564α4−0.2537α5)fapp=3.44ReLw/2dhRe, Lw=2L1+γ2π2πE(Lπ1+γ2π2)$
$j=(jwavy,T5+jLBL5)1/5jwavy,T=7.541RePr1/3(1−2.610α+4.970α2−5.119α3+2.702α4−0.548α5)jLBL=0.664Re1/22dhLw$
2013Dong et al. [28]; based on own dataa$f=3.865Re−0.416(2(S+t)H)−0.138γ1.098(2A2(S+t))−0.506(Lλ)−0.45j=0.0864RePr1/3Re0.914(2(S+t)H)−0.301γ0.7875(2A2(S+t))−0.226(Lλ)−0.254$
2014Aliabadi et al. [29]; based on computational simulation results.For laminar regime (Re < 1900), $f=38.7488Re−0.3840(Sdh)−1.4790(Hdh)−0.3696(λdh)−1.4542(tdh)0.1016(2Adh)1.0903(Ldh)−0.1549j=0.2951Re−0.1908(Sdh)0.7356(Hdh)0.1378(λdh)−0.3171(tdh)0.0485(2Adh)0.2467(Ldh)−0.4976$
For turbulent regime (Re > 1900), $f=52.2375Re−0.3524(Sdh)−1.6277(Hdh)−0.3529(λdh)−1.7484(tdh)0.1034(2Adh)1.2294(Ldh)−0.2371j=0.7293Re−0.3637(Sdh)0.7966(Hdh)0.2398(λdh)−0.4979(tdh)0.0402(2Adh)0.2012(Ldh)−0.3026$
2018Kuehndel et al. [31]; based on own dataa$f=0.526Re−0.221(2AS+t)0.725γ1.226(λL)−1.083j=0.210RePr1/3Re0.638(2AS+t)0.133γ0.641(λL)−0.413$
a

these have not been suitably vetted or validated for minimizing measurement uncertainties and data reduction.

The comparison of these correlations with the experimental data from Kays and London [1], as presented in Fig. 2, is instructive in highlighting the large discrepancies in each case as well as among them. The predictions from each of the different correlations deviate dramatically from the experimental data for both f and j, with the exception of those of Awad and Muzychka [32]. The latter case is within a relative error of ±20%, but then the respective equation was developed by regression-fitting this dataset. Moreover, they [32] have completely ignored the swirl that is generated in the wavy channels and only consider the longer effective flow length. That too is included in a developing boundary-layer length scale, similar to that in external flat plate flows, without providing credible justification. The correlation by Kuehndel et al. [31] provides reasonable predictions for j factors, but significantly over predicts f factors. In this case, as well as in two other ones [27,28], arbitrary combination of α, γ, ε, and $(λ/L)$ have been used in simple power-law curve fits without credible analysis of flow physics. The expressions provided by Aliabadi et al. [29] are each an inexplicable multiparameter regression fit through computationally simulated results.

Fig. 2
Fig. 2
Close modal

It is evident that a rational set of predictive correlations, which are devised from and based on a phenomenological scaling of the forced convection in wavy-fin channels, is needed. The available experimental data sets are also very limited and incomplete, especially for wavy-fin cores with high corrugation severity. Additionally, the demarcation of flow regimes in the wavy channels needs to be resolved based on a careful parametric analysis of the convection behavior. All of these issues and unresolved questions are addressed in this experimental study, where the parametric influences on the transport mechanisms in wavy-plate-fin cores are carefully examined. Extended thermal-hydrodynamic performance data are provided for a judicious selection of fin core with different geometrical features (S, H, λ, and A). These results are analyzed to determine the defining demarcation of transition from laminar to turbulent flows. So also is the attendant enhancement in each regime identified, and a new scaling parameter is proposed to account for the surface-waviness-induced swirl flows. Finally, a new set of correlations are developed for predicting the f and j performance. These are devised by appropriately scaling the flow physics, and their predictions are shown to be in very good agreement with all the available but verifiable data in the literature.

## 2 Experimental Method

Carefully controlled experiments were conducted in this study so as to parametrically characterize the thermal-hydrodynamic performance of wavy plate-fin cores. With air (Pr ∼ 0.71) as the fluid medium, the flow rates spanned the range of 50 ≤ Re ≤ 4000, where Re is based on the interfin channel hydraulic diameter dh. The experimental measurement setup for this is schematically shown in Fig. 3(a). It primarily consists of a low-pressure wind tunnel, in which steady and continuous flow of compressed air is provided through an air filter (IMI Norgren F46-600-A0DA). This ensures that dry and clean (particulate contamination free) air flows into the system. The flow rate can be adjusted by a metering valve and measured by a mass and volumetric gas flowmeter (OMEGA FMA-1611A or OMEGA FMA-1623A; their respective usage depends upon the magnitude of the range of flow rates). As the flow passes through a long inlet expansion section, a honeycomb flow straightener (of a length > 6× honeycomb cell dh) at its entrance ensures uniform and steady airflow to the heated test section. Free stream velocity measurements in the flow straightening section, upstream of test section inlet, indicated steady flow in all such sample checks. The inlet and outlet temperatures are, respectively, averaged from multiple precalibrated T-type thermocouples (four locations at the inlet and ten locations at the outlet), which have a precision of ±0.5 °C. The pressure drop across the test section is measured by a differential pressure transducer (Dwyer 607–01 and Setra model 239) or a precision-scale U-tube manometer (Dwyer 1227), depending upon its magnitude, which are connected to pressure taps at the inlet and outlet of the test section. All pressure and temperature measurements are linked to a data acquisition system (National Instruments NI-9214) and are recorded and stored in real-time. For each set of measurements and recorded data, it is ensured that the convection conditions are stable and the steady-state energy balance is maintained.

Fig. 3
Fig. 3
Close modal

The test section is electrically heated with controllable strip heaters, as shown in Fig. 3(b). It was specially designed and built with adjustable features so as to accommodate fin cores of different heights. With a frontal width of 6 in. (152.4 mm), the height of the core section can be adjusted from 0.375 in. (9.53 mm) to 0.75 in. (19.05 mm). Likewise, there are two units each with a flow length of 1.5 in. (38.1 mm) and 3 in. (76.2 mm) so as to accommodate different fin flow lengths. Moreover, the duct cross section of the upstream flow-straightening segment matches that of the fin-core frontal area in all cases, such that the heated top and bottom plates of the test section are completely insulated from any entry-flow infringement. The wavy plate-fin coupons were all made of Al 3033 (kAl = 190 W/m·K), and each core can be sandwiched in between two 5/8 in.-thick (15.88 mm) copper plates. A thin layer (∼50 μm) of thermal paste (OMEGA OT-201, kTP = 2.31 W/m·K) was applied between each copper plate and plate-fin surfaces so as to reduce the contact resistance. Two flexible surface heaters (Mod-Tronic Kapton heater) are affixed on each of the top and bottom outer surfaces of the copper plates so as to provide the requisite heat input to the test section. A variable transformer (Staco Energy 3PN1510B) is connected to the heaters to control and adjust the power input to the test section.

The heated surface temperature is measured from six T-type thermocouples placed inside predrilled holes in each copper plate and the arrangement of the measurement points is sketched in Fig. 3(b). The thick copper plates (kCu = 391 W/m·K) served as good heat spreaders and help maintain a constant and uniform surface temperature across the test section (within 2 °C). To obtain the wall temperature of wavy plate-fins, the measurements from 12 thermocouples are averaged and 1D conduction was applied to estimate the surface temperature as follows:
$Tw=[112(T1+T2+⋯+T12)−q2Ab(δCukCu+δTPkTP)]$
(2)
In Eq. (2), $T1,T2, … ,T12$ represent temperature measurements from each of the 12 different thermocouples, Tw is the obtained wall temperature, q is the heat load on the copper plates, Ab is the base area of the copper plate, δCu represents the distance from the measurement point to the bottom of the copper plate, δTP is the thickness of the thermal paste, and kCu and kTP are the respective thermal conductivity of copper and thermal paste. Moreover, the entire test section is wrapped in 2 in. (50.8 mm) thick fiberglass insulation in order to minimize heat loss. The fin-core pressure drop was similarly measured with precision pressure transducers, and the measured ΔP was corrected for the entrance and exit losses to obtain $ΔPcore$ as follows:
$ΔPcore=[ΔP−1ρ(Kc+Ke)ρu22]$
(3)

In this equation, Kc and Ke, are the coefficients of pressure loss at the entrance and exit of wavy plate-fin core, respectively, and are evaluated as per the method proposed by Webb [34].

From these steady-state measurements, the thermal-hydrodynamic performance metrics can be readily determined, and the dimensionless Fanning friction factor f can be calculated as
$f=(ΔPcoreL)(dh2ρu2)$
(4)
In Eq. (4), ρ is the fluid density, u is the mean air velocity, L is the flow length of the fin core in the streamwise direction, and dh is the hydraulic diameter. The latter for the interfin flow channel is defined as
$dh=4ACPw=4HS2(H+S)$
(5)
The heat transfer rate q is given by the enthalpy change in the air flow as it enters and leaves the heated test section or
$q=m˙cp(To−Ti)$
(6)
Here, $m˙$ is the mass flow rate, and Ti, To represent inlet and outlet mean temperature of the test section, respectively. The steady-state energy balance, or the difference between Eq. (6) and the electrical heat input or q = VI, was maintained such that the difference from the latter was always < 10%. The heat transfer coefficient h can thus be obtained from its definition for uniform wall temperature conditions [35] given by
$h=qASΔTlm$
(7)
In Eq. (7), AS is the total heat transfer area, and $ΔTlm$ is the log-mean temperature that is defined as
$ΔTlm=(Tw−To)−(Tw−Ti)ln[(Tw−To)/(Tw−Ti)]$
(8)
Moreover, the experimentally determined heat transfer coefficient needs to be corrected by accounting for the fin effects in the core-flow convection [36]. This is done by considering the overall surface area effectiveness η, and the corrected heat transfer coefficient h0 can be obtained iteratively from the following:
$1ηh0AS=1hAS$
(9)
The overall fin surface efficiency is given by
$η=1−AfAS(1−ηf)$
(10)
where Af is the fin area. In the case of two-side heating of the fin core, the fin efficiency ηf can be expressed as
$ηf=tanh(mH/2)mH/2; m=2h0kAlt$
(11)
Finally, therefore, the Nusselt number Nu and hence the Colburn factor j are defined, respectively, as
$Nu=h0dhk, and j=NuRePr1/3$
(12)
All measurements were made with carefully calibrated precision instruments, and the uncertainties in the calculated variables, Re, f, and j, were determined as per Ref. [37] and the following articulation
$δΘ=(∂Θ∂aea)2+(∂Θ∂beb)2+(∂Θ∂cec)2+⋯$
(13)

In this equation, $a,b,c,…$ represent different sensor measurements or a derived parameter, as the case may be, with the given precision or calculated uncertainties $ea,eb,ec,…$, and $Θ=Θ(a,b,c,…)$ represents the calculated parameter. Based on this single-sample analysis of the entire dataset, the calculated maximum uncertainties in Re, f, and j in the experiments are within ±3%, ±15%, and ±8%, respectively. Moreover, the validity and fidelity of measurements, experimental method and its apparatus were established by measurements for plain plate-fin cores. As presented in greater detail elsewhere [38], the performance data for both f and j were in excellent agreement with other reliable dataset as well as predictions. A sample comparison of data for a representative plain-plate-fin core with predictions of the Lin et al. [38] correlation is given in Fig. 4.

Fig. 4
Fig. 4
Close modal

## 3 Results and Discussion

To reiterate the inadequacy of the correlations reported in the literature, and as pointed out in Sec. 1 (see Fig. 2), one wavy-fin dataset from this study's measurement is compared with their predictions. This is presented in Fig. 5, where results for a fin geometry with γ = 0.272, α = 0.187, and ζ = 0.368 are graphed. The shaded area around the f and j data, and their variation with Re, enclosed with dotted lines, represents the ±20% band. The overall disagreement and incompatibility of the predictive equations listed in Table 1 are clearly evident in Fig. 5. Not only do the predictions fall outside the ±20% scatter, but in many cases incorrect trends are indicated. This indubitably underscores the compelling need for phenomenologically derived correlations based on extended experimental data.

Fig. 5
Fig. 5
Close modal

The geometrical attribute of the sinusoidal-wavy-fin surface that has a dominant effect on forced convection in interfin channels is the severity of the fin-surface corrugation. This is scaled by the waviness aspect ratio γ (= 2 A/λ; see Fig. 1(a)), and its effect is seen in Fig. 6. Two different data sets for Fanning friction factor f and Colburn j-factor and their variation with Re are presented. One date set is from this study's experimental measurements (See Table 2), and the second is from Kays and London [1]. They both have nearly the same values for α and ζ, but their corrugation aspect ratios are γ = 0.133 and 0.208. The latter is 1.56 times higher, and its substantive influence on the enhancement of convection is evident in the plots.

Fig. 6
Fig. 6
Close modal
Table 2

Geometric details and dimensions of wavy-fin cores

Data sourceFin tagfpiH (in)t (in)S (in)A (in)λ (in)αγεζ
This studyCore 1120.3690.0060.07730.0250.3750.2100.1331.5460.206
Core 2110.4940.0060.08490.0500.3750.1720.2670.8490.226
Core 370.4940.0060.13690.0500.3750.2770.2671.3690.365
Core 4100.7420.0080.09200.0340.2500.1240.2721.3530.368
Core 5100.4920.0080.09200.0340.2500.1870.2721.3530.368
Core 6110.7420.0080.08290.0340.2500.1120.2721.2190.332
Kays and London [1]11.44-3/8W11.440.4070.0060.08140.0390.3750.2000.2071.0440.217
11.5-3/8W11.50.3650.0100.07700.0390.3750.2110.2080.9870.205
17.8-3/8W17.80.4070.0060.05020.0390.3750.1230.2070.6440.134
Data sourceFin tagfpiH (in)t (in)S (in)A (in)λ (in)αγεζ
This studyCore 1120.3690.0060.07730.0250.3750.2100.1331.5460.206
Core 2110.4940.0060.08490.0500.3750.1720.2670.8490.226
Core 370.4940.0060.13690.0500.3750.2770.2671.3690.365
Core 4100.7420.0080.09200.0340.2500.1240.2721.3530.368
Core 5100.4920.0080.09200.0340.2500.1870.2721.3530.368
Core 6110.7420.0080.08290.0340.2500.1120.2721.2190.332
Kays and London [1]11.44-3/8W11.440.4070.0060.08140.0390.3750.2000.2071.0440.217
11.5-3/8W11.50.3650.0100.07700.0390.3750.2110.2080.9870.205
17.8-3/8W17.80.4070.0060.05020.0390.3750.1230.2070.6440.134

It is also seen that with increasing Re the flow regime changes and the transition is not discontinuous. As a guide, results from computational simulation, reported in a different study [39], are graphed. Implicit in these simulations is the use of distinct laminar and turbulent flow models, which, of course, leads to a distinct and abrupt change. The experimental data however depicts a smooth transition from laminar to the turbulent regime (Fig. 6). This is primarily due to swirl behavior induced by the fin-surface curvature in the confined inter-fin channel, which tends to suppress the flow instabilities and has been observed in other swirl-dominated flows as well [4042]. Nevertheless, scaling of the transition from one regime to another is incumbent on devising a physics-based correlation.

### 3.1 Demarcation of Flow Regimes.

Forced convection performance data, as measured by Fanning f-factor and Colburn j-factor, for air flows in six different wavy-fin cores, as listed in Table 2, were obtained in this study. Also included for the ensuing analysis are the dataset for three other wavy-fin cores in the literature [1]. These results, given by the variations in f and j with Re, are graphed in Figs. 7(a) and 7(b), respectively. The change in flow regimes with increasing Re is notably evident from the change in slopes of the f and j plots. What is also seen from the data are that the severity of fin-surface waviness has a pronounced effect on flow transition. In some cases, it may be noted, the fin-core results have been extended by using discrete simulation results for Re ≤ 300 [39] so as to fill in the unavailable data in this range, and provide additional clarity in the trends.

Fig. 7
Fig. 7
Close modal

The progression from laminar to turbulent regime with increasing Re has a distinctly continuous transition region. The critical Re that marks the onset of transition, and the size of the interval till the change to turbulent flow are both seen to vary with γ. This transition region is demarcated with dashed lines in Fig. 7, and the critical Re at either end of the changeover in flow conditions is seen to decrease with increasing γ. With gentler fin-surface waviness or smaller corrugation aspect ratio γ, the flow transition shifts to higher Reynolds number. Moreover, because of the wavy-surface-induced flow instability, the transitional Reynolds numbers in wavy interfin channels are in general much smaller than that in plain rectangular ducts. In latter case, it may be noted that flow transition typically occurs at Re ∼ 2000 [38].

In order to resolve this and find a normalized or common transition criteria for all γ, the curvature induced swirl in the wavy-channels needs to be scaled. The axial flow field has recirculating or vortical flow superimposed on it, primarily in the trough regions of the channels. As proposed by Manglik and Bergles [4244], a general scaling of the effects of such flows can be obtained from the following force balance:
$inertia force × centrifugal force(viscous force)2$
(14a)
In this balance, the ratio of inertial force to viscous force is the well-known Reynolds number. By considering a one-wave-long (or λ) unit volume in the interfin channel and its hydraulic diameter dh, this part of the force balance yields
$inertial froceviscous force∼ρdh2λ(u2/dh)μdhλ(u/dh)$
(14b)
The cross-stream recirculation is induced by the curvature of the wavy or periodically undulating flow passage. Thus, the force balance to account for this can be expressed as
$centrifugal forceviscous force∼ρdh2λ(u2/rc,min)μdhλ(u/dh)$
(14c)
Because the curvature of a sinusoidal wavy channel varies along the flow path, the minimum radius of curvature $rc,min$ corresponding to the radius of curvature at the trough or peak point of the undulations has been applied. This radius can therefore be expressed as
$rc,min=|(1+y′2)3/2y″|=|[1+(2πA/λ)2 cos2(2πx/λ)]3/2(4π2A/λ2)sin(2πx/λ)|x=0.25λ or 0.75λ=λ2π2γ$
(15)
Consequently, by combining Eqs. (14) and (15), the dimensionless Swirl number can be defined and expressed as follows:
$Sw∝1/2(inertial force)(centrifugal force)(viscous force)2=Redh2rc,min=πRe2γζα+1$
(16)

Note that the square root of the force balance is taken because the magnitude (due to u2) would otherwise be very large. Furthermore, this force-balance has been contended to have universality in axial flows with superimposed recirculation [45]. With a different characteristic velocity, or curvature length, or inclusion of buoyancy force in place of centrifugal force, this scaling and nondimensional variable definition yields the twisted-tape-insert flow Swirl number, curved-tube-flow Dean number, and mixed-convection Grashof number, respectively.

The efficacy of the Swirl number, as defined by Eq. (16), in characterizing the flow behavior in wavy-fin channels is articulated in Fig. 8. The experimental data for f and j factors are regraphed with respect to Sw, and the flow regime transitions are seen to begin and end at approximately the same Sw values for all wavy-fin cores. These ends, marked by vertical dashed lines in Fig. 8, span the region 300 ≤ Sw800. Correspondingly, the flow is considered to be laminar when Sw<300, and turbulent when Sw>800 with the intermediate flow rates in the transition region. The progression from one flow regime to the other is not only continuous, but is effectively scaled by Sw that also maps the development of flow recirculation in the trough region. Swirl evolution is further seen to be influenced by changes in γ and ζ.

Fig. 8
Fig. 8
Close modal

### 3.2 Scaling of Wavy-Fin Geometry Effects.

As indicated previously the wavy-fin geometry can be described by the dimensionless variables α, ζ, and γ. These geometric features influence both the forced-convective attributes and their magnitude. The scaling of their commensurate effects on the hydrodynamic and thermal performance, in terms of dimensionless descriptors, is instructive in explicating the associated mechanisms and needed for devising generalized correlations.

Consider the variation of f and j in plain fins of rectangular flow cross section with its aspect ratio α. In the case of laminar fully developed flow and heat transfer, the effect of α is quite pronounced, and fPF,fd and jPF,fd can be obtained, respectively, from the following [38,46]:

$fPF,fd,lam=24Re(1−1.355α+1.947α2−1.701α3+0.956α4−0.254α5)$
(17a)
$jPF,fd,lam=7.541RePr1/3(1−2.61α+4.97α2−5.119α3+2.702α4−0.548α5)$
(17b)

In laminar developing flow (small L/dh), however, the effect of α tends to be less substantive [38], and is essentially obviated by the flow periodicity in wavy-fin channels. So also is the case in turbulent flow convection [3,35,38], and the fully developed f and j in this regime can be represented by the following [35,47]:

$fPF,fd,tur=0.1268Re−0.3$
(18a)
$jPF,fd,tur=0.023Re−0.2$
(18b)

To evaluate the influence of α, selected f and j results for wavy-fin channels [39], normalized by those for fully developed flow in equivalent plain-fin cores are graphed in Fig. 9. Two flow regimes, represented by Sw=100 and 2000, are considered and the variation with α are graphed. In the laminar regime with Sw=100, the near constant j and f ratios imply that the cross section aspect ratio α affects the thermal-hydrodynamic behavior in wavy channels in a similar way as that in plain rectangular channels. So also is the case in the turbulent flow regime with Sw=2000. The very small negative gradient with respect to α, as shown subsequently, needs no correction for and is a second-order effect. It thus follows that the influence of α can be effectively accounted for by the corresponding plain-fin fully developed flow correlations given by Eqs. (17) and (18).

Fig. 9
Fig. 9
Close modal

Another geometrical feature of wavy-fin cores that affects the convective flow field is the interfin separation S (see Fig. 1). In previous two-dimensional modeling studies (H  S; α → 0) [21], as well as some other investigations [27,28,31], this was scaled by the dimensionless ε = (S/2A). The underlying rationale was that when ε < 1, viscous effects dominate and swirl tends to be suppressed [21] as the interfin spacing is less than the peak-to-valley dimension (2 A, or twice the amplitude of sinusoidal waviness) of the fin corrugation. With ε ≥ 1, on the other hand, trough-region recirculation prevails in the flow field. This nonetheless has scaling limitations as seen in Fig. 10(a), where variations in f and j with ε for Sw=2000 and α = 0.19 are graphed. While convection is seen to be enhanced with increasing ε, it tapers off and decreases after a relatively large interfin spacing for a given γ. In essence the large spacing negates the influence of the fin-wall corrugation. This threshold in ε, however, is seen to be a function of γ and the transition in performance (f and j) occurs at different values of ε.

Fig. 10
Fig. 10
Close modal

A better scaling parameter that readily accounts for this behavior is ζ (= S/λ), in which the interfin spacing is normalized by the fin-corrugation wavelength. Its efficacy is self-evident in Fig. 10(b), where the f and j data are regraphed with respect to ζ. The performance transition, for the case of fins with α = 0.19, but varying γ and Sw=2000, is seen to occur at ζ ∼ 0.5 for all γ and in both f and j. Moreover, both large ζ (or ε) and α are impractical geometric features of wavy or corrugated fins. A large interfin spacing S is obtained when the linear fin density (number of fins/inch) is small. This effectively nullifies the primary application of fins, which is to enlarge the thermal surface area and larger linear fin densities are typically used [14].

### 3.3 Development of Correlations.

With the parametric evaluation of the mechanistic influence of geometric or surface features of wavy-fin cores in Secs. 3.1 and 3.2, the following thus are the primary convection enhancement expressions: (i) surface-curvature-induced swirl or vortical circulation in the trough regions, and (ii) the surface-area enlargement or longer flow residence time in the corrugated-fin channels. The former, as will be delineated in the ensuing, scales with the grouping (Sw, γ, ζ), and the latter with κ. Based on this phenomenological assessment of the convection behavior, the performance prediction model for wavy-plate-fin cores can be expressed as follows:
$f(or j)=f(or j)PF,fd︸Fully developed flow in plain−plate−fin cores×κ︸Surface area enlargement×(1+Φf(or j)(Sw, γ, ζ))︸Enhancement by flow modulation$
(19)

The first term on the RHS of Eq. (19) represents the f or j factor for fully developed flow in a plain rectangular duct. This term is the baseline or limiting case for wavy-plate-fin cores for which the corrugation amplitude A → 0 and/or wavelength λ → ∞. For this limit the two other terms in Eq. (19) become unity (κ → 1; Sw → 0, γ → 0, and ζ → 0), and the f or j factors approach that of a respective plain rectangular plate-fin core.

Moreover, the dimensionless ratio κ, which is the ratio of actual convoluted or wavy surface area to the projected plain or flat surface area given by Eq. (1), can be more readily resolved numerically to excellent accuracy by the following polynomial [48]:
$κ=1+0.11γ+2.04γ2−0.85γ3$
(20)

It is evident here that the effective surface area increases with γ or greater severity of fin ruffles or corrugations. This heat transfer surface area enhancement also implicitly provides for a larger fluid-flow residence time; higher γ yields a longer flow length over a fixed linear footprint.

In wavy-fin channels the flow streamlines undulate with the surface corrugations, and depending upon γ and Re vortices are generated in the wave trough regions. This surface-curvature-induced swirl and the consequent fluid mixing enhances the convective heat transfer. Its magnitude can be scaled and represented by the function Φf(or j), which is directly related to the Swirl number Sw, corrugation aspect ratio γ, and fin spacing ratio ζ. The consequent functionality of Φf(or j) with Sw, γ, and ζ has a power-law-type dependence and can be expressed in the following manner:
$Φf(or j)=C1SwC2γC3ζC4$
(21)

The leading coefficient and exponents, $C1, C2, ⋯, C4$ can be determined from the respective experimental dataset by the least square method for each flow regime.

Because there are dissimilarities in the magnitude and trends for flow friction and thermal convection behavior in the laminar, transitional, and turbulent regimes, piecewise equations are first developed for each case. These correlations of f and j in the laminar and turbulent regime, respectively, are as follows:

For laminar flow (Sw<300),
$flam=fPF,fd,lamκ(1+0.6Sw0.58γ1.27ζ0.45)$
(22a)
$jlam=jPF,fd,lamκ(1+0.2Sw0.23γ0.90ζ0.15)$
(22b)
For turbulent flow (Sw > 800),
$ftur=fPF,fd,turκ(1+884Sw0.11γ2.80ζ1.20)$
(23a)
$jtur=jPF,fd,turκ(1+274Sw−0.34γ1.11ζ0.74)$
(23b)
In these expressions, fPf,fd, lam, jPf,fd,lam, fPf,fd,tur and jPf,fd,tur are given by Eqs. (17)(18). As for the flow in the transition regime (300 ≤ Sw800), because of the complexity of the convective behavior, it may be recognized that no simple model avails itself to predict the thermal-hydraulic behavior even in plain channels. However, in the present case, the following power-law model describes the measured f and j factors (300 ≤ Sw800) rather well
$ftran=33.1Sw−0.14α−0.07γ2.20ζ0.98$
(24a)
$jtran=0.32Sw−0.17α−0.13γ0.89ζ0.38$
(24b)
The efficacy of these expressions in representing the experimental data is seen in Fig. 11. Variation in f and j with Sw are presented for the wavy-fin core 5 (Table 2) that has geometrical features described by α = 0.187, γ = 0.272, and ζ = 0.368. Also indicative in the graph is that the piecewise expressions for the three flow regimes can be combined into a single equation by asymptotic matching [33,43]. The smooth changeover from one flow regime to the other, without any obvious discontinuity, facilitates this. The consequent generalized correlations over the entire domain of laminar, transition, and turbulent regimes are, respectively, given by
$f=[flam5+(ftran−10+ftur−10)−0.5]0.2$
(25a)
$j=[jlam10+(jtran−20+jtur−20)−0.5]0.1$
(25b)
Fig. 11
Fig. 11
Close modal

The predictive validity of these correlation is seen in the scatter plots of Fig. 12, where they are compared with experimental results for nine different wavy-plate-fin cores. These include those employed in this study (Table 2) as well as the ones in Kays and London [1]. The dataset covers a wide range of airflows (50 ≤ Re ≤ 4000) in wavy-plate-fin cores with 0.11 < α < 0.28, 0.13< γ < 0.27, and 0.13 < ζ < 0.37. As seen from Fig. 12, the predictions for f and j from the new correlations are, respectively, within ± 20% and ± 15% of the experimental data. They therefore provide rational and reliable predictive equations for the thermal and hydrodynamic performance of wavy-plate-fin cores with sinusoidal corrugations. The inherent parametric scaling in these expressions properly map the convection mechanisms, and thereby are effective design tools for the optimization of both heat transfer enhancement in and sizing of compact heat exchangers with wavy fins.

Fig. 12
Fig. 12
Close modal

## 4 Conclusions

New generalized correlations that phenomenologically represent and scale airflow (Pr ∼ 0.71) convection in sinusoidal wavy-plate-fin cores are devised and presented in this study. They are based on extended experimental results for a wide range of flow rates (50 ≤ Re ≤ 4000) and ruffled-or corrugated-fin-surface geometrical features. The latter are described by the dimensionless waviness aspect ratio γ, interfin separation or spacing ratio ζ, and cross-sectional aspect ratio α. For scaling of fin-surface-curvature-induced swirl or recirculating flow behavior a Swirl number Sw is introduced. As expressed in Eq. (16), it is based on the primary force balance of viscous, inertial, and centrifugal forces, where the radius of curvature of the peak or trough corrugation is employed as the characteristic length for the centrifugal force effect. The Swirl number, or $Sw=πRe2γζ/(α+1)$, effectively demarcates the laminar (Sw<300), transition (300 ≤ Sw800), and turbulent (Sw>800) regimes at approximately the same values for all wavy-fin cores with different α, γ, and ζ. It is also noted that the changeover from one regime to the other, as manifest in the f and j data and their variation with Re or Sw, is smooth and without the typical discontinuity seen in duct flows.

The correlations for predicting the Fanning friction factor f and Colburn j factor inside wavy-plate-fin channels with rectangular cross section are, respectively, described by Eqs. (25a) and (25b). The development of these equations is based on the fully developed behavior in plain-plate-fin channels and a supposition of convection enhancement effects that are induced by the corrugated fin surfaces. The fully developed plain-fin f and j performance can be readily scaled by the cross section aspect ratio α in the laminar regime and Re in the turbulent regime. The enhanced convection of wavy-fin surfaces, on the other hand, has two components, and they are shown to be described or scaled by (a) the surface-area enlargement ratio κ, and (b) swirl flow effects given by Φf (or j), which is a function of Sw, γ, and ζ. Contiguous expressions for f and j are then devised by asymptotic matching of the respective convection performance in the laminar, transition, and turbulent flow regimes. The f and j predictions from these correlations are, respectively, within ± 20% and ± 15% of all available experimental data; both from this study as well as those vetted from the literature. The proposed generalized correlations are thus the much-needed design tools for sizing and performance optimization of compact heat exchangers employing sinusoidal wavy-plate-fin cores.

## Funding Data

• This study was carried out with support from ARPA-e (# DOE DE-AR0000577; ARID program) for the development of novel dry-cooling systems. Moreover, the technical discussions and manufacturing support from Robinson Fin Machines, Inc., as well as for providing the needed plate-fin coupons is gratefully acknowledged.

## Nomenclature

• A =

amplitude of wavy fin (m)

•
• Ab =

area of base plates (m2)

•
• Ac =

area of cross section (m2)

•
• As =

total surface area (m2)

•
• cP =

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

•
• dh =

hydraulic diameter (m)

•
• e =

uncertainty from measurements

•
• f =

Fanning friction factor, Eq. (19)

•
• h =

heat transfer coefficient (W/m2 K)

•
• H =

fin height (m)

•
• h0 =

heat transfer coefficient for 100% overall surface efficiency (W/m2 K)

•
• j =

Colburn factor, Eq. (12)

•
• k =

thermal conductivity (W/m K)

•
• Kc =

coefficient of entrance pressure loss

•
• Ke =

coefficient of exit pressure loss

•
• L =

flow length or fin length (m)

•
• $m˙$ =

mass flow rate (kg/s)

•
• Nu =

Nusselt number, Eq. (17)

•
• P =

pressure (Pa)

•
• Pr =

Prandtl number

•
• q =

heat transfer rate (W)

•
• Re =

Reynolds number

•
• S =

fin spacing (m)

•
• Sw =

Swirl number

•
• t =

fin thickness (m)

•
• T =

temperature (K)

•
• ub =

bulk velocity (m/s)

### Greek Symbols

Greek Symbols

• α =

cross section aspect ratio

•
• γ =

corrugation aspect ratio

•
• ε =

fin spacing ratio

•
• ζ =

fin spacing ratio

•
• η =

overall efficiency

•
• κ =

surface area ratio

•
• λ =

corrugation wavelength or wave pitch

•
• μ =

dynamic viscosity (Pa·s)

•
• ρ =

density (kg/m3)

### Subscripts

Subscripts

• f =

fin

•
• fd =

fully developed flow

•
• i =

at flow inlet

•
• lam =

laminar flow

•
• lm =

logarithmic-mean

•
• o =

at flow outlet

•
• PF =

plain fin

•
• tran =

transitional flow

•
• tur =

turbulent flow

•
• w =

at wall conditions

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