Inclination Angle Optimization for “Inclined Projected Winglet Pair” Vortex Generator

Mohammad Oneissi

Mechanical Engineering Department, Lebanese International University,

P.O. Box 146404, Mazraa, Beirut 1105, Lebanon;

Industrial Energy Department, IMT Lille Douai, Douai 59500, France e-mail: mohammad.oneissi@liu.edu.lb

Charbel Habchi

Mem. ASME Thermofluids Research Group, Notre Dame University–Louaize, P.O. Box 72, Zouk Mikael 1200, Lebanon e-mail: charbel.habchi@ndu.edu.lb

Serge Russeil

Industrial Energy Department, IMT Lille Douai, Douai 59500, France e-mail: serge.russeil@imt-lille-douai.fr

Daniel Bougeard

Industrial Energy Department, IMT Lille Douai, Douai 59500, France e-mail: daniel.bougeard@imt-lille-douai.fr

Thierry Lemenand

LARIS, ISTIA, Angers University, LARIS EA 7315, Angers 49000, France e-mail: thierry.lemenand@univ-angers.fr

Inclination Angle Optimization for “Inclined Projected Winglet Pair” Vortex Generator

Vortex generators (VG) are widely used in multifunctional heat exchangers/reactors for augmenting the heat transfer from fin plates to the working fluid. In this study, numerical simulations for longitudinal VGs are performed for both laminar and turbulent flow regimes. The shear-stress transport (SST)j–xmodel is used for modeling turbulence.

Inclination angle for a new streamlined VG configuration called inclined projected wing- let pair (IPWP) was varied to study the effect of this angle on the heat transfer enhance- ment and pressure drop. Response surface methodology (RSM) was used to deduce the inclination angle effects on heat transfer, pressure drop, and vorticity from both local and global points of view. Such study highlights the optimization for this VG configura- tion for better heat transfer intensification, based on thermal enhancement factor (TEF).

Finally, it is found that the VG with inclination angle ranging from 30 deg to 35 deg exhibits the best global performance compared to other inclination angles. This type of studies is important for the enhancement of the thermal performance of heat exchangers and static mixers in various engineering applications.[DOI: 10.1115/1.4041438]

Introduction

Heat transfer intensification has been an essential subject for many years and for several industrial applications, due to increas- ing demands and energy savings. Heat exchangers are as consid- ered one of the main challenges in this domain since they are widely used in power systems, air conditioning, aerospace indus- try, and many other applications. In heat exchangers, enhancing the heat transfer between the fin and the working fluid is crucial for increasing efficiency and compactness [1]. Convective heat transfer enhancement in a heat exchanger is accomplished by either active or passive methods. Due to many reasons such as efficiency, industry, economy, and maintenance supremacy of passive methods, vortex generators (VG) are widely used in this field [2,3]. Various types of vortex generators exist such as helical and twisted inserts [4], dimples or protrusions [5], cylindrical tubes [6], transverse and longitudinal vortex generators [7–9], plane or curved surface of VG [10,11]. It has been shown that the longitudinal vortex generators have better performance relative to transverse vortex generators in terms of heat transfer [12,13].

Finding the optimum performance of a vortex generator depending on specific criteria is also crucial. In fact, many param- eters may alter the performance of the heat exchanger, thus vary- ing the VG shape, angle of attack, position, composition with other types, Reynolds number, and many others is a field of con- sideration for many researchers [14].

Saliviano et al. [15] studied the optimization of VG position and angles in a fin-tube compact heat exchanger using the genetic algorithm. The results were evaluated using direct optimization and response surface methodology (RSM). Lei et al. [16] studied the effect of changing the angle of attack for delta-winglet vortex generator between 10 deg and 50 deg, in addition to changing VG aspect ratio from 1 to 4 and the Reynolds number from 600 to 2600. It was found that the delta-winglet vortex generator with an attack angle of 20 deg and an aspect ratio of 2 provides the best global performance over the range of Reynolds number computed.

Heat transfer enhancement by punched winglet-type VG arrays in fin-and-tube heat exchangers using numerical studies was investigated by He et al. [17]. They varied the VG angle of attack and spacing between staggered VGs and concluded that the heat transfer and pressure drop increase as angle of attack increases.

Lemouedda et al. [18] examined an optimization of the angle of attack of delta-winglet VGs in a plate-fin-and-tube heat exchang- ers using a combination among numerical analysis, genetic

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OFTHERMALSCIENCE ANDENGINEERINGAPPLICATIONS. Manuscript received September 28, 2017; final manuscript received September 1, 2018; published online October 23, 2018. Assoc. Editor: Ting Wang.

Journal of Thermal Science and Engineering Applications FEBRUARY 2019, Vol. 11 / 011014-1

algorithm and RSM. Angle of attack was varied between90 deg and 90 deg for Reynolds numbers between 200 and 1200. They found that for the inline arrangement the common-flow-down con- figuration is more suitable than the common-flow-up.

Wu and Tao [19] studied experimentally and numerically the convective heat transfer on the top and bottom surfaces of a plain plate and four plates with delta winglet VG pair in laminar flow.

The VG was directly punched from the plates at different angle of attacks (15 deg, 30 deg, 45 deg, and 60 deg). They deduced that the average Nusselt number of the plate with 60 deg angle of attack is slightly higher than that of 45 deg attack angle.

Tanaka et al. [20] optimized a double-winglet VG by varying both the inclination angle and angle of attack for Reynolds num- bers from 360 to 3600. It was found that the best thermal perform- ance was obtained for an inclination angle of 60 deg and an angle of attack of 45 deg.

Khan and Li [21] optimized the shape and angles of VG by adopting a Bayesian regularized artificial neural network with trained data from numerical simulations and used to drive a multi- objective optimization algorithm. The parameters are the attack and inclination angle and the shape of VG which could vary from triangular to trapezoidal and rectangular.

As a developing work of Oneissi et al. [22], the aim of this study is to analyze the effects of varying the inclination angle of the novel inclined projected winglet pair (IPWP) configuration on the flow characteristics and thus on the heat transfer process over a wide range of Reynolds number. These effects are presented with RSM as a function of incidence angle (a) and Reynolds number (Re). RSM is a collection of statistical and mathematical techniques useful for developing, improving thus optimizing a case study, showing the response for changing several input varia- bles. Direct optimization can be imposed on RSM showing the maximum or minimum of the function.

Mathematical Models and Numerical Methods

Problem Description.In this study,xis defined as the stream- wise direction,yas the spanwise direction, andzfor the fin pitch direction. Dimensions of the channel simulated are similar to Oneissi et al. [22] case taken from a previously designed experi- mental benchmark. A heightH¼38.6 mm, breadthB¼1.6H, and lengthL¼13Hare the main dimensions of this channel. An iso- metric view with the boundary conditions is presented in Fig.1.

Figure2shows the dimensions of the delta winglet pair (DWP), the 30 deg inclined delta winglet pair (named as IPWP-30), and the 60 deg IPWP (named as IPWP-60) vortex generators. The same base and height dimensions are conserved for all VGs while the incident angle is varied for each IPWP. Note that the incident angle is the complementary angle of the inclination angle

(inclination angle (a)þincident angle¼90 deg), since angle of incidence is measured from the base plane while angle of inclina- tion is measured from the DWP vertical plane. Only the inclina- tion angle was varied in this study, all other dimensions and attack angle are kept constant. These geometries are obtained by the frontal projection of the DWP on inclined planes from the base axis. The studied inclination angles are 0 deg for the DWP case (DWP corresponding to IPWP-0), 25 deg, 30 deg, 32 deg, 35 deg, 38 deg, 40 deg, 45 deg, 50 deg, 55 deg, and 60 deg for the IPWP configurations. Studied VGs length ranges between 1.4Hfor the DWP till 5Hfor the IPWP-60.

Subsequently the lengths of the VGs are different and therefore, in practical applications, the number of VGs applied to a surface of fixed length would be different. This could be investigated in future work where the heat transfer and friction characteristics will be investigated for a given surface with different number of VGs.

The Reynolds number is calculated based on the hydraulic diameter Dh¼2H generally used for flow between two parallel plates. The flow is assumed to be hydrodynamically and thermally developing. Uniform inlet velocity and inlet temperature profiles are adopted. The inlet temperature is equal to 300 K for all studied Reynolds numbers (270, 540, 1080, 2800, 4600, 7700, 10,800, 16,200, 21,600, and 30,000) while the velocity magnitude varies for each Reynolds number. A constant zero gauge pressure condi- tion is set for the outlet. Isothermal upper and bottom walls with a constant temperature of 350 K are chosen for all the cases. The VGs are assumed adiabatic in order to highlight the effect of the flow structure on the convective heat transfer. Turbulence inten- sity at the inlet is assumed equal to 3%, which is common for this type of studies [22–24] especially that it is in the range of the moderate wind tunnel intensities (1–5%).

The aim of this study is to analyze the effect of VG inclination angle on the heat transfer enhancement, pressure drop, and vortic- ity while maintaining the same frontal area of the DWP case. This means that if any of the VGs are projected on a transvers plane normal to thex-axis, it will have thus the same projected shape and the same area. An area projection on 25 deg, 30 deg, 32 deg, 35 deg, 38 deg, 40 deg, 45 deg, 50 deg, 55 deg, and 60 deg inclined planes of the DWP are studied.

Numerical Model.Three-dimensional numerical simulations of the flow and heat transfer in a parallel plate channel with new vortex generators is carried out for laminar and turbulence regimes. The following assumptions are used for the numerical simulations:

Steady three-dimensional fluid flow and heat transfer.

The flow is incompressible Fluid properties are constant.

Fig. 1 Isometric view of the computational domain and the boundary condi- tions. The upper wall was hidden for the sake of clarity.

Radiation heat transfer, body forces, and viscous dissipation are neglected.

Based on the above assumptions, the channel flow is governed by the 3D steady-state Reynolds-averaged Navier–Stokes equa- tions. The continuity and linear momentum equations for an incompressible Newtonian fluid are

@u_i

@x_i¼0 (1)

u_j@u_i

@x_j¼ 1 q

@p

@x_iþv @²u_i

@x_j@x_j@u⁰_iu⁰_j

@x_j (2)

where the termu⁰_iu⁰_jis the Reynolds stress tensor resulting from the averaging process on the nonlinear convective terms in the Navier–Stokes equations. The heat transfer is computed by solv- ing the energy equation:

qc_p u_i@T

@x_i

¼ @

@x_i keff

@T

@x_i

(3)

where E is the total energy, and keff is the effective thermal conductivity.

All computations are performed using ANSYS FLUENT15 CFD solver based the cell-centered finite volume discretization method [25]. Shear-stress transport (SST)j–xmodel developed by Menter [26] is used in this study for the turbulent flow cases.

This model computes two additional partial differential equa- tions; a modified version of the turbulence kinetic energy equa- tion j and a transport equation for the specific dissipation x.

The shear stress transport uses thej–xformulation in the inner parts of the boundary layer and thej–eformulation in the free- stream. This model shows a good behavior in adverse pressure gradients and separating flows, and it has been widely used by many researchers showing a fair agreement with experimental results [27,28]. Therefore, the SST j–x turbulence model is adopted in the present study.

This approach necessitates the dimensionless wall distancey^þ to be less than 4, ensuring that the viscous sublayer is well computed.

The SSTj–xmodel transport equations are q @

@x_iðku_iÞ ¼ @

@x_j C_k@k

@x_j

þG~_kY_k (4)

q @

@x_iðxu_iÞ ¼ @

@x_j C_x@x

@x_j

þG_xY_xþD_x (5)

whereG~_k is the production of turbulence kinetic energy due to mean velocity gradients, Gx is the generation of x, C_k is the effective diffusivity of j;C_k¼lþ ðl_t=r_kÞ, C_x is the effective diffusivity ofx;C_x¼lþ ðl_t=r_xÞ,Ykis the dissipation ofjdue to turbulence,Yxis the dissipation ofxdue to turbulence,Dxis the cross-diffusion,r_k;r_xare the turbulent Prandtl number forj andx, respectively,l_tis the turbulent viscosity.

A detailed discussion of the above equations is given in Ref. [29]. Double precision and second-order upwind is used to compute the flow equations for spatial discretization of the con- vective terms [30]. Central-difference and second-order accuracy are used for the diffusion terms. The Coupled algorithm is used for the pressure–velocity coupling with the pseudo transient option which is a form of implicit under-relaxation for steady- state cases.

The mesh used in this work is based on a previous study done by Oneissi et al. [22] where a nonuniform polyhedral mesh was used. All solid surfaces (walls and VG surfaces) are refined with ten inflation layers with 40lm first layer thicknesses in order to obtain acceptabley^þvalues which should be less than 4 to ensure accurate modeling of the near wall region. Mesh independency is carried out for an empty channel turbulent flow with Re¼21,600.

The characteristics for the mesh sensitivity are presented in Table1.

The procedure conducted to reach the mesh sensitivity conver- gence is the following: arbitrary maximum element size is selected then decreased by a factor of 1.3 until the percent error based on both global Nusselt number and friction coefficient Fig. 2 Delta winglet pair (or IPWP-0), IPWP-30 and IPWP-60 geometric dimensions and

parameters

reaches a value less than 2%. Thus, mesh-3 is used for all the sim- ulations. Also, mesh validation with the experimental results are already studied and presented in Oneissi et al. [22] for the same reference configuration. The grid convergence index (GCI) is also calculated to access the uncertainty of discretization due to mesh sensitivity. As shown in Table1, the GCI is around 0.52% for the most refined mesh confirming the accuracy of the present numeri- cal simulations.

Results and Discussion

Quantitative Parameters and Experimental Validation.The evaluation of the thermal performance for the different cases is divided into two main categories, global and local analyses.

Global Nusselt number is given by Nu¼D_hh

k ¼2Hh

k (6)

where h¼mC_ _pðT_oTinÞ=A_fðT_sTavgÞ is obtained from the energy balance on a control volume enclosing the channel, and Tavg¼ðT_oþTinÞ=2 is the average temperature between the inlet and outlet.Afis the heat exchange surface area, i.e., area of the heated upper and bottom walls.

Global friction factor is given by f ¼ D_hDP

2LqU²¼ HDP

LqU² (7)

The thermal enhancement factor (TEF) is given by TEF¼ Nu

Nu0

f f0

1=3

(8) where Nu0is the global Nusselt number for empty channel, Nu is the global Nusselt number for channel with VG,f0 is the global friction coefficient for empty channel, andfis the global friction coefficient for channel with VG.

Local Nusselt number at a given xlocation in the channel is given by

Nux¼D_hh_x

k ¼ 2Hq⁰⁰_x

kðT_sT_x;bÞ (9)

whereq⁰⁰_xis the span-averaged wall heat flux at locationx.

The local bulk temperature is obtained as follows:

Tx;b¼ 1 UA

ð

A

uTdA (10)

Local friction at a givenxlocation in the channel is given by

f_x¼ DP Pdynamic

¼2ðPinP_xÞ

qU² (11)

The results for the laminar flow case are validated for global Nusselt number using Stephan correlation [31]. It is the integra- tion of the local Nusselt number (Nux) on the channel length, and it is defined based on the bulk temperature at each positionTx;b. This correlation is valid in the range 0:1Pr1000 for parallel plate channels.

The local Nusselt number is given by Nu¼D_hh

k ¼ 2Hq⁰⁰

kðT_sT_x;bÞ (12)

whereq⁰⁰is the heat flux at positionxand Tb¼ 1

UA ð

A

uTdA (13)

Nu0x¼1 x

ðx 0

Nuð Þ_xdx (14)

and

Nu0x¼7:55þ 0:024x^1:14

1þ0:0358x^0:64Pr^0:17 (15) where

x¼ x

PeD_h¼ x

Re_DhPrD_h (16)

The local Nusselt number is expressed using the following ana- lytical correlation, derived from the Ref. [31], based on the bulk temperature

Nux¼7:55þ0:0240:0179x^1:78Pr^0:170:14x^1:14

½1þ0:0358x^0:64Pr^0:172 (17) The relation that combines the Nusselt number based on the bulk temperature (Nu_x) and that based on the entrance temperature ðNu_eðxÞÞis

Nu_{e x}ð Þ¼Nuxe

kNux Dh:_m Cpx

(18) Table2represents the values and percent relative error of the global Nusselt numbers obtained from the present simulations and compared to correlation of Stephan [31] for laminar flow.

A second approach is adopted to validate the numerical results for the case with VG. This approach depends on the proximity of the simulation results to Tiggelbeck’s experimental results [32].

Table 2 Comparison of the global Nusselt number between present simulation and Stephan’s correlation [31]

Reynolds number Simulation Stephan’s correlation [31] Error (%)

270 7.85 8.42 6.80

540 9.41 9.26 1.60

1080 11.58 10.78 7.40

Table 1 Characteristics of the mesh sensitivity analysis

Mesh 1 2 3

Maximum element size (mm) 2.30 1.70 1.30

Wall sizing (mm) 1.50 1.15 0.90

Type of mesh Polyhedral Polyhedral Polyhedral Number of elements 815,265 1,344,799 2,572,124

Inflation thickness (lm) 20 30 40

Number of inflation layers 18 12 10

Maximumy^þ 0.90 0.80 0.80

Error (%) — 2.00 1.00

GCI (%) — 1.07 0.52

Table 3 Normalized global Nusselt number and friction factor comparison

Simulation Experiment [32] Error (%)

Nu=Nu0 1.56 1.49 4.70

f=f0 1.95 1.91 2.00

TEF 1.25 1.20 4.20

Tiggelbeck et al. [32] in their experimental study found that the values of the normalized global Nusselt number and friction factor at Re¼4600 are equal to 1.49 and 1.91, respectively. The same channel dimensions, boundary conditions, and postprocessing of Tiggelbeck’s et al. [32] experiment are computed in this study.

Table 3 compares the numerical results obtained here to those obtained experimentally by Tiggelbeck et al. [32] at Re¼4600.

The first term in this table corresponds to the normalized Nusselt number, the second one is the normalized friction factor, and the third term corresponds to the thermal enhancement factor, TEF which is defined in Eq.(8). The results are in a good agreement with a maximum percent relative error less than 5%. This ensures that the present numerical model is reliable to predict the flow and heat transfer characteristics.

Since numerical simulations are close to ideal experiment, exhibited discrepancy between numerical simulations and experi- mental work can be mainly caused from the setup process differ- ence between numerical simulation and experiment, in addition to data acquisition uncertainties and other factors.

Global Performances.The results obtained from more than 100 simulations covering a wide range of Reynolds numbers and different inclined projections of the VGs are plotted using RSM.

The Nusselt number response surface representing the heat transfer due to IPWP different configurations is presented in Fig.3. This figure shows a monotonic increase in Nusselt number as a function of Reynolds number. The Nusselt number increases from a value around 30 for Re¼5000, 50 for Re¼10,000 and 127 for Re¼30,000. Otherwise, the Nusselt number is almost independent of the inclination angle, one can see nearly no change in the Nusselt number value when varying the inclination angle from 0 deg to 60 deg, i.e., the IPWP configuration does not increase the heat transfer compared to DWP configuration.

Figure4presents the RSM for the friction factor as a function of Reynolds number and angle of inclination (a). The plot clearly shows that the friction factor decreases as Reynolds number increases. Very high values of friction factor are depicted for rela- tively low laminar flow reaching a value of 0.16 at Re¼270 and sharply decreases to reach a value of about 0.05 at Re¼2800. For turbulent flow, the friction factor decreases gently from about 0.04 at Re¼4600 until 0.025 at Re¼30,000.

In addition, Fig.4also shows that the friction factor is so sensi- tive to the VG inclination (a). Friction factor decreases as VG inclination (a) increases; this is further inspected locally in the

local performance section. The reason behind the diminishing of the friction factor with the increase of inclination angle is due to the more aerodynamic profile of the VG. When increasing inclina- tion angle (a) (or decreasing the angle of incidence), profile or parasitic drag decreases which is a combination of the pressure or form drag and the skin friction drag. The pressure drag is caused by the separation flow from the VG body, while skin friction drag is caused due to the viscous shear boundary layer over the VG body. When inclining the VG body, while maintaining the same frontal area, the pressure drag tends to decrease due to the more aerodynamic form, while the skin friction drag tends to increase due to the increase in immersed body surface area. The decrease of the form drag is more dominant than the increase of skin drag, leading to global decrease in the drag thus lower friction factor.

In addition to the Nusselt number and friction factor graphs, Fig. 5shows the response surface methodology performance of the IPWP based on the thermal enhancement factor (TEF defined in Eq.(8)) criterion as a function of Reynolds number and inclina- tion angle (a). The RSM plot shows that the TEF increases rapidly

Fig. 3 Response surface methodology representing the Nus- selt number of the IPWP function of Reynolds number and incli- nation angle (a)

Fig. 4 Response surface methodology representing the fric- tion factor of the IPWP function of Reynolds number and incli- nation angle (a)

Fig. 5 Response surface methodology representing the ther- mal enhancement factor of the IPWP function of Reynolds num- ber and inclination angle (a)

with the increase of Reynolds number to reaches its maximum around Re¼5000 zone. Then, TEF decreases gradually in slower manner to reach ineffective thermal enhancement after a Reynolds number about Re15,000. Clearly, the zone of intensification for Reynolds number appears to be in the range (500–15,000).

In addition, the RSM illustrates that the TEF is quietly affected by the inclination angle of the vortex generator reaching its maxi- mum value in the region between 30 deg and 35 deg. It can be shown clearly that the maximum TEF for all the configurations studied is achieved in the near region of Re¼5000. The global maximum TEF for the RSM, reaching a value of 1.33, is attained at an inclination angles between 30 deg and 35 deg at Re4600.

Local Performance.After demonstrating the performance of the IPWP with various angles of inclination (a) from a global point of view, this section illustrates the performance from a local point of view leading to better understanding of the flow charac- teristics. The condition Re¼4600 is chosen to illustrate the local performance, due to its domination from the global performance study.

From a local scope, the performance of the IPWP with various inclinations is demonstrated throughout this section, Figs. 6–8,

which represent, respectively, the normalized Nusselt number, the friction factor, and the local helicity for four configurations of IPWP (30 deg, 40 deg, 50 deg, and 60 deg) compared to DWP case. All VGs possess the same leading edge with the bottom fin and signified on each figure by the lower filled triangle on thex- axis located at a distancex¼0.06 m from the channel inlet. The other vertical lines with an upper filled triangle are the VGs trail- ing edges attached to the upper fin and denoted by the line type of each configuration.

Figure 6 shows the streamwise evolution of the normalized spanwise-averaged local Nusselt number throughout the channel (Eq.(9)). It can be seen from this figure that the normalized Nus- selt numbers do not show any enhancement through the channel for the IPWP configurations compared to the reference case DWP, except for IPWP-30 which exhibits higher values most of the time. The DWP configuration shows a smooth increasing curve for the Nusselt number over the channel length, while inclined configurations behave in a fluctuating manner. These fluctuations are due to the multiscale vortices interaction which sometimes may built up and other times attenuate each other. Table4repre- sents the normalized Nusselt number area weighted average values calculated from VG leading edge to the end of the channel.

It can be seen that IPWP with inclinations in the range of 25 deg to 35 deg have the highest values of averaged Nusselt numbers, that represents an increase of 3.1% compared to DWP configura- tion. In addition, further increase of the inclination angle is seemed to be disadvantageous for the Nusselt number, since from an inclination of 40 deg the Nusselt number is diminished com- pared to DWP case. This Nusselt number decrease follows a mon- otonic tendency: the higher the inclination angle, the lower the heat transfer.

Normalized friction coefficients are illustrated in Fig.7. DWP friction factor peak is severely suppressed when using IPWP con- figurations, showing a decrease by 50% for the IPWP-30. It can be shown that the more the inclination angle increases, the more the friction factor decreases, to reach about a 70% peak decrease for the IPWP-60. The decrease in the friction factor while the inclination angle increases, is also observed along all longitudinal positions. Table5gives the peak and averaged values of the nor- malized friction factors of different VG configurations in addition to the percentage differences of these values compared to DWP configuration. Note that the friction factor average values are cal- culated from VG leading edge to the end of the channel. The area weighted average friction factor of DWP case is reduced by 18%

for IPWP-30, reaching 43% decrease for the IPWP-60. The decrease of the friction factor with the increase of the inclination Fig. 6 Normalized Nusselt number for the cases DWP and

IPWP with inclinations Fig. 8 Local helicity for the cases DWP and IPWP with

inclinations

Fig. 7 Normalized friction factor for the cases DWP and IPWP with inclinations

angle is a monotonic function under the studied operating conditions.

The investigation of the flow effect is carried out by the study of helicity in this work, in order to remove the effect of high vor- ticity values near the walls. Helicity of IPWP configurations with four inclination angles and also DWP case is demonstrated in Fig. 8. It shows that IPWP-30 configuration submits the highest helicity almost downstream the VG, always higher than DWP case. IPWP-60 provides the lowest values, lower than DWP case

and IPWP-40 and IPWP-50 shows an increase of the helicity downstream the VG compared to DWP case over a short distance, untilx0.3 m.

Table6shows the helicity profiles averaged from VG leading edge located at a distancex¼0.06 m to the end of the channel for the different VG configurations, in addition to the percentage dif- ferences of helicity compared to DWP case. IPWP-30 provides the highest helicity profile along the channel of about 16.8 m/s², with an augmentation of 35.5% it dominates over all other Table 4 Average normalized Nusselt number of different VG configurations and the percentage difference compared to DWP case

DWP IPWP-25 IPWP-30 IPWP-35 IPWP-40 IPWP-45 IPWP-50 IPWP-55 IPWP-60

Average value 1.94 2.00 2.00 2.00 1.88 1.84 1.74 1.68 1.58

Difference (%) — 3.1 3.1 3.1 3.1 5.2 10.3 13.4 18.6

Table 5 Normalized friction values of different VG configurations and the percentage difference compared to DWP case (peak value and average value)

DWP IPWP-25 IPWP-30 IPWP-35 IPWP-40 IPWP-45 IPWP-50 IPWP-55 IPWP-60

Peak value 5.25 2.94 2.60 2.34 2.13 1.98 1.82 1.71 1.62

Difference (%) — 44 50 55 59 62 65 67 69

Average value 2.51 2.06 1.96 1.90 1.77 1.67 1.57 1.49 1.42

Difference (%) — 18 22 24 29 33 37 41 43

Table 6 Helicity profiles averaged from VG leading edge located at a distancex50.06 m from inlet of different VG configurations and the percentage difference compared to DWP case

DWP IPWP-25 IPWP-30 IPWP-35 IPWP-40 IPWP-45 IPWP-50 IPWP-55 IPWP-60

Average value 12.4 16.3 16.8 16.2 14.9 13.8 12.3 10.9 9.0

Difference (%) — 31.5 35.5 30.6 20.2 11.3 0.8 12.1 27.4

Fig. 9 Velocity streamlines for the IPWP-30, IPWP-40, IPWP-50, and IPWP-60 at Re54600 at different planesx/Hbetween 2.6 and 11.7

configurations in this aspect. From this inclination angle, when inclination angle increases averaged vorticity rapidly decreases until reaching lower values than for DWP case from IPWP-50.

The decrease attains 27.4% for the IPWP-60 relative to DWP performance.

Flow Structure.From the global and local performances stud- ies in the “Global Performances” and “Local Performance” sec- tions, it is obvious that the IPWP-30 and IPWP-35 configurations show performances that are prominent especially at Re4600. A study of the flow structure is established in this section at this Reynolds number to better understand the heat transfer process linked to the streamwise vortices.

According to Jeong and Hussain [33], a vortex corresponds to a region where two eigenvalues of the symmetric part of the square of the velocity gradient are negative; this is the concept of k₂ parameter which is used here in order to locate the streamwise vortices. For further details concerning the vortex identification using thek₂criterion, reader may refer to Jeong and Hussain [33].

Transversal section planes are performed to show the velocity streamlines andk₂contours in different locations downstream the VG. Four configurations that visualize the differences among are selected in this study,a¼30 deg, 40 deg, 50 deg, and 60 deg.

Figure 9 shows the velocity 2D streamlines for the IPWP-30, IPWP-40, IPWP-50, and IPWP-60 configurations at different longi- tudinal planes. The 2D streamline of a 3D flow are obtained from the secondary velocity components inyandzdirections. This repre- sentation is useful to better understand the flow structures.

Figure 10shows the k₂ contours for the IPWP-30, IPWP-40, IPWP-50, and IPWP-60 configurations at different longitudinal

planes. Figures 9 and 10 enable to see the high intensity and multiscale vertical structure predominance for the IPWP-30 over other configurations. Figures9and10also enable to high- light the vortices location inside the channel for the IPWP configurations.

The interaction among the multiscale vortices created by the IPWP-30, leads to the permanence effect of the vortices in the downstream direction. Figures 9 and10 show that the vortices produced by IPWP-30 maintain their intensity for further distance compared to other inclined configurations.

In addition, Fig.9illustrates that the number of multiscale vor- tices for IPWP-30 reaches eight (two main and six induced vorti- ces) at a distance x/H¼11.7. IPWP-40 and IPWP-50 reach a number of four vortices (two main and two induced vortices), while IPWP-60 decreased to only two vortices at a distance x/H¼11.7. Table7summarizes the number of primary and sec- ondary vortices developed for each IPWP configuration. Reader may refer to Oneissi et al. [22] for further detailed description of the multiscale interaction mechanism among vortices.

Fig. 10 Vortex core detected byk2criterion for the IPWP-30, IPWP-40, IPWP-50, and IPWP-60 at Re54600 at different planes x/Hbetween 2.6 and 11.7

Table 7 Primary and secondary vortices for IPWP configura- tions (for a pair of VGs) atx/H511.7

Primary vortices Maximum secondary vortices

DWP 2 2

IPWP-30 2 6

IPWP-40 2 2

IPWP-50 2 2

IPWP-60 2 0

Conclusion

Several three-dimensional numerical simulations for more than ten configurations of vortex generators are performed to analyze heat transfer enhancement in parallel plate-fin heat exchanger.

Laminar flows are computed for Reynolds numbers 270, 540, and 1080, while turbulent flow is modeled with using thej–xSST model and validated with correlations and experimental data. The DWP configuration studied experimentally by Tiggelbeck et al.

[32] was used to validate the results obtained from the present simulations.

The objective of this work is to identify the optimum inclina- tion angle configuration that enhances the heat transfer and reduces the pressure drop. IPWP configurations with different inclination angles (25 deg, 30 deg, 32 deg, 35 deg, 38 deg, 40 deg, 45 deg, 50 deg, 55 deg, and 60 deg) are then studied by manipulat- ing the winglets’ geometry and orientation.

In order to explain the effects of inclination angle on heat transfer and pressure drop, RSM is used from a global scopes. Local performance is weighed based on the stream- wise distribution of Nusselt number and friction coefficient criteria in addition to helicity. It is found that the IPWP with inclination angle in the range of 30–35 deg bestow better global performance than all other inclinations. The peak fric- tion magnitude of IPWP-30 configuration is 50% less than that recorded by the DWP and further inclination of the VG may reach about a 70% peak decrease for the IPWP-60 case.

The area-weighted average friction compared to DWP case decreases 18% for IPWP-30 configuration, reaching about 43% decrease for the IPWP-60 case. IPWP-30 configuration exhibits the maximum intensification of averaged helicity with an augmentation of 35.5% compared to DWP case. In addition, it is illustrated that the number of multiscale vorti- ces for IPWP-30 configuration reaches eight (two main and six induced vortices) while for other configurations it may suffice with two main vortices.

To summarize, it is found that the thermal enhancement factor was increased up to 6% for IPWP-30 and IPWP-35 vortex genera- tors compared to DWP case. Such study highlights the optimiza- tion for IPWP involved in the convective heat transfer intensification, based on TEF, by varying the VG’s inclination angle.

Nomenclature

A¼cross-sectional area, m² Af¼heat transfer surface area, m²

B¼channel width, m

Cp¼specific heat at constant pressure, J kg¹K¹ Dh¼hydraulic diameter, m

f¼friction factor

h¼convective heat transfer coefficient, W m²K¹ H¼channel height, m

k¼thermal conductivity, W m¹K¹ L¼channel length, m

_

m¼ ¼mass flow rate, kg s¹ Nu¼Nusselt number

Pe¼Peclet number¼Re Pr Po¼Poiseuille number

Pr¼Prandtl number q¼heat flux, W m² Re¼Reynolds number

s¼distance between tips of winglet pair, m Tb,x¼bulk temperature at positionx, K

Ti¼inlet temperature, K To¼outlet bulk temperature, K

u¼flow velocity inxdirection, m s¹ U¼mean flow velocity, m s¹

v¼flow velocity inydirection, m s¹ w¼flow velocity inzdirection, m s¹

Ts¼surface temperature, K DP¼pressure drop, Pa Greek Symbols

l¼dynamic viscosity, Pa s ¼kinematic viscosity, m²s¹ q¼fluid density, kg m³

x¼specific turbulence energy dissipation rate, m²s³ Abbreviations

CFD¼computational fluid dynamics DWP¼delta winglet pair

IPWP¼inclined projected winglet pair RSM¼response surface methodology

SST¼shear-stress transport TEF¼thermal enhancement factor

VG¼vortex generator

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