1. Introduction
Fin-and-tube heat exchangers (FTHEs) find numerous applications such as in automobile radiators, air conditioning, petrochemical industries, and many more. Conventional heat exchangers suffer a major drawback of high space requirements and low airside heat transfer coefficient. Several heat transfer enhancement techniques, viz. the use of nanoparticles, corrugated surfaces, elliptical tubes, wavy fin, and vortex generators in the flow field, have been reported in the last few years to overcome these setbacks and increase the overall thermo-hydraulic performance of heat exchangers.
Recently, the application of vortex generators (VGs) to increase the heat transfer coefficient on the airside has become popular to enhance HEs’ efficiency [
1,
2]—these VGs produce intense longitudinal vortices without many frictional losses [
3]. Actually, VGs installed on the airside increase heat transfer by redirecting the main fluid flow toward the recirculation zone. However, the increase in heat transfer is accompanied by enhanced pressure losses and this sets the foundation for the optimization of vortex generator geometry (in our case, curved trapezoidal winglets). VGs can be arranged in common flow-up and flow-down configurations; the former is known to be more efficient in accelerating the flow and delaying flow separation [
3,
4,
5,
6,
7,
8]. The tubes can be arranged in an inline or staggered array arrangement in an FTHE [
9,
10]. The shape of the tubes in an FTHE is generally circular, but elliptical tubes have also been reported in the open literature [
5].
The geometry of the VG plays a major role in heat transfer by creating additional turbulence in the flowing fluid downstream. A comparative study was performed by [
11] between curved rectangular winglets, sinusoidal wavy rectangular winglets, and rectangular winglets in an FTHE, and the authors reported that wavy rectangular VGs provided the highest heat transfer enhancement. In an FTHE, the fin pitch, longitudinal tube pitch, transverse tube pitch, length of vortex generator, the height of vortex generator, and attack angle of the delta winglet type VG were found to be the most important geometric entities in the view of thermal performance [
12].
Apart from the straight profiles of the winglet, curved profiles have also gained popularity [
13,
14]. An experimental investigation was reported on the comparison of the shapes of the VG, viz. rectangular, delta, and trapezoidal with straight as well as curved profiles, by Zhou et al. [
15] and via numerical simulation by Lu et al. [
16] and Esmailzadeh et al. [
17]. They reported that curved profiles are more promising to increase the thermal performance of the HE. In addition, a similar study was reported with punched holes near the base in VGs, wherein the pressure losses were reduced with a punched hole configuration [
18,
19,
20]. The effect of the position and geometry of a curved VG in an FTHE was studied by Oh et al. [
21]. A CFD study on an FTHE having a curved rectangular VG and staggered tube arrangement was reported by Gong et al. [
19], with an aim of predicting the optimum design parameters, viz. circumferential position, radial position, base length, edge height, and fin spacing. Apart from a surface-mounted VG, some modified fin structures such as brazed plate HE [
22], double-pipe HE [
23], finned tube [
24], and microchannel heat sink [
25] have also been reported. One such surface modification in the form of arc-shaped baffles, studied and optimized by Promvenge et al. [
26], reported that a 90° angle of arc-shaped baffles and a pitch ratio of 8 offered the highest heat transfer enhancement. The impact of tube diameter and fin pitch on the Nusselt number (
Nu) in an FTHE with a punched delta winglet VG and punched rectangular VG was investigated by Wu et al. [
27] and Tian et al. [
28]. The experimental results revealed that the fin pitch had a more prominent effect on
Nu compared to the tube diameter.
Apart from analyzing the impact of variations in an individual design parameter, multi-objective optimization studies have also been reported in the open literature. The latter is advantageous to achieve the global optimized value(s) of the design parameters to enhance the overall thermal performance of the HE. Different optimization techniques have been proposed and used by various researchers to achieve the goal, viz. RSM (response surface methodology) [
29], ANN (artificial neural network), and GA (genetic algorithm). An optimization study utilizing RSM was reported by [
30] for the optimization of the Ranque-Hilsch vortex tube refrigerator and found that the number of intake nozzles and orifice diameter largely influenced the rate of heat transfer. Hosseinirad et al. [
31] performed a CFD investigation for the analysis of wavy VG and used RSM for its optimization. Kanaris et al. [
32] employed RSM and developed a correlation between the design parameters and response variables based on the CFD results, and later on predicted the optimal design for the plate HE. Khaljzadeh et al. [
33] performed a similar study using RSM on vertical ground HE.
A comparative study between the RSM- and ANN-based optimization techniques applied in a V-down rectangular channel was reported by Chamoli et al. [
34]. It was further stated that RSM is more accurate in predicting, with a maximum deviation of 5%. Apart from RSM, several researchers performed combined analyses using CFD, RSM, and GA. For instance, Lemouedda et al. [
35] optimized delta winglet VGs in a fin-and-tube HE for an optimum VG shape and attack angle. A similar investigation was performed by Sun et al. [
36] but for elliptical tubes instead of circular ones. Gholap et al. [
37] optimized the energy consumption and material cost in an HE for refrigerators. The effect of fin spacing, the ratio of longitudinal to transverse tube pitch, and the attack angle of the delta winglet in an FTHE were studied by Wu et al. [
38] using RSM, and optimized using GA. Similar optimization techniques have been used by several authors, for instance, for the analysis and optimization of a novel helical coil tube in a tube HE by Liu et al. [
39], and in a conical strip VG by Zheng et al. [
40]. A comparative study between RSM and direct optimization (DO) was performed by Salviano et al. [
41] to optimize the VG angle and position in an FTHE and they also studied the influence of VG roll angle on heat transfer. They reported that the DO approach delivered better results than the RSM approach.
The artificial neural network (ANN) approach has gained immense popularity in the last few decades due to its capability to tackle non-linearity with great accuracy [
42,
43]. The work of Haykin [
44] and Eeckman [
45] provides a detailed history and mathematical background of neural networks. Berber et al. [
46] used ANN for the prediction of heat transfer in a circular tube. A double-pipe HE with delta-wing-type inserts was studied by Kocyigit et al. [
47] and Khan et al. [
48] and they used ANN to validate the results with experimental data. Several authors used ANN as a surrogate model and GA for optimization. For instance, CFD, ANN, and GA have been clubbed together for the optimization of wavy fin-and-tube HE by Li et al. [
6], who reported that an increase in corrugation height and angle of attack have a positive impact on the heat transfer. Additionally, a wavy fin height of 1.8 mm and an attack angle of 60° produced a maximum heat transfer effect. In a similar study, Abdollahi et al. [
3] optimized the winglet geometry. Khan et al. [
49] studied the trapezoidal and rectangular VG, attack, and attach angle in a plate-fin channel and concluded that an attack angle between 45 and 60° and an attached angle of 60° are the most suitable for maximum heat transfer and minimum pressure losses. Song et al. [
8] studied a wavy fin with a circular tube HE equipped with a curved VG, and a maximum enhancement in heat transfer of 30.6% was achieved. Instead of surface modifications, some researchers opted for an active heat transfer enhancement method, introduced nanofluid for increasing the heat transfer performance [
50], and achieved a maximum enhancement of 82%. Ke et al. [
51] considered winglet attack angle for optimization to enhance the performance of the HE.
From the literature, it can be confirmed that numerous experimental and numerical studies have been performed to analyze and ultimately enhance the heat transfer performance of heat exchangers equipped with vortex generators. The experimental and numerical studies on curved winglets [
52,
53] are limited and the scope for multi-objective optimization is vast. Thus, the present focus is on a multi-objective genetic algorithm (MOGA) with an intention of increasing the heat transfer capability and minimizing the pressure losses, and thus generating a set of optimal data points (Pareto fronts) that best satisfies the objective.
Although RSM/ANN has been used in some of the multi-objective studies, the present work incorporates both of these approaches (separately) for establishing a relation between design parameters and objective function and then applying MOGA to compute Pareto fronts. Although Sarangi et al. [
54] performed multi-objective optimization of the curved trapezoidal winglets, they considered only three parameters for optimization, viz. trailing edge height, arc radius, and the angle subtended on the tube center. Apart from these parameters, the present work accounts for the winglet front edge height and the eccentricity of winglets for analysis. Furthermore, the most sensitive parameters are considered for optimization.
To summarize, the theme of the present study is to enhance the overall thermal performance of the FTHE by optimizing the curved trapezoidal winglet geometry. To achieve this, the work is planned as follows: first, the flow domain is modeled using curved trapezoidal winglets on alternate fin walls placed on either side. Second, different levels of mesh are generated and tested to ensure the results are grid-independent, followed by a validation study to justify the solver settings and boundary conditions used. Third, before proceeding with optimization, the critical parameters having a significant effect on the performance of the HE are identified. Fourth, based on these parameters, the design of the experiment (DoE) is prepared using the factorial design method. Fifth, we make use of response surface methodology as well as the Bayesian-regularized ANN surrogate model as the two separate surrogate models. Sixth, a multi-objective optimization study is performed using GA to maximize the heat transfer and minimize the pressure losses, and two separate Pareto fronts are generated using RSM and ANN surrogate models. Finally, a few Pareto fronts, selected from both of the surrogate models, are selected and compared with the CFD results to measure the relative error and explore the flow physics of the optimized models.
2. Model Description
2.1. Geometry of Fin-and-Tube Heat Exchanger
In the present work, we used the HE geometry proposed by Joardar and Jacobi [
55] as the reference model—the same was also used for validation. Different experiments were carried out by Joardar and Jacobi [
55] on HEs, having seven inline rows of tubes. The first HE did not incorporate any winglets and this model was taken as the baseline model (BLM) here. They also considered one and three pairs of the delta winglets mounted on one and three (alternate) tubes, respectively, and these models were compared to the optimized models. Regarding the geometrical details, the tubes have a diameter (
D) of 10.67 mm, transverse pitch (
Pt), and longitudinal pitch (
Pl) of 25.4 mm each. The length (
L) of the FTHE was 177.8 mm, the fin pitch (
Fp) was 3.63 mm, and the fin thickness (
t) and winglet thickness were 0.18 mm each. Instead of delta winglets—as taken by Joardar and Jacobi [
55]—we considered curved trapezoidal winglets mounted alternately on either side of the fin surface as well as on the alternate tubes. A detailed view of the geometry is presented in
Figure 1.
Figure 1a represents the geometry of the entire model. Owing to its symmetrical nature, we considered a small part of the entire domain (marked by the enclosure). Due to the repetitive nature of the geometry about the plane containing the fin surface (cf.
Figure 1b), a periodic boundary condition was applied to these plates (with half the thickness of the original fin plate). The implementation of the two approaches largely reduced the size of the geometry, and hence the computational overheads. As mentioned earlier, the winglets were mounted on either side of the fin plate (cf.
Figure 1b) and the alternate tubes (cf.
Figure 1c).
2.2. Equations Used for Calculating the Performance of HE
The performance of an HE is expressed in a dimensionless form and computed based upon several parameters, in accordance with Joardar and Jacobi [
55], as follows:
Here,
is the mass flow rate in kg/s,
is the specific heat capacity in J/kgK,
and
are the area-weighted average temperature at the outlet and inlet, respectively, in Kelvin (
K),
is the wall temperature,
is the density of air in kg/m
3,
is the inlet velocity in m/s,
is the kinematic viscosity in Pa·s,
and
are the pressure at the inlet and outlet, respectively, in Pascal (Pa), and
is the Prandtl number defined as the ratio of kinematic viscosity (
μ) and thermal diffusivity (
α).
B is the width of the computational domain and is equal to 25.4 mm. The Colburn factor (
j) was used to calculate the heat transfer and the friction factor (
f) was used to represent the pressure drop. The enhancement factor, given by Equation (11), was adopted from Webb and Kim [
56], where
j and
f are calculated based on the selected Reynolds number (
Re), and
jo and
fo are reference values taken from the baseline model under similar operating conditions. Using these reference values in the equation makes it a non-dimensional parameter, useful for calculating the rating.
2.3. Description of Numerical Simulation
Across the last few decades, computational fluid dynamics (CFDs) have gained tremendous popularity and have been widely used as a robust tool for thermal and fluid flow analysis [
57]. CFD is a well-proven numerical technique and the best alternative to experimentations due to significantly fewer capital investments, higher accuracy, and lesser time requirement. If performed carefully, the CFD calculations are as good as the experimental results. Furthermore, with CFD, we have the advantage to visualize and extract the flow field details at any time. It also provides the flexibility to make changes in the geometry during design optimization with no additional cost factor involved, other than the power consumption by the hardware.
The present study makes use of Menter’s transition shear stress transport (SST) model—a four-equation model—together with curvature correction for modeling turbulence. This transition model utilizes the two equations of the SST turbulence model along with two additional transport equations, viz. gamma (
γ) for determining intermittency and
Reθ for determining laminar–turbulent transition process determination [
58].
The following governing equations are solved for the fluid zone:
Here, E signifies the total energy, and the effective thermal conductivity is given by = + , where is the turbulent thermal conductivity.
The transition SST model uses the standard SST model as the base, given in terms of the turbulent kinetic energy (
k) and specific rate of turbulence dissipation (
ω), as follows:
The two transport equations relating to the intermittency and transition momentum thickness Reynolds number, coupled with the original SST model, i.e., Equations (15) and (16), respectively, are given as follows:
The energy equation in the solid regime reads as follows:
Here, refers to the conductivity matrix.
For further details on the production term (
Pk), destruction term (
blending function (
F1), the turbulent viscosity (
µt), and all of the other terms, including the details of the model constants, we referred to [
59,
60].
2.4. Discretization of the Solution Domain and Solver Settings
The first step in CFD is to discretize the entire domain into small control volumes (cells) over which governing equations are solved to generate a flow field. In the present work, we used hexahedral cells to discretize both fluids as well as solid domains. Ansys workbench was used to generate the Cartesian mesh over the entire volume and prism layers were provided over the cold tube walls to accurately capture the temperature gradients. The Cartesian cells were orthogonal, and hence, of high quality—they quickly filled up the entire domain and were trimmed off to conform to the solid boundaries. Furthermore, these cells are less diffusive in nature compared to the tetrahedral cells. Local grid refinement was conducted on the winglet walls using the octree method.
Figure 2 shows the mesh details over the fins, cooling tubes, and VGs. The fin walls were made separately in ICEM CFD and appended as a case file in FLUENT, and interfaces were made between the solid and fluid domains manually so that conjugate heat transfer could take place.
Next, we performed a detailed study on the dependency of grid count on the performance parameters of the HE. Seven different levels of mesh were considered for analysis, and the percentage deviation from the experimental results is shown in
Table 1. It becomes apparent that after the fourth level of mesh, the deviation is less than 3%—the closest agreement is observed with the finer mesh consisting of more than 1 million cells. To obtain a good balance between the accuracy and the computational time, the grid topology corresponding to 332 K cells was selected for all future work.
Air was taken as a working medium with a density of 1.225 kg/m3 and viscosity of 1.885 × 10−5 kg/ms. At the inlet, the velocity inlet boundary condition was used, and a uniform velocity profile was prescribed with a mean velocity of 2.14 m/s, corresponding to a Reynolds number of Re = 634.5 at a temperature of 310.6 K. On the outlet plane, a pressure outlet boundary condition was used with the gauge pressure amounting to zero, and for backflow, a normal to boundary method was quoted. The temperature of the cold tubes was kept constant at 291.77 K. For aiding the conjugate heat transfer between fluid, fin walls, and winglets, the coupled boundary condition was implied. A no-slip boundary condition (u⸱ = 0) was applied for the tube walls, fin surface, and winglets.
Referring to the schematic diagram of the HE (
Figure 1), and as discussed earlier, a symmetry boundary condition was applied to the fluid zone over the plane described in
Figure 1a—no flow takes place across these surfaces. Owing to the repeatability across the fin walls (cf.
Figure 1b), a periodic boundary condition was applied. The application of these boundary conditions together reduces the size of the computational domain significantly with no loss in accuracy. The SIMPLEC (semi-implicit method for pressure-linked equation consistent) method was applied for the pressure velocity coupling. Standard discretization was implied for pressure. The QUICK scheme was implied for the discretization of turbulent kinetic energy, intermittency, momentum, specific dissipation rate, and momentum thickness Re and energy equations [
61,
62].
2.5. Validation and Methodology
The numerical model was validated against the experimental data of Joardar and Jacobi [
55], consisting of a 3 VG pair array (cf. Figure 1, p. 1158).
Figure 3 shows the variations in the heat transfer coefficient with different Re against the experimental data. A close agreement exists between the numerical calculations and the reference data—a maximum deviation of 8.94% (only at the highest Re) and a minimum deviation of 0.46% are observed.
2.6. Selection Criteria of Geometric Entities for Optimization
The effect of radius, the angle subtended by the winglet on tubes, the winglet leading as well as trailing edge heights, and eccentricity provided to the winglets upon heat transfer and pressure drop were studied. Here, one parameter is varied at a time while imposing three different values (to all of the other geometrical entities), namely, lower, middle, and upper bounds (also, cf.
Table 2), as shown in
Figure 4. The conclusive results indicate the following:
With an increase in radius (r), both j and f values increase for the upper bound values of other parameters and this change is significant. For the middle bound values, the change in both j and f values is mild. At the lower bound, the j as well as f values decrease moderately until r = 9.5 mm and then increase with a further increase in r. This indicates the prominent impact of r at the upper bound values.
The j and f values both increase with an increase in θ at the upper bound until θ = 60°, followed by nearly no change in the j value while the f value still increases. A similar trend is seen for the j value at the middle bound—initially, it increases until θ = 60° and then decreases by a small amount, while the f value increases consistently. At the lower bound, the j value decreases until θ = 45° and then increases mildly with θ, whereas the f value first undergoes reduction up until θ = 45° and thereafter it is nearly a constant.
With an increase in the leading edge height (h1), at the upper bound both the j and f values increase. At the middle bound, the j values first increase until h1 = 2 mm and then remain constant while the f value continues to increase mildly. At the lower bound, the j value undergoes a more significant reduction than the f values do.
The values of j and f increase with an increase in trailing edge height (h2) at the upper bound and middle bound while this increase is significant at the upper bound. At the lower bound, the variations are quite small.
The effect of eccentricity on the j and f values is as follows: At the upper bound, the variations in the j values resemble a sinusoidal pattern of very low amplitude, while the f values elucidate a mild but continuous increment. At the middle bound for the j value, there is a slight increase up to e = 5 mm, followed by a decreasing pattern, and again a mild increase beyond e = 7 mm. The f values increase slightly with the eccentricity at the upper bound, while beyond e = 7 mm, a slight reduction is observed. However, in contrast to the upper bound results, at the lower bound, the f value (although small) reduces continuously with an increase in eccentricity values.
Figure 4.
Variation in j and f values with changes made in the design parameters.
Figure 4.
Variation in j and f values with changes made in the design parameters.
Table 2.
Settings for the DoE.
Table 2.
Settings for the DoE.
Variable | Quantity | Lower Bound | Upper Bound |
---|
r | Winglet radius (in mm) | 8 | 11 |
| The angle subtended by winglet (in degrees) | 30 | 75 |
h1 | Winglet leading edge height (in mm) | 0.7 | 3.3 |
h2 | Winglet trailing edge height (in mm) | 0.7 | 3.3 |
A general trend shows that with an increase in the values of the design parameters, the heat transfer and pressure drop both increase, particularly at the upper bounds. This is due to the fact that for the upper bound values of the geometrical parameters, a large amount of fluid from the mainstream is directed toward the wake region and helps in improving the mixing of the fluid. At the lower bound values, there is a slight decrease in heat transfer since the smaller geometrical values may not be able to influence the flow field to the extent of the upper bound. Interestingly, with the middle bound option, the performance undergoes a change after a certain value (as discussed earlier in detail).
The conclusive results indicate that the eccentricity does not contribute significantly to the heat transfer process, whereas the other geometrical entities demonstrate a remarkable effect on the performance of the HE (particularly at the upper bounds and to a certain value in the middle bounds). Hence, we consider four geometrical entities, viz. the winglet radius, the angle subtended by the winglet, and the height of the leading edge and trailing edge of the winglets. The range of values considered for optimization has been represented in
Table 2.
2.7. Design of Experiments
Numerous researchers have performed optimization studies taking into consideration the effect of single design parameters while keeping the others fixed. This is a time-consuming and local approach, and the results obtained may not reach a globally optimized value. Therefore, the most widely accepted and adopted technique is to design the experiments in accordance with some standard methods. This approach ensures the participation of all of the chosen design parameters, and hence, is interactive in nature. The values of the dependent variables for each combination of independent variables are calculated by making use of either experimentations, numerical simulations, or (semi-)empirical relations. The next step is to fit the surface over these data to generate the equations, which are then subjected to optimization; hence, this step (to generate equation(s) prior to optimization) is often called a surrogate method. The accuracy of the results is highly dependent on the way that the experiment is designed and calculated, the method used for surface fitting, and the approach used for optimization.
Since the design space is large, a three-level factorial design—that divides each factor into three levels, viz. lower, middle, and upper levels—was used in the present study to analyze the non-linear interaction of the design variables. Considering the design parameters, a sample space consisting of 3
4 = 81 design points was utilized to capture the quadratic relationship for each design variable. As mentioned earlier, four design parameters were chosen for optimization, and the extent of their variation is listed in
Table 2. The chosen scheme fills the entire design space uniformly. The values of the Colburn factor (
j) and friction factor (
f) computed using CFD are listed for all of the design points in
Table 3—since there are a large number of runs, only the first nine and last nine points are shown.
4. Conclusions
The present study was undertaken to optimize the performance of the FTHE with curved trapezoidal winglets installed on alternate tubes to maximize j and minimize f. The critical design parameters were first identified before proceeding to DoE using CFD simulation. The radius (r), the angle subtended on cold tubes (θ), the front edge height (h1), and the trailing edge height (h2) emerged as the most influential geometric entities for optimization. Considering these parameters, the DoE was prepared using a three-level factorial design which cumulated to a total of 81 design points. The resulting data were used to train RSM and ANN. The R2 values for the j and f values were 97% and 90 %, respectively. The deviation in the ANN prediction from the target data was limited to ±3% of the target data. The RSM and ANN data acted as surrogate models for the genetic algorithm (GA) for optimizing the four independent design variables. GA provided two separate Pareto fronts for both surrogate models in a wide prospect.
Some Pareto fronts were selected from the models to evaluate the accuracy of Pareto with respect to the CFD simulations. The maximum deviation in j for Variant R and Variant A was 3.7 and 1.7%, respectively. Similarly, the maximum deviation in f for Variant R and Variant A was 6.45 and 3.12%, respectively. The percentage deviation for Variant A was less than Variant R and hence Variant A is more accurate than Variant R. Additionally, the enhancement factor of Variant A is higher than Variant R.
It was observed that with the increase in the winglet’s size, a larger amount of mainstream flow was diverted toward the thermal isolation zone, enhancing the thermal intermixing, and suppressing the large eddies in that zone. The winglet’s front edge height is always smaller than the trailing edge height.
The CFD study based on the selected models from Pareto fronts confirmed that the heat transfer increased significantly along with the reasonable expense of pressure losses. Thus, the Pareto fronts obtained in this study would be beneficial to academicians as well as researchers across industries. The optimized models were also tested under the off-design fluid flow conditions, wherein their performance was still better for all values of Re.