Thermal-hydraulic performance optimization of inline and staggered fin-tube compact heat exchangers applying longitudinal vortex generators

Accepted Manuscript Title: Thermal-hydraulic performance optimization of inline and staggered fintube compact heat exchangers applying longitudinal vortex generators Author: Leandro O. Salviano, Daniel J. Dezan, Jurandir I. Yanagihara PII: DOI: Reference:

S1359-4311(15)01315-0 http://dx.doi.org/doi: 10.1016/j.applthermaleng.2015.11.069 ATE 7350

To appear in:

Applied Thermal Engineering

Received date: Accepted date:

14-10-2015 19-11-2015

Please cite this article as: Leandro O. Salviano, Daniel J. Dezan, Jurandir I. Yanagihara, Thermal-hydraulic performance optimization of inline and staggered fin-tube compact heat exchangers applying longitudinal vortex generators, Applied Thermal Engineering (2015), http://dx.doi.org/doi: 10.1016/j.applthermaleng.2015.11.069. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Thermal-hydraulic performance optimization of inline and staggered fin-tube compact heat exchangers applying longitudinal vortex generators Leandro O. Salviano*1, Daniel J. Dezan*, Jurandir I. Yanagihara* * Department of Mechanical Engineering, Polytechnic School, University of São Paulo, Av. Prof. Luciano Gualberto, 380, São Paulo, SP, 05508-970, Brazil

HIGHLIGHT     

Simple approach based on trigonometric decomposition to treat the constraints. The optimum ratio between the vortex generator height and fin-pitch is 0.6. Heat transfer for staggered tube arrangement is more pronounced than for inline. The optimum vortex generator shapes are similar to rectangular-winglet. Optimum configurations are more influenced by tube arrangement than by Re number.

ABSTRACT Optimization is now recognized as essential in the design of modern compact heat exchanger since the augmentation of heat transfer applying longitudinal vortex generator depends on different interacting parameters such as its streamwise and spanwise locations, attack and roll angles and shape. For this work, an optimization procedure based on the SIMPLEX method was conducted through a fluid-solid conjugated heat transfer modeling considering a fin-tube compact heat exchanger with two rows of tubes in staggered and inline arrangements, in order to find an optimum configuration of the vortex generators for two objective functions related to Colburn factor (j) and Friction factor (f). Two Reynolds numbers (250 and 650), performing as a function of fin-pitch, were evaluated. Moreover, a simple approach based on trigonometric decomposition to treat the constraints is presented, which allows great flexibility to vortex generators to cover the design space solution. Seven independent input parameters for each vortex generator in optimization procedure were considered, totaling fourteen independent variables. The results indicate that this work optimized configuration for the vortex generators achieved higher heat transfer augmentation than those in previous works in the open literature, for both objective functions, Reynolds number and tube arrangements, and it is more pronounced for staggered tube arrangement than for inline tube arrangement. Moreover, suitable vortex generator shapes to maximize the objective functions are more similar to rectangular-winglet type than delta-winglet type, and the optimum ratio between the vortex generator height and fin-pitch is 0.6. Although several trends could be defined for the optimum points, optimized configurations of the vortex generators were found to be different for each Reynolds number, tube arrangement and objective functions, indicating strong asymmetry between the vortex generators to achieve higher heat transfer augmentation. Keywords: Heat exchanger, Heat Transfer Augmentation, Vortex Generator, SIMPLEX Method, Optimization

1. Introduction Compact Heat Exchangers are widely applied to several engineering fields, such as automotive and chemical industries, heating, residential air-conditioning and refrigeration. For applications in which the physical space available and weight are concerns, for example for automotive and aerospace industries, the search for solutions that could increase the ratio between heat transfer and pressure drop has become a key part during the thermal system development process. It is known that the heat exchanger performance could be enhanced providing a decrease of the airside convection resistance, which is usually dominant due to the thermo-physical properties of the air. Vortex Generators (VG), such as wing and winglet types, are passive enhancement technique able to provide this effect and to improve the heat transfer performance in fin-tube heat exchangers. In this passive technique, the heat transfer surface is intentionally modified to introduce secondary vortices into the main flow [1-3]. Although the surface total area of the heat transfer is not significantly changed due to the VG surface, the fluid flow dynamics may be strongly disturbed. In fact, VG not only disturbs the flow field and disrupts the growth of the boundary layer, but also causes fluid swirling and heavy exchange of the fluid core and walls, leading to enhancement of heat

1

Corresponding author. Email address: [email protected] (L. O. Salviano).

Page 1 of 30

transfer with small pressure drop penalty [3-5]. Therefore, to identify an optimal configuration for the geometric and spatial parameters of the VG that could provide higher heat transfer with loss pressure penalty is still a great challenge. Heat transfer augmentation by VG is affected by different factors, such as attack angle and roll angle [6-7], aspect ratio, spanwise and streamwise locations, tube arrangement and flow patterns. Lemouedda et al. [8] investigated the optimal attack angle for a delta-winglet VG based on the Pareto optimal strategy considering the inline and staggered tube arrangements for VG placed on common-flow-down configuration (CFD) and common-flow-up configuration (CFU). The results showed that the use of delta-winglets as enhancement devices increased the heat exchanger performance due to the increase of attack angle. Lei et al. [3] studied the effects of the VG aspect ratio and attack angle on hydrodynamics and heat transfer for a compact heat exchanger in staggered tube arrangement. They concluded that the heat transfer coefficient and Friction factor increased due to the increase of the VG attack angle and aspect ratio. Moreover, a better ratio between the heat transfer and pressure loss for VG aspect ratio 2 was found. Torii et al. [9] proposed a vortex generator arrangement able to provide simultaneous increase in heat transfer and reduction in pressure loss for a fin-tube heat exchanger for both inline and staggered arrangement, considering VG aspect ratio 2. Different levels of enhanced heat transfer for each arrangement were found. Jang et al. [10] conducted an optimization of the VG attack angle and transverse location by a simplified conjugate-gradient method for an inline and staggered tube arrangements with rectangular winglet type mounted behind the tubes. Zeng et al. [11] provided a comparative study of effects of the VG attack angle, VG length and VG height, fin material, fin thickness and fin pitch and tube pitch on fin-tube heat exchanger performance using the Taguchi method. A parametric analysis for inlet velocity range, indicating higher intensity of heat transfer due to increase of attack angle, length and height of the vortex-generator, followed by increase of pressure drop, was presented. Wu and Tao [12] studied the influences of the main parameters of the longitudinal vortex generator on the heat transfer enhancement and flow resistance in a rectangular channel, including the location of VG in the channel, geometric sizes and shape. As in Jang et al. [10], the researchers also analyzed their results from a parametric approach. In fact, only one VG parameter is evaluated while other important parameters were fixed and some conclusions were drawn from this particular case. The heat transfer enhancement was reduced with increase of VG distance from the channel inlet. The location of VG did not significantly influence the flow loss. Increasing the VG area, both the heat transfer and the pressure loss was increased. Hwang et al. [13] stated that the efficiency of the delta winglet vortex generator applied to tubular heat exchanger is also strongly affected by its size, shape and location on the fin surface. Salviano et al. [6] investigated the impact of VG parameters (spanwise and streamwise locations, attack and roll angles) on the Colburn factor and Friction factor, considering a fin-tube compact heat exchanger in staggered arrangement with VG aspect ratio 2. Moreover, they considered the VG parameters independent for each VG totaling eight independent variables. They indicated that each VG parameter influenced Colburn factor and Friction factor at different levels and important interactions among the first and the second vortex generators were identified. The researchers also reported an optimization procedure through Response Surface Methodology for the VG parameters indicating the independence of the VG parameters to reach maximum heat transfer augmentation for different objective functions. These conclusions were re-evaluated by Salviano et al. [7] considering two different approaches: Direct Optimization and Response Surface Methodology; indicating that the Direct Optimization presented a better performance than the Response Surface Methodology. This study also confirmed that the best ratio between heat transfer and pressure drop was obtained when the vortex generators were not symmetrical. In these studies, the VG aspect ratio was kept unchanged. He et al. [14] evaluated the impact of a rectangular winglet pair on heat transfer enhancement and pressure loss penalty for fin-tube heat exchangers. They claimed that this type of vortex generator could enhance the thermal mixing of the fluid, delay the boundary layer separation, and reduce the size of tube wake, indicating significant improvement in the heat transfer performance of the fin-tube heat exchangers with moderate pressure loss penalty. Undoubtedly, the heat transfer augmentation in compact heat exchangers through vortex generators has been successfully developed. However, finding optimal VG parameters to reach the maximum heat transfer with small pressure loss penalty is still a great challenge, especially due to the complex flow pattern [1,14,15] produced by longitudinal vortex generator. Thus, our work proposes an optimization of the VG parameters (attack and roll angles, streamwise and spanwise locations and shape related to VG rear and front height and chord) in order to increase the ratio between Colburn factor (j) and Friction factor (f). It is considered a fin-tube compact heat exchanger with two rows of tubes for staggered and inline arrangements, two Reynolds numbers and two objective functions which are separately optimized. Reynolds numbers of 250 and 650 were chosen because they represent two typical operating conditions for automotive radiators. A simple approach based on trigonometric decomposition is presented to treat the constraints, which allows great flexibility for the vortex generator to move along the fin in order to cover the solution design space better than the previous strategies used in former studies [6,7,10]. Therefore, seven input parameters for each VG are considered, totaling fourteen VG variables submitted to optimization procedure for two objective functions.

2. Methodology 2.1. Governing equations of the fluid flow and heat transport Ferrouillat et al. [16] and Fiebig [17] provided a broad discussion about the suitable approach for numerical analysis regarding the phenomenon of interest in heat transfer enhancement with longitudinal vortex generator. Ferrouillat et al. [16] tested the k-epsilon and

Page 2 of 30

LES turbulence models for a rectangular winglet pair, which for LES turbulent model an unsteady simulation were carried out. The researchers showed that the k-epsilon and the LES models satisfactorily predict the heat transfer enhancement and drag coefficient. A result comparison from LES computation leads to the conclusion that the generation of longitudinal vortices is actually a quasi-steady phenomenon, and there is no significant difference between these two approaches in terms of heat transfer and drag effect. Fiebig [17] also stated that the enhancement by swirl is the most important mechanism in steady flow and concluded that the longitudinal vortices enhance the heat transfer in steady flow. Therefore, according to these assessments, regarding the numerical modeling of the fluid flow and heat transport in a compact heat exchanger with longitudinal vortex generator and from works by Salviano et al. [6,7], the airflow is assumed to be steady-state, turbulent, three-dimensional and incompressible with constant physical properties. Hence, for a Newtonian fluid with constant properties, finite volume-based commercial software [18] was used to solve the governing equations. The mass conservation, momentum and energy are respectively as follows:  x j 

u j   0

(1)

p

x j

  u j u i   ij     x

  T  u jh  k  x j  x j

(2)

i

 ui p   ij   u j   x  xj j 

(3)

Second Order Upwind discretization scheme was used for the convection terms. The turbulence closure method used is the k-omega Shear-Stress Transport (SST) model, which considers the enhanced wall treatment as default. Important characteristics of this closure model make this method more accurate and robust when high adverse pressure gradients in main flow are verified, as those present on circular tubes. A robust algorithm called Coupled Algorithm to perform the pressure-velocity coupling was also considered. This coupled algorithm solves the momentum and pressure-based continuity equations together. The computational convergence is ensured making the residuals lower than 10-5 for continuity and momentum equations and 10-8 for energy equation. 2.2. Thermal-hydraulic parameters The parameters to perform the heat transfer and pressure drop in heat exchangers depend on its geometry and flow conditions. Reynolds number (Re), Colburn factor (j) and Friction factor (f) are considered to describe the thermal-hydraulic behavior, respectively: Re 

j

f 

 U c  2 FP



(4)



h U ccP

2

Pr

3

(5)

Ac 2  p

(6)

Ao  U c2

The total heat transfer, pressure drop, log-mean temperature difference and convective heat transfer coefficient are defined as follows: .

.



Q  m c p  Tln  m c p T in  T out  p  p in  p out



(7) (8)

Page 3 of 30

 TW

 T ln 

h

 

 T in  TW  T out



 T  T in  ln  W   TW  T out 

Q Ao  Tln

(9) (10)

where:

  pdA p

A



(11) dA

A

  uTdA T 

A

  udA

(12)

A

2.3. Computational domain and boundary conditions Fig. 1, Fig. 2 and Fig. 3 show the main geometric dimensions for the heat exchanger evaluated here. Table 1 shows the geometric dimensions that correspond to a typical heat exchanger applied to the automotive industry. Due to the periodic characteristics and geometrical symmetry of the heat exchanger, only one-half of the heat exchanger was computed, as shown in Fig. 1 and Fig. 2. Streamwise and spanwise directions shown in Fig. 3 correspond to the x-axis and y-axis, respectively, and the z-axis corresponds to the fin pitch direction. Fig. 2 also indicates the reference point (0,0,0), which is important to the vortex generator displacement in streamwise and spanwise directions for both inline and staggered tube arrangements. Fig. 3 shows the computational domain divided into three parts: upstream-extended region, core region and downstream-extended region. The upstream region is extended one time of the core region to ensure the inlet velocity uniformity and the downstream region is extended 7 times in order to avoid reversed flow. At the core region, noslip condition on fins and tubes is assumed. The tube temperature is set to be constant and, due to the conjugated nature of the problem, the fin surfaces are considered part of the solution domain. Thus, the boundary condition in the core region is defined according to Tao et al. [19], which defined  T  z on the upper and lower fin and solid-fluid interface. Symmetry boundary conditions in upstream extension regions, in downstream extension regions and at side planes were applied. At the channel inlet, uniform airflow velocity and temperature are taken to be constant. At the outlet, streamwise gradient (Neumann boundary conditions) for all variables are set to be zero. Wu and Tao [20] analyzed the effects of the punched vortex generator and its thickness on the flow and heat transfer. They concluded that the punched holes and VG thickness influenced the profile of the Nusselt number and Friction factor near the holes, with small effect considering the whole channel. Thus, to simplify the automated workflow regarding mesh generation, the vortex generators are assumed to be with zero thickness, attached to bottom fin (without punch hole) and adiabatic. Table 1 summarizes the geometric dimensions of the compact heat exchanger for the present work.

2.4. Grid independence and numerical validation The mesh validity verification is based on the systematic method proposed by Celik et al. [21], from which the Grid Convergence Index (GCI) may be obtained. Three different mesh densities are given in Table 2. As the heat transfer and fluid shear-stress are strongly influenced by the near-wall refinement, the non-dimensional wall distance parameter (y+) is also checked, indicating the region of Law of Wall in which the governing equations are solved (it is well accepted that this value for k-omega SST model should be close to unity). Vortex generators of the delta-winglet type (aspect ratio, attack angle and position) mounted on common-flow-down configuration, as proposed by Lemouedda et al. [8], were considered at this mesh verification step. According to Celik et al. [21], the grid refinement factor (r) is desirable to be greater than 1.3. Thus, as shown in Table 2, this recommendation is followed regarding mesh densities. The procedure to determine the GCI for Friction factor (f) and Colburn factor (j)

Page 4 of 30

was applied, and the results are presented in Table 3. From this table, it can be concluded that the higher uncertainty from fine-grid solution is about 1.00% for Colburn factor (j) and Re = 250. Therefore, the grid independence is checked and the refined Grid 1 may be used as a reference mesh density for other vortex generators configurations. In order to provide a numerical validation, the results of the present study regarding Colburn factor and Friction factor were compared to experimental correlations proposed by Wang et al. [22]. These correlations have widely been applied for numerical validation of thermal-hydraulic performance of the compact heat exchanger [6,7,13,23], which are most accurate and reliable with wide applicable ranges considering a staggered tube arrangement (without vortex generators). In fact, the mean deviations of Wang's correlations are 8.31% and 7.53% for the Friction factor and Colburn factor, respectively. Fig. 4 shows a comparison between Wang's correlations and numerical results for the refined mesh (Grid 1). For Friction factor and Re = 250 and Re = 650, the differences are 4.2% and 7.7%, respectively. For the Colburn factor, the differences are 0.3% and 6.3%, respectively. Thus, for both thermal-hydraulic factors and Reynolds number, the differences are lower than the mean deviations of Wang's correlations, as indicated by the bars in Fig. 4. This results indicate that the numerical approach adopted herein is reliable.

3. Vortex generator 3.1. Input parameters for the vortex generator According to previous reports, the influence of the input parameters of the vortex generator on Colburn factor and Friction factor has been investigated. However, few input parameters are simultaneously evaluated or submitted to optimization methods, as shown in Lei et al. [3], Lemouedda et al. [8] and Jang et al. [10]. As shown in Fig. 5 and Fig. 6, our work considers seven input parameters for each vortex generator, which cover the most significant geometric and spatial variables reported by previous works regarding heat transfer augmentation. Fig. 5 shows points xi and yi indicating the vortex generator displacement in the streamwise direction and spanwise direction, respectively. This point is assumed to be in the middle of the chord of the vortex generator, which is represented by ci (Fig. 6). Roll angle φi, proposed by Salviano et al. [6,7], is also considered, Fig. 5. The roll angle represents the rotation of the vortex generator around the x-axis. Similarly, attack angle θi is also analyzed, Fig. 6, considering the vortex generator rotation around the z-axis. Finally, input parameters ai and bi, Fig. 6, correspond to rear and front height of the vortex generator, respectively. Parameters ai, bi and ci define the shape of the vortex generator, which may vary from delta-winglet to rectangular-winglet. These input parameters of the vortex generator cover the major variables evaluated by previous researches regarding common-flow-down and common-flow-up configurations and aspect ratio. Therefore, the Colburn factor and Friction factor may be represented as a function of these variables and are calculated according to equations (5) and (6): j  j  a1 ,b1 ,c1 ,  1 , 1 , x1 , y1 ,a 2 ,b2 ,c 2 ,  2 ,  2 , x 2 , y 2 

(13)

f  f  a 1 ,b1 ,c1 ,  1 , 1 , x1 , y 1 ,a 2 ,b2 ,c 2 ,  2 ,  2 , x 2 , y 2 

(14)

3.2. Constraint for the Vortex Generator It is necessary to define an approach to treat the vortex generators constraints due to computational domain boundaries, Fig. 3. Lemouedda et al. [8] defined the vortex generator position in function of the attack angle to avoid the interference with tubes. Jang et al. [10] also mounted the vortex generators behind tubes in order to study the attack angle and spanwise displacement. However, in these works, the streamwise position of the vortex generator was set to be unchanged to avoid the interference with tubes. Both approaches make the solution design space narrower, which may result in losses of important features regarding heat transfer augmentation in compact heat exchangers through vortex generators. A simple approach based on trigonometric decomposition to treat the constraints is presented, which provides vortex generators with great flexibility to rotate and to displace within the solution design space, avoiding interference with tubes and computational domain boundaries. This approach defines some check-nodes (cn) on vortex generator, Fig. 6, in order to control the displacement of each checknode of the vortex generator within the solution design space shown by Fig. 2. The position of each cn for Cartesian coordinates (x,y,z) as a function of the input parameters is defined and the distance from the center point of the tube to each cn is calculated. For feasible design, all distances must be greater than d (Fig. 2). For an optimization procedure, identifying the unfeasible designs is relevant to drive the optimizer to find an optimal solution, rather than simply removing geometries when the CAD or meshing software shows an error message. In Fig. 6, subscript i identifies the first vortex generator (i = 1) and the second vortex generator (i = 2), for example, cn81 is related to check-node 8 for VG1, and cn32 is check-node 3 for VG2. Thus, the Cartesian coordinates (x,y,z) of each check-node are calculated from equation (15) to equation (23).

Page 5 of 30

c c   cn1i  x , y , z    x i  i cos   i  , y i  i sin   i  , 0  2 2  

(15)

c c   cn 2 i  x , y , z    x i  i cos   i  , y i  i sin   i  , 0  2 2  

(16)

c c   cn 3 i  x , y , z    x i  i cos   i   a i cos  90   i  sin   i  , y i  i sin   i   a i cos  90   i  cos   i  ,a i cos   i   2 2  

(17)

c c   cn 4 i  x , y , z    x i  i cos   i   bi cos  90   i  sin   i  , y i  i sin   i   bi cos  90   i  cos   i  ,bi cos   i   2 2  

(18)

a  bi a  bi a  bi   cn 5 i  x , y , z    x i  i cos  90   i  sin   i  , y i  i cos  90   i  cos   i  , i cos   i   2 2 2  

(19)

c c   cn 6 i  x , y , z    x i  i cos   i  , y i  i sin   i  , 0  4 4  

(20)

c c   cn 7 i  x , y , z    x i  i cos   i  , y i  i sin   i  , 0  4 4  

(21)

c 3b  a i c 3b  a i 3b  a i   cn 8 i  x , y , z    x i  i cos   i   i cos  90   i  sin   i  , y i  i sin   i   i cos  90   i  cos   i  , i cos   i   4 4 4 4 4  

(22)

c 3 a  bi c 3 a  bi 3 a  bi   cn 9 i  x , y , z    x i  i cos   i   i cos  90   i  sin   i  , y i  i sin   i   i cos  90   i  cos   i  , i cos   i   4 4 4 4 4  

(23)

Considering the Cartesian coordinates of input parameters (xi,yi) and each check-node (cn), the distances between the check-nodes and the center of the tube (Ci) are calculated according to equations (24) and (25), respectively. dx

i

,y i



d cn  ji

 C ix

2 2 D   x i    C iy  y i   1 . 1  o   2 

 C ix  cn jix 

2

2 D    C iy  cn jiy   1 . 1  o   2 

(24) (25)

where subscript j is related to the number of check-node, i.e., j  1,..., 9 . For example, cn42x is related to check-node 4 for VG2 in the streamwise direction. All check-nodes in direction x and y must be within the input parameter range shown by Table 4, defining additional constraints as follows, 0 . 2  cn j 1 x  0 . 99 Pl

(26)

1 . 01  cn j 2 x  1 . 99 Pl

(27)

P  0 . 2  cn jiy  0 . 99  l   2 

(28)

The constraints are determined by equations (24) to (28), which identify unfeasible designs. This simple approach to treating the constraint allows vortex generators to be closer to boundaries and tubes than the approach applied by Salviano et al. [6,7], as well as the narrow design space defined by Lemouedda et al. [8] and Jang et al. [10] regarding the displacement of the vortex generators. 4. Optimization

Page 6 of 30

According to Thevenin and Janiga [24], optimization means: "The design and operation of a system or process to make it as good as possible in some defined sense. As a consequence, the best possible solution constrained by appropriate conditions should be found, and not simply a “better” one". Concerning fluid dynamics, the first applications of optimization are found for aeronautical problems, particularly to improve wing profile and flight properties (typically, reduce drag). Several works regarding optimization of the vortex generator parameters have also been conducted [6,7,8,10,25-27]. Among the algorithms for optimization, Genetic Algorithm (GA) has been successfully applied to optimize compact heat exchangers [6,7,8,25-27]. A robust optimization algorithm must be able to avoid stalling [24]. In fact, it is one of the main advantages of GA, which can detect a global extreme in a problem with multiple local extremes. However, evolutionary algorithms, such as GA, require a large number of evaluations to obtain a refined solution [28]. Either it is a major problem when the evaluation is computationally expensive or when there are many input variables. One way to speed-up optimization is to improve the computational efficiency of the software (for CFD the mesh refinement is usually made coarser) or to try another optimization algorithm able to find an optimal solution faster than GA. Thus, the method known as Nelder and Mead downhill Method [29] - SIMPLEX - is evaluated. SIMPLEX is an algorithm for nonlinear optimization problems and should not be confused with the Simplex method for linear programming. The SIMPLEX method is a single-objective optimizer and does not require derivatives of the objective function, i.e., it is hence more robust than the algorithm based on local gradients. Garrison et al. [30] and Cavazzuti et al. [31] applied the SIMPLEX algorithm to optimize a heat exchanger. A brief comparison between SIMPLEX and GA was conducted in the present work. The analyses are related to a compact heat exchanger in staggered tube arrangement (Fig. 2), Reynolds number of 650 and for objective function shown by Eq.(29). The SIMPLEX algorithm reached the convergence of the objective function with 1/3 of runs required by GA method. Thus, considering the mesh refinement defined by GCI and the number of input variables evaluated by the present work, the method chosen for remaining analyzes is SIMPLEX. Note that above conclusions are based on a simple comparison between these optimization methods and should be taken as a reference for the particular heat exchanger evaluated herein. For other thermal problems we encourage the researchers to conduct a distinct analysis to define the more suitable algorithm method for their optimization studies considering the computational resources and numerical modeling. This method uses the Simplex concept, which is a polyhedron containing n + 1 points in a n dimensional space. For example, in two dimensions, there is a triangle and, in three dimensions, there is a tetrahedron. The SIMPLEX method compares the values of the objective function at the n + 1 vertices and then moves this polyhedron gradually towards the optimum point during the iterative process. The aim of this algorithm is to move by replacing vertices. The movements towards the optimal point are driven by three operations: Reflection, Expansion and Contraction. These operations are widely available in the open literature [28,29,32] and are not presented in detail herein. 4.1. Statement of optimization problem Many performance evaluation criteria (PEC) have been developed to evaluate the performance of heat exchangers [33]. In fact, PEC can be considered an objective function as far as optimization is concerned. The choice for an objective function is governed by the nature of the problem and may be one of the most important decisions in the whole optimum design process. For heat exchangers that apply enhanced heat transfer techniques, such as vortex generators, the main goal is to maximize the heat transfer and to minimize the pressure loss. In fact, this approach considers a multi-objective optimization problem and the analysis would employ the concept of Pareto optimality, which indicates a set of Pareto optimal solutions (so-called Pareto-Frontier). This approach to optimization analysis uses large computational resources to characterize the Pareto-Frontier. Therefore, the objective functions or PEC are here defined as the ratio between heat transfer and pressure loss, which indicates a single-objective optimization problem (required by method SIMPLEX). This approach finds an optimum point for each objective function with feasible computational resource. Thus, the objective functions, evaluated separately in our work, to evaluate the thermal hydraulic performance of a heat exchanger based on equations (5) and (6) are as follows,  j   j  o   JF   f   f  o  

1

JF

3



(29)

 j   j  o   1

(30)

 f 3  f  o  

Page 7 of 30

where jo and fo are the Colburn factor and Friction factor for the heat exchanger without vortex generator (Fin-tube), respectively. Consider the design vector Z (Eq. 31), which arranges all input variables evaluated herein. Z   a1 ,b1 ,c1 ,  1 , 1 , x1 , y 1 ,a 2 ,b 2 ,c 2 ,  2 ,  2 , x 2 , y 2 

(31)

Thus, according to Rao [28], a

Z 

dx

i

,y i

d cn

ji

 

J F (or JF

 Do  ,  2 

i  1, 2

 Do  ,  2 

i  1, 2

 Z   1 .1   Z   1 .1 

1 3

)

j  1,..., 9

5. Results and discussion The optimization procedure followed herein was the Direct Optimization (DO), as discussed by Salviano et al. [7]. With this DO approach, the objective function is evaluated directly by a numerical model without surrogate modeling. Table 5 shows the input parameters of the vortex generators for the present work and for some important researches in the open literature considering the reference point indicated by Fig. 2. These works in the open literature present some peculiarities with regard to Reynolds number and geometric parameters. Thus, in order to provide an adequate base for comparison, the input parameters of the vortex generator by Torii et al. [9], Jang et al. [10] and Salviano [7] as a function of the tube diameter, longitudinal and transverse pitch, and fin pitch of the compact heat exchanger, were parameterized. In Lemouedda et al. [8] both common-flow-up (CFU) and common-flow-down (CFD) arrangements of the vortex generator were also compared. Table 6 shows the vortex generator shape evaluated by Torii et al. [9], Jang et al. [10], Lemouedda et al. [8] and Salviano [7]. Among these works, Jang et al. [10] considered rectangular-winglet vortex generator while the other researchers evaluated delta-winglet vortex generator. The aspect ratio (c/a) for each vortex generator could be performed through input parameters presented in Table 5. Fig. 7 and Fig. 8 show the optimum configurations of the vortex generator for the present work found by optimization procedure for inline and staggered tube arrangements, respectively, for both Reynolds number and objective functions. 5.1. Analysis of the general trend for optimized configurations For optimized configurations the results shown in Table 5, Fig. 7 and Fig. 8 indicate that the vortex generator shapes are more similar to rectangular-winglet type than delta-winglet type, independently of the Reynolds number, tube arrangements and objective functions. In general, for both Reynolds number, the VG1 aspect ratio (performed as a ratio of the Chord and the higher height (a or b) of the vortex generator) is higher than the VG2 aspect ratio, except to inline tube arrangement for objective function JF. For objective function JF⅓, the optimized geometries present higher aspect ratios than for objective function JF, especially for VG1, since the pressure loss for this objective function is weighted by exponent ⅓ indicating that the heat transfer is a priority goal as compared to pressure loss penalty. A general-guide, most used by researchers for rectangular-winglet type, recommends that the ratio between vortex generator height and finpitch should be half that of the channel (z/Fp = 0.5), similarly as used by Jang et al. [10]. However, this ratio for the optimum points shown in Fig. 7 and Fig. 8, calculated based on average height of the vortex generator in direction-z (cn3z and cn4z), is closer to z/Fp = 0.6. Furthermore, the attack angles (θ) for optimum configuration were found to be higher for JF⅓ than for JF, independently of the Reynolds number and tube arrangements. For the objective function JF, the optimum configuration of the vortex generators is more influenced due to the tube arrangement than to the Reynolds number, Fig. 7a-b and Fig. 8a-b. The same trend for objective function JF⅓ was also observed, Fig. 7c-d and Fig. 8c-d. On the other hand, regarding the optimum configurations of the vortex generator found for JF and JF⅓, the difference is more pronounced for inline tube arrangement than for staggered tube arrangement, as can be seen from Fig. 7a-c and Fig. 8a-c (Re = 250) and Fig. 7b-d and Fig. 8b-d (Re = 650).

Page 8 of 30

For staggered tube arrangement, common-flow-up vortex generator configuration was found to be more suitable to provide higher enhancement of heat transfer than common-flow-down vortex generator configuration, as indicated by Torii et al. [9], independently of the Reynolds number and objective function, Fig. 8. However, for inline tube arrangement, this trend is not readily observed, since the VG1 for JF (Fig. 7a-b) and VG2 for JF⅓ (Fig. 7c-d) present unusual configurations. Nevertheless, VG2 for JF (Fig. 7a-b) and VG1 for JF⅓ (Fig. 7c-d) follow the same trends for staggered tube arrangement. Still regarding the vortex generator configuration, when the commonflow-up configuration is found as an optimum point, the position of the vortex generator in streamwise direction is set to be closer to the minimum flow area, as discussed by Salviano et al. [7]. The influence on the thermal-hydraulic behavior due to unusual configuration of the vortex generators are discussed in the next sections. 5.2. Global thermal-hydraulic performance The global thermal-hydraulic performance of a compact heat exchanger is evaluated by JF and JF⅓, which are calculated based on average pressure, Eq. (11), and average temperature, Eq. (12), at the inlet and outlet in the core region (Fig. 3). Fig. 9 and Fig. 10 show a comparison between the present work and researches by Torii et al. [9], Jang et al. [10], Lemouedda et al. [8] and Salviano et al. [7]. Letters from A to F in these graphs concern the Code column shown in Table 5 and in Table 6. Fig. 9a-c and Fig. 10a-c are related to the objective function JF, whereas Fig. 9b-d and Fig. 10b-d regard to objective function JF⅓. Fig. 9 and Fig. 10 allow verifying that the global performance results for our work is higher than other works independently of the objective function, Reynolds number and tube arrangement. Regarding the difference between Salviano et al. [7] and the present work, both researches applied a direct optimization approach; however, we consider in the present work that the vortex generator shape varies during the optimization procedure, while in Salviano et al. [7], it is unchanged, as shown in Table 6. In general, significant heat transfer augmentation for the low Reynolds number (Re = 250) is more difficult to be achieved than for the higher Reynolds number (Re = 650), as shown in Fig. 9 and Fig. 10. For JF, the maximum value of heat transfer augmentation for Re = 250 and Re = 650 is 15.5% (Fig. 9c) and 36.9% (Fig. 10c), respectively, both results for staggered tube arrangement. The same behavior is observed for objective function JF⅓, in which the heat transfer augmentation is 26% (Fig. 9d) for Re = 250 and 42.3% (Fig. 10d) for Re = 650, also for staggered tube arrangement. Additionally, the heat transfer augmentation is higher for JF⅓ than for JF, which is due to the weighting of Friction factor (f), allowing the optimizer to find more aggressive vortex generator configurations especially looking for aspect ratio and attack angle, Table 5. This is more evident for inline tube arrangement than for staggered tube arrangement, Fig. 7 and Fig. 8. Heat transfer enhancement is associated with pressure loss penalty, although an important feature of the vortex generator is to provide heat transfer augmentation with low additional pressure loss, as compared to other passive techniques, such as louvers and wave fins. Fig. 9 and Fig. 10 indicate that this difficulty in achieving higher heat transfer than pressure loss is more pronounced for the inline tube arrangement. For the inline tube arrangement, Fig. 9a and Fig. 10a show that for all previous works compared herein, the heat transfer enhanced by vortex generator is lower than its pressure loss penalty, except for results presented herein. In fact, for Re = 250, the heat transfer increased 6.1% with pressure loss penalty of 1.3% (Fig. 9a), while for Re = 650 (Fig. 10a) it was 21.6% and 4.2%, respectively. For staggered tube arrangement, Torii et al. [9] and Salviano et al. [7] were able to provide heat transfer augmentation higher than its pressure loss penalty for both Reynolds numbers. However, the optimized configurations of the present work, for objective function JF, achieved a much more favorable relation between them. For Re = 250, the heat transfer increased 22.6% with pressure loss penalty of 6.2% (Fig. 9c), while for Re = 650 (Fig. 10c) it was 38.9% and 1.5%, respectively. The optimized configurations, for objective function JF⅓, follows the same trends as JF only for the staggered tube arrangement for both Reynolds numbers; for Re = 250, the heat transfer increased 37.9% with pressure loss penalty of 31.1% (Fig. 9d), while for Re = 650 (Fig. 10d), it was 51.7% and 21.3%, respectively. For the inline tube arrangement, the heat transfer augmentation (39.9%) is smaller than the pressure loss penalty (74,2%) for Re = 250 (Fig. 9b), while for Re = 650 it was 68.2% and 96.7%, respectively. 5.3. Temperature and velocity distributions As well as achieving good global thermal-hydraulic performance from the optimization procedure, it is also necessary to identify the main thermal-hydraulic phenomena able to provide higher heat transfer augmentation, according to the results presented in Fig. 9 and Fig. 10, and discussed in the previous section. The analysis of the interface between thermal behaviors due to flow dynamic in compact heat exchanger is in fact a great challenge due to the phenomena complexity, especially when vortical systems are introduced in the first flow as those generated by longitudinal vortex generators. Thus, the analysis is based on results presented by Table 7, Table 8 and Table 9, which show the temperature distributions in several y-z cross sections and velocity distribution in three x-y cross sections with streamlines for Re = 250 and Re = 650, respectively. The temperature distribution and velocity distribution for the Fin-tube configuration (without vortex generator), taken as reference flow, is also presented and analyzed. 5.3.1 Fin-tube (without vortex generator)

Page 9 of 30

Important thermal-hydraulic behaviors for the Fin-tube are firstly discussed since the vortex generators are inserted in the flow and should interact and distort the natural phenomena present in fin-tube heat exchangers in order to achieve high heat transfer with low pressure penalty. The results from Fin-tube in Table 7 show that the boundary layer for Re = 250 is thicker than for Re = 650. In other words, the wall effect is more easily propagated into the flow for Re = 250 than for Re = 650, evidencing the difficulty to achieve heat transfer augmentation with longitudinal vortex generator for the low Reynolds number. In fact, for the low Reynolds number, the vortex generator could provide greater increase in pressure drop than increase in heat transfer; therefore, the ratio between them is lower than one unit, as shown in Fig. 9a-c for the works by Lemouedda et al. [8] and Jang et al. [10]. This is even more evident for the inline tube arrangement. Still with the Fin-tube configuration, a slight distortion in temperature distribution indicated by iso-values lines, between inline and staggered tube arrangement, is observed. For Re = 250, this distortion is verified closer to the second tube, which is more evident for Re = 650 for staggered tube arrangement. The horseshoes vortices are more strongly generated for the staggered tube arrangement than for the inline tube arrangement near the second tube. This is due to flow acceleration near the second tube, as can be seen in Table 8 (Re = 250) and in Table 9 (Re = 650). Therefore, for Fin-tube configuration, the heat transfer for the staggered tube arrangement is higher than for the inline tube arrangement. The heat transfer for the former is 13.9% higher than the latter for Re = 250, and 26.7% for Re = 650. Although the stronger horseshoes for the staggered tube arrangement provide boundary-layer delay, which is an important feature to mitigate the pressure loss penalty, the effect on the flow due to the position of the second tube for staggered configuration is higher than the mitigation effect due to boundary-layer delay. For inline tube arrangement, the second tube is positioned behind the first tube, which decreases its effect on the flow. Thus, the pressure loss penalty for the staggered tube arrangement is 27.5% higher than for the inline tube arrangement for Re = 250, and 54.4% for Re = 650. The ratio between heat transfer and pressure drop (j/f) for the inline tube arrangement is 12% higher than for the staggered tube arrangement for Re = 250, and 21.8% for Re = 650. Although the staggered tube arrangement provides higher heat transfer, the inline tube arrangement showed higher j/f ratio. Evaluating the ratio j/f⅓, the staggered tube arrangement is 5% higher than the inline tube arrangement for Re = 250, and 9.7% for Re = 650. Thus, for a Fin-tube compact heat exchanger (without vortex generator), a key-step to define the best choice regarding the tube arrangement is more dependent on the ratio (j/f or j/f⅓) than the Reynolds number. Another important flow feature in a Fin-tube compact heat exchanger regards the wake formed behind the tubes, as can be seen in Table 8 and Table 9 for Re = 250 and Re = 650, respectively. The wake is observed to be different for the first tube and for the second tube for both tube arrangements. The wake zones are also different between the tube arrangements. In fact, the transverse vortex diameter in the wake zone of the first tube for the staggered tube arrangement is smaller than for the inline tube arrangement, due to the tube arrangement. However, the wake formed behind the second tube is similar for both tube arrangements. These tube wake zones corresponding to the low heat flux, which is due to the low velocity of the transverse vortex (Table 8 and Table 9) that is almost isolated from the main flow, and the small temperature difference between the fluid and the wall, Table 7. The thermal-hydraulic flow characteristics for a Fin-Tube compact heat exchanger (without vortex generator) discussed above suggest that the vortex generators optimal configurations should be searched independently from each other, indicating an input parameters asymmetry between the first and the second vortex generators, as pointed out by Salviano et al. [6,7] and confirmed here. 5.3.2 Optimized configuration for the vortex generators According to the results shown in Fig. 9 and Fig. 10 for the compact heat exchanger with vortex generator, the optimized configurations for the present work present higher global performance than those from previous works compared herein. The secondary flow generated by the vortex generator is strongly three-dimensional, transporting heat convectively and leading to a spiraling motion that distorts the temperature profiles and increases the heat flux between the wall layer and the flow core, as shown in Table 7. The distorted temperature profiles are strongly dependent on the strength of the secondary flow, which is a function of the input parameters of the vortex generator and of the Reynolds number. In fact, the temperature profiles present higher distortions for Re = 650 and for objective function JF⅓, Table 7, in which the input parameters were more aggressive mainly due to the attack angle and to the vortex generator shape. Although the temperature of the core of the main vortex remains close to the inlet temperature, which is more evident for Re = 650, significant distortion in the thermal boundary layer due to the strong swirling flow decreases the thermal resistance between the flow and the walls, increasing the global heat transfer. This feature is smoother for Re = 250 especially due to the thickness of the boundary layer, which is thicker for Re = 250 than for Re = 650. The optimized configurations found by our work, independently of the tube arrangement, Reynolds number and objective functions, generate longitudinal vortex from VG1 in such a way that it avoids the collision directly onto VG2, as claimed by Salviano et al. [7]. The authors [7] considered a small and unchanged aspect ratio winglet-type vortex generator in staggered tube arrangement and this collision could be readily avoided. This unwanted effect is also avoided herein, although the optimized configurations reported show higher aspect ratio of the vortex generators even for inline tube arrangement, Table 5. This is evidenced in Table 7 following the main vortex path from VG1 and its feature regarding the core of the main vortex, as discussed previously. The position of the vortex generators in the spanwise direction (y-axis), Table 5, is important to avoid this collision since y1 and y2 are unaligned for optimized configurations. These

Page 10 of 30

conclusions are valid except for the optimal vortex generator configuration for objective function JF⅓ and inline tube arrangement for both Reynolds numbers, since the optimized parameters of the vortex generators lead to unusual configurations, Fig. 7c-d. Friction factor (f) is differently weighted, as can be seen in Eq. (30). Among the works compared herein, Lemouedda et al. [8] and Torii et al. [9] present this collision of the main vortex generated from VG1 on VG2 for both tube arrangements, while for the vortex generator configuration by Jang et al.[10], this unwanted flow pattern is observed only for the inline tube arrangement. Regarding the distortion in temperature profile, the optimized vortex generator configurations provided additional mixing between the main flow and the flow in the wake zone, as seen in Table 8 and Table 9 for Re = 250 and Re = 650, respectively. In these tables, all the velocity fields indicate that the flow is changed by vortex generators, influencing the heat flux in both bottom and top fins. This is actually expected, since the vortex generator heights are comparable with the channel height (Fp). For vortex generator height comparable to viscous sublayer, only the channel side where vortex generators are attached is influenced. For inline tube arrangement, objective function JF and Re = 250, the optimized configuration, the VG1 is placed in the fore region of the first tube, which provides a slight distortion in temperature profile and imperceptible changes in the aft region of the first tube (wake region). The effect of the secondary flow generated by VG1 is almost completely dissipated by the viscous effects. On the other hand, the VG2 causes flow mixing in the wake region and main flow, besides creating a constricted passage between the vortex generator and the second tube, increasing the flow velocity, which causes boundary layer separation delay. Narrowing the wake and suppressing vortex shedding are the obvious outcomes that reduce pressure loss. Since the fluid is accelerated in this passage, the zone of poor heat transfer is also diminished at the near-wake of the tube. This effect is evident in three positions at plane-z. The same behaviors are also observed for Re = 650. However, the secondary flow generated by VG 1 is stronger than for Re = 250 and the viscous effects are not able to completely dissipate the vortices up to the computational domain outlet. Similarly, for Re = 650, the VG1 did not influence the wake flow behind the first tube, either. The flow acceleration between VG2 and the second tube is also evident; however, this effect is more pronounced up to z/Fp = 0.50. Fig. 7a and Fig. 7b show that VG2 has a similar shape. This lack of influence of the VG2 at z/Fp = 0.75 for Re = 650 is due to the boundary layer thickness, which is thinner for Re = 650 than for Re = 250. For objective function JF⅓ and for the inline tube arrangement, the vortex generator shape, its positions and angles are completely different from those found by the optimizer for objective function JF, Fig. 7. Table 7 indicates that the optimum parameters of the vortex generator found for JF⅓ generate stronger distortion in temperature profile for both Reynolds numbers than for objective function JF. Table 8 and Table 9 allow observing that the synergy between VG1 and VG2 significantly improve the flow mixture in the wake region between the first and the second tubes. For Re = 250, the wake region is more reduced by this synergy than for Re = 650, especially at z/Fp = 0.50 and z/Fp = 0.75. Since the vortex generator shapes are very similar for both Reynolds number (Fig. 7c and Fig. 7d), the root cause of this result is the thinner boundary layer for Re = 650. This synergy between VG1 and VG2, besides the accelerated flow in the wake region, impinges the main flow on the adjacent downstream tube and results in a high temperature gradient, achieving local heat transfer enhancement. For both Reynolds numbers, the vortex generators do not cause significant changes in the wake region behind the second tube, differently from the results discussed for objective function JF. Overall, the optimum parameters found for the vortex generators perform different functions regarding the thermalhydraulic flow behavior in the inline tube arrangement. While for objective function JF VG1 and VG2 do not influence the large wake region formed between the first and the second tube, for JF⅓, this wake region is strongly changed by the synergy between VG1 and VG2. For the staggered tube arrangement, the optimum parameters found for the vortex generators present slight change in function of the Reynolds number and of the objective function. However, differently from the conclusions for the inline tube arrangement, each vortex generator performs similar functions. In fact, both VG1 and VG2 distort the temperature profile by generating the secondary flow (Table 7) and improve the flow mixture in the wake region behind the first and the second tubes (Table 8 and Table 9), independently of the Reynolds number and objective function. The wake region behind the first and the second tubes is more reduced for the objective function JF⅓ than for JF since the gap between the vortex generators and its tubes is narrower for JF⅓ than for JF, which provide high flow acceleration and, consequently, the flow separation delay. For both objective functions, the vortex generators influences three planes-z, even for Re = 650, in which the boundary layer is thinner than for Re = 250. Regarding the streamwise position (x1 and x2), optimized configurations tend to be close to the minimum flow area, where an important flow acceleration is verified analyzing the Fin-tube (without vortex generator), Table 8 and Table 9 and Table 5. Although the temperature profiles for both Reynolds number and objective functions are different for optimum points, these difference is smoother for the staggered tube arrangement than for the inline tube arrangement, Table 7. 5.4. Span-average Colburn factor and relative pressure profile Evaluating the distribution of span-average Colburn factor (j) and relative pressure profile allow compiling the thermal-hydraulic effects due to the longitudinal vortex generator, previously discussed in detail, regarding the distortion in temperature profile of the secondary flow, increase of heat flux in the wake region, separation delay of the boundary layer and pressure drop penalty. This approach to calculate the Colburn factor (j) is a "local" analysis based on the span-average temperature difference at plane n and n-1 along the fin in the streamwise direction. The span-average relative pressure is evaluated at the plane in the streamwise direction. Fig. 11 and Fig. 12 present the distribution of span-average Colburn factor (j) and Relative pressure for Re = 250 and Re = 650, respectively, for both tube

Page 11 of 30

arrangements and objective functions. Only the better results in Fig. 9 and Fig. 10 were plotted. The Fin-tube regards compact heat exchanger without vortex generators. Considering the Fin-tube configuration, the span-average Colburn factor profile presents different behavior for inline and staggered tube arrangements, independently of the Reynolds number. For the inline tube arrangement, this profile indicates a downward profile up to the front of the second tube; from this point, up to the minimum flow-area, this profile increases due to flow acceleration. From the minimum flow-area up to the domain outlet, the profile becomes downward again due to the wake region behind the second tube and flow deceleration due to the increase in span-area. This span-average Colburn factor profile is similar for both Reynolds number. For the staggered tube arrangement, the span-average Colburn factor profile also indicates a downward profile up to x/L = 0.4 for Re = 250 and x/L = 0.5 for Re = 650. From these points up to the front of the second tube, these profiles abruptly increase due to the horseshoes formation, as can be seen in Table 7, which is more evident for Re = 650. From the front of the second tube up to the domain outlet, these profile becomes downward again in the streamwise direction. The Fin-tube span-average relative pressure profiles for both tube arrangements and Reynolds number present a similar downward trend along the fin in the streamwise direction. Evidently, the flow dissipation for Re = 650 is higher than Re= 250 and, consequently, the pressure drop measured in the inlet and in the outlet of the core domain is also higher, comparing the results in Fig. 11b and Fig. 12b for the inline tube arrangement, and Fig. 11d and Fig. 12d for the staggered tube arrangement. Still considering the Fin-tube configuration, the pressure drop for the staggered tube arrangement is 19.5% higher than inline tube arrangement for Re = 250, while for Re = 650, this percentage increases by 32.5%. For objective function JF, the inline tube arrangement and Re = 250, Fig. 11a-b, the optimum configuration found for VG1 (in front of the first tube) causes a small increase in Colburn factor and relative pressure, compared to Fin-tube span-average profiles, up to the leading edge of VG2. From this point onwards, the Colburn factor is increased due to higher heat flux in the wake zone and the pressure drop penalty is mitigated due to flow separation delay in the second tube. Among the previous works, Jang et al. [10] has also presented satisfactory global performance, Fig. 9a. However, the higher span-average Colburn factor in the wake zone between the tubes and the minimum flow-area were not high enough compared to the global results reached herein. This result is due to the lack of boundary layer delay on the second tube, although VG2 is placed at the tube aft, according to Jang et al. [10]. For Re = 650, Fig. 12a-b, the same behaviors were also observed, yet the effect of VG1 on the span-average Colburn factor is clearer compared to the Fin-tube configuration, particularly from the minimum flow-area up to the domain outlet of the second tube. For objective function JF⅓, inline tube arrangement and Re = 250, Fig. 11a-b, the synergy between VG1 and VG2 cause higher spanaverage Colburn factor than in Torii et al. [9] along the fin up to the domain outlet. The increase due to the flow mixture improvement in the wake zone is deemed important, but the downstream propagation of the longitudinal vortex is an important advective mechanism to provide heat transfer augmentation. Evidently, the associated pressure drop penalty herein is significantly higher than in other works. However, the weighting factor (⅓) applied to objective function JF⅓, Eq.(30), mitigates its effect on global performance. For Re = 650, Fig. 12a-b, the same behaviors were also observed since the optimum parameters of the vortex generators indicate the same trend. For objective function JF, the staggered tube arrangement and Re = 250, Fig. 11c-d, the span-average Colburn factor profile for the present work is significantly higher than Torii et al. [9], which is the best work among the literature papers compared herein, as shown in Fig. 9c. Our results are higher from the leading edge of the vortex generator up to the domain outlet. Although our results regarding the heat transfer are significantly higher than in Torii et al. [9], the associated pressure drop penalty is almost equivalent along the fin, with smooth difference near the minimum flow-area of the second tube. This difference is due to the position of VG2 since the fluid is accelerated in the gap formed between the vortex generator and the second tube, verified in Table 8. The effect of the boundary layer delay due to VG2 may be observed in the second tube since the relative pressure profile becomes flat from x/L = 0.8 up to x/L = 1.0, while the relative pressure profile for the Fin-tube is still downward within this range. For Re = 650, Fig. 12c-d, the same behaviors were also observed, yet the heat transfer augmentation is still more pronounced on the second tube due to VG2, since the heat transfer is significantly higher than in Torii et al. [9] and the effect of the boundary layer delay is also improved. This feature is relevant because the optimum configuration for objective function JF, Fig. 8b, increased the heat transfer by 38.9% while the pressure loss penalty was 1.5%, compared to the Fin-tube. For objective function JF⅓, staggered tube arrangement and Re = 250, Fig. 11c-d, the span-average Colburn factor profile is significantly increased due to the high aspect ratio and attack angle of the vortex generator. Moreover, the abrupt increases of the Colburn factor in front of the second tube is due to the synergy between the second flow generated by VG 1 and the horseshoes, which is more pronounced for staggered tube arrangement on the second tube. Although the pressure drop is weighted for the objective function JF⅓, the optimum configuration of VG2 was able to provide boundary layer delay on the second tube from x/L = 0.8 up to x/L = 1.0, as well as observed for objective function JF. The same behaviors for Re = 650 were also observed, with small changes in span-average Colburn factor profile due to VG1. In fact, the profile for Re = 650 between x/L = 0.15 and x/L = 0.5 is flatter than that indicated for Re = 250, which is due to aspect ratio of the VG1, which is lower for Re = 650. Indeed, the aspect ratio and Reynolds number influences the strength of the longitudinal vortex generator and, therefore, the combination of high Reynolds number and high aspect ratio could generate large longitudinal vortexes from VG1 making possible its collision onto VG2. Moreover, large longitudinal vortices from VG1 may interfere with horseshoes that are strong on the second tube for staggered tube arrangement, as shown in Table 7 for Fin-tube configuration.

Page 12 of 30

6. Conclusions An optimization of the input parameters of the longitudinal vortex generators by the SIMPLEX method is presented. The optimization is conducted through a three-dimensional numerical simulation of a fin-tube compact heat exchanger with two rows of tubes in staggered and inline arrangements for Re = 250 and Re = 650. Fourteen input parameters of the vortex generators were considered (seven for each vortex generator) in order to optimize two objective functions JF and JF⅓ (separately analyzed) which are based on Colburn factor (j) and Friction factor (f). Moreover, a simple approach based on trigonometric decomposition is presented to treat the constraints, which allows great flexibility for vortex generators to move along the fin and to cover the solution design space more than other strategies used in previous works. The major findings are summarized as follows:  Horseshoes vortices for Fin-tube compact heat exchanger (without vortex generator) is stronger for staggered tube arrangement than inline tube arrangement near the second tube and, consequently, the heat transfer for staggered tube arrangement is 13.9% higher than for inline tube arrangement for Re = 250, and 26.7% for Re = 650.  Optimized configurations (with vortex generators) for the present work achieved higher global performance than those results from previous works in the open literature compared herein, for both objective function JF and JF⅓, independently of the Reynolds number and tube arrangements.  Heat transfer augmentation for low Reynolds number (Re = 250) is more difficult to be achieved than for higher Reynolds number (Re = 650) for both objective functions. Moreover, the augmentation of heat transfer is more pronounced for staggered tube arrangement than for inline tube arrangement.  The suitable vortex generator shapes to maximize the objective functions are more similar to rectangular-winglet type than delta-winglet type, independently of the Reynolds number and tube arrangements.  Optimum vortex generator configurations indicated that the ratio between the vortex generator height (defined by checknodes at direction-z) and the fin-pitch (Fp) is 0.6 instead of 0.5 as suggested by previous works.  Optimum vortex generator configurations produced longitudinal vortices from VG1 that avoided the collision directly on VG2.  For both objective functions JF and JF⅓, the optimum configurations of the vortex generators are more influenced by the tube arrangement than by the Reynolds number.  For staggered tube arrangement, common-flow-up vortex generator configuration was found to be more suitable to provide higher enhancement of heat transfer than common-flow-down vortex generator configuration and the position of the vortex generator in the streamwise direction was set to be closer to the minimum flow area.  For inline tube arrangement, while for the objective function JF, the VG1 and VG2 do not influence the wake region formed between the first and second tube, for JF⅓, this wake region is strongly changed by synergy between VG 1 and VG2. This was observed for both Reynolds numbers. However, for staggered tube arrangement, the optimum parameters found for the vortex generators showed slight changes in function of the Reynolds number and objective functions; the vortex generators perform similar functions and do not present synergy effect. Although some trends for the optimum points of the longitudinal vortex generators could be identified, the optimized configuration is still dependent on the Reynolds number, tube arrangement and objective function so an optimization procedure should be conducted for each compact heat exchanger requirement. Thus, many of the aforementioned conclusions are stressed to be valid for the geometric characteristics of the Fin-tube compact heat exchanger and for the inlet velocity assumed here.

Acknowledgment The authors acknowledge CNPq (Brazilian National Research Council) for Grant No. 307421/2014-7 and ESSS (Engineering Simulation and Scientific Software) for technical support.

Page 13 of 30

Nomenclature a Ac Ao b c cn cp d Do Dc Fp h L m

N p Pl Pt Q r T u Uc x y y



∆P ∆T

Rear height of the vortex generator Minimum free flow area Total flow area Frontal height of the vortex generator Chord of the vortex generator Check-node Specific heat Distance up to center of the tubes Tube diameter Fin collar outside diameter Fin pitch Heat transfer coefficient Depth of the heat exchanger Mass flow rate Number of tube rows Pressure Longitudinal pitch Transverse pitch Heat transfer Refinement factor Temperature Velocity component Velocity at inlet domain Streamwise direction Spanwise direction Mean value of y-plus Pressure drop Temperature difference

(mm) (m2) (m2) (mm) (mm) (J/kg.K) (mm) (m) (m) (mm) (W/m2.K) (mm) (kg/s) (Pa) (mm) (mm) (W) (K) (m/s) (m/s) (mm) (mm) (Pa) (K)

Dimensionless numbers f Friction factor j Colburn factor Pr Prandtl number Re Reynolds number Greek symbols δ Fin thickness φ Roll angle μ Dynamic viscosity θ Attack angle ρ Density

(mm) (°) (kg/m.s) (°) (kg/m3)

Subscript in inlet f fin out outlet ln logarithmic mean W Wall i Related to VG1 (i = 1) and VG2 (i = 2) Abbreviation CFD Common-flow-down CFU Common-flow-up

Page 14 of 30

DO GA GCI IN RSM ST VG

Direct Optimization Genetic Algorithm Grid Convergence Index Inline Response Surface Methodology Staggered Vortex Generator

References [1] Yanagihara, J.I. and Torii, K., Enhancement of laminar boundary layer heat transfer by a vortex generator, JSME International Journal (Ser. II) 35, pp. 400-405, 1992. [2] Jacobi, A. M., Shah, R. K., Heat transfer surface enhancement through the use of longitudinal vortices: A review of recent progress, Experimental Thermal and Fluid Science 11, pp. 295-309, 1995. [3] Lei, Y.G., He, Y.L., Tian, L.T., Chu, P. and Tao, W.Q., Hydrodynamics and heat transfer characteristics of a novel heat exchanger with delta-winglet vortex generators, Chem. Eng. Sciences 65, pp.1551–1562, 2010. [4] Yang, K. S., Li, S. L., Chen, I. Y., Chien, K. H., Hu, R. and Wang, C. C., An experimental investigation of air cooling thermal module using various enhancements at low Reynolds number region, International Journal of Heat and Mass Transfer 53, pp. 5675–5681, 2010. [5] Tian, L. T., He, Y. L., Lei, Y. G. and Tao, W. Q., Numerical study of fluid flow and heat transfer in a flat-plate channel with longitudinal vortex generators by applying field synergy principle analysis, International Communication in Heat Mass Transfer 36, pp. 111–120, 2009. [6] Salviano, L. O., Dezan, J. D., Yanagihara, J. I. Multi-objective optimization of vortex generators positions and angles in fin-tube compact heat exchanger at low Reynolds number using neural network and genetic algorithm, Proceedings of the 15th International Heat Transfer Conference, IHTC-15, Kyoto-Japan, 2014. [7] Salviano, L. O., Dezan, J. D., Yanagihara, J. I. Optimization of winglet-type vortex generator positions and angles in plate-fin compact heat exchanger: Response Surface Methodology and Direct Optimization, International Journal of Heat and Mass Transfer 82, 373-387, 2015. [8] Lemouedda, A., Breuer, M., Frans, E., Botsch, T., and Delgado, A., Optimization of the angle of attack of delta-winglet vortex generators in a plate-fin-and-tube heat exchanger, International Journal of Heat and Mass Transfer 53, pp. 5386–5399, 2010. [9] Torii, K., Kwak, K. M. and Nishino, K., Heat transfer enhancement accompanying pressure-loss reduction with winglet-type vortex generators for fin-tube heat exchangers, International Journal of Heat and Mass Transfer 45, pp. 3795-3801, 2002. [10] Jang, J-Y., Hsu, L-F., Leu, J-S., Optimization of the span angle and location of vortex generators in a plate-fin and tube heat exchanger, International Journal of Heat and Mass Transfer 67, pp. 432–444, 2013. [11] Zeng, M., Tang, L.H., Lin, M., Wang Q. W. Optimization of heat exchangers with vortex-generator fin by Taguchi method, Applied Thermal Engineering 30, 1775-1783, 2010. [12] Wu, J.M., Tao, W.Q. Numerical study on laminar convection heat transfer in a channel with longitudinal vortex generator. Part B: Parametric study of major influence factors, International Journal of Heat and Mass Transfer 51, 3683-3692, 2008. [13] Hwang, S. W., Kim, D. H., Min, J. K., Jeong, J. H. CFD analysis of fin tube heat exchanger with a pair of delta winglet vortex generators, Journal of Mechanical Science and Technology 26, 2949-2958, 2012. [14] Fiebig, M., Kallweit, P., Mitra, N. and Tiggelbeck, S., Heat transfer enhancement and drag by longitudinal vortex generators in channel flow, Experimental Thermal and Fluid Science 4, pp. 103-114, 1991. [15] Yanagihara, J. I. and Torii, K., Heat transfer characteristics of laminar boundary layers in the presence of vortex generators, Proceedings of the 9th International Heat Transfer Conference - IHTC-1990, Vol. 6, pp. 323-328, 1990. [16] Ferrouillat, S., Tochon, P., Garnier, C., Peerhossaini, H., Intensification of heat-transfer and mixing in multifunctional heat exchanger by artificially generated streamwise vorticity, Applied Thermal Engineering 26, 1820-1829, 2008. [17] Fiebig, M., Embedded vortices in internal flow: heat transfer and pressure loss enhancement, Int. J. Heat and Fluid Flow 16, 376388, 1995. [18] Fluent Inc., Ansys Fluent Theory Guide, 2010. [19] Tao, W. Q., Cheng, Y. P., Lee, T. S., 3D numerical simulation on fluid flow and heat transfer characteristics in multistage heat exchanger with slit fins, Heat Mass Transfer 44, 125-136, 2007. [20] Wu, J. M., Tao, W. Q. Numerical study on laminar convection heat transfer in a rectangular channel with longitudinal vortex generator. Part A: Verification of field synergy principle, International Journal of Heat and Mass Transfer 51, 1179–1191, 2008. [21] Celik, I. B., Ghia, U., Roache, P. J., Freitas, C. J., Coleman, H., Raad, P. E. Procedure for estimation and reporting of uncertainty due to discretization in CFD applications, J. Fluids Eng. 130 (7). 078001-078001, 2008.

Page 15 of 30

[22] Wang, C. C., Chi, K.Y. and Chang, C. J., Heat transfer and friction characteristics of plain fin-and-tube heat exchangers, part II: Correlation, International Journal of Heat and Mass Transfer 43, pp. 2693-2700, 2000. [23] Appa, A., Gulhane, N. P. CFD analysis of fin tube heat exchanger using rectangular winglet vortex generator, International Journal of Emerging Technologies in Computational and Applied Sciences, IJETCAS 14-462, 2014. [24] Thevenin, D., Janiga, G., Optimization and Computational Fluid Dynamics, Institut für Strömungstechnik und Thermodynamik (ISUT), Springer-Verlag Berlin Heidelberg, Germany, 2008. [25] Godazandeh, K., Ansari, M. H., Godazandeh, B., Ashjaee M., Optimization of the angle of attack of delta-winglet vortex generators over a bank of elliptical-tubes heat exchanger, Proceedings of the ASME 2013 International Mechanical Engineering Congress and Exposition IMECE2013, San Diego, California, USA, 2013. [26] Jang, J-Y, Hsu, L-F, Leu, J-S, Optimization of the span angle and location of vortex generators in a plate-fin and tube heat exchanger, International Journal of Heat and Mass Transfer 67, 432-444, 2013. [27] Abdollahi, A., Shams, M., Optimization of shape and angle of attack of winglet vortex generator in a rectangular channel for heat transfer enhancement, Applied Thermal Engineering 81, 376-387, 2015. [28] Rao, S. S., Engineering Optimization: Theory and Practice, Fourth Edition, Published by John Wiley & Sons, Inc., Hoboken, New Jersey, 2009. [29] Nelder, J. A., Mead, R., Computer Journal, 7, 308, 1965. [30] Garrison, S. L., Hardy, B. J., Gorbounov, M. B., Tamburello, D. A., Corgnale, C., vanHassel, B. A., Mosher, D. A., Anton, D. L., Optimization of internal heat exchangers for hydrogen storage tanks utilizing metal hydrides, International Journal of Hydrogen Energy 37, 2850-2861, 2012. [31] Cavazzuti, M., Agnani, E., Corticelli, M. A., Optimization of a finned concentric pipes heat exchanger for industrial recuperative burners, Applied Thermal Engineering 84, 110-117, 2015. [32] Lagarias, J. C., Reeds, J. A., Wright, M. H. and Wright, P. E., Convergence properties of the Nelder-Mead Simplex method in low dimensions”, SIAM J. Optim., 9(1): 112-147, 1998. [33] Webb, R. L., Principles of Enhanced Heat Transfer, John Wiley & Sons, New York, 1994.

Page 16 of 30

Fig. 1. Periodic characteristic of the heat exchanger.

Fig. 2. Symmetry characteristics of the heat exchanger for both inline and staggered arrangements.

Fig. 3. Computational domain including the core and extended regions.

Page 17 of 30

Fig. 4. Numerical verification based on Wang’s correlations [22] for Friction factor (f) and Colburn factor (j).

(a)

(b)

Fig. 5. Input parameters of the vortex generator, (a) Roll angle, and (b) Attack angle.

Fig. 6. Representation of each check-node (cn) for a generic vortex generator.

Page 18 of 30

(a) Present work J F , Re = 250

1

(c) Present work JF 3 , Re = 250

(b) Present work J F , Re = 650

1

(d) Present work JF 3 , Re = 650

Page 19 of 30

(a) Present work J F , Re = 250

1

(c) Present work JF 3 , Re = 250

(b) Present work J F , Re = 650

1

(d) Present work JF 3 , Re = 650

Page 20 of 30

(a)

(b)

(c)

(d)

Page 21 of 30

(a)

(b)

(c)

(d)

Page 22 of 30

(a)

(b)

(c)

(d)

Fig. 11. Distribution of span-average Colburn factor (j) and Relative Pressure for both tube arrangements and Re = 250.

Page 23 of 30

(a)

(b)

(c)

(d)

Fig. 12. Distribution of span-average Colburn factor (j) and Relative Pressure for both tube arrangements and Re = 650.

Page 24 of 30

Page 25 of 30

Table 1 Geometric dimensions for the compact heat exchanger. Fin thickness Fin pitch Tube diameter Fin collar outside diameter Transverse pitch Longitudinal pitch Depth Number of tube rows

δf FP Do Dc = Do + (2δf) Pt Pl L N

0.11 mm 1.50 mm 7.00 mm 7.22 mm 18.00 mm 12.50 mm 25 mm 2

Table 2 Mesh characteristics with VG. On Core region, Fig. 1 Cells number, n

Grid 1 3,023,595

Grid 2 1,153,386

Grid 3 517,535

Mesh size, h  V n  Refinement factor, r

0.0450

0.0621

0.0811

0.7 1.1 0.4 0.8

1.38 1.0 1.6 0.6 1.1

1.31 1.2 2.0 0.8 1.5

Maximum y Average y 



1 3

Re = 250 Re = 650 Re = 250 Re = 650

Table 3 Grid Convergence Index (GCI) for the finest mesh with VG.

GCI (%)

Friction factor, f Re = 250 Re = 650 0.99 0.06

Colburn factor, j Re = 250 Re = 650 1.00 0.34

Table 4 Input parameter ranges for the solution design space (Fig. 2). Input parameter range 0 < a1 < 0.97Fp

VG1

VG2

(mm)

0 < b1 < 0.97Fp 1.0 < c1 < Pt 2

(mm)

-60 < φ1 < +60

(º)

(mm)

-75 < θ1 < +75

(º)

0.2 < x1 < 0.99Pl 0.2 < y1 < 0 . 99 Pt 2

(mm)

0 < a2 < 0.97Fp

(mm)

0 < b2 < 0.97Fp 1.0 < c2 < Pt 2

(mm)

-60 < φ2 < +60

(º)

-75 < θ2 < +75

(º)

1.01Pl < x2 < 1.99Pl 0.2 < y2 < 0 . 99 Pt 2

(mm)

(mm)

(mm)

(mm)

Page 26 of 30

Table 5 Summary of the vortex generators positions and angles according to the reference point shown in Fig. 2. Input variables Code Researches

Re

x1 IN 7.1 250/650 ST 7.1

A

Torii [9]

B

Jang [10]

250/650

C

Lemouedda [8] CFU

250/650

D

Lemouedda [8] CFD

250/650 250/650

JF

E

Salviano [7] 1

JF

3

250

JF 1

F

Present Work

JF

3

250 650

JF 1

JF

250/650

3

650

Vortex Generator 1 y1 θ1 φ1 a1 b1 6.3 -15 0 1.4 0.0 6.3 -15 0 1.4 0.0

c1 7.2 7.2

x2 19.6 19.6

Vortex Generator 2 y2 θ2 φ2 a2 b2 6.3 -15 0 1.4 0.0 2.7 15 0 1.4 0.0

c2 7.2 7.2

IN ST IN ST IN ST IN ST IN ST IN ST

10.9 10.9 10.4 10.4 10.4 10.4 7.9 7.9 5.3 5.3 1.2 7.7

4.2 4.2 4.4 4.4 4.4 4.4 5.6 5.6 5.6 5.6 3.2 5.6

42 42 -60 -60 60 60 -28 -28 -32 -32 -15 -27

0 0 0 0 0 0 -5 -5 11 11 -40 -13

0.5 0.5 1.5 1.5 1.5 1.5 1.3 1.3 1.3 1.3 1.1 1.1

0.5 0.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.1 0.3

2.8 2.8 3.0 3.0 3.0 3.0 2.6 2.6 2.6 2.6 1.3 6.0

23.4 23.4 22.9 22.9 22.9 22.9 23.3 23.3 23.3 23.3 21.4 20.6

4.2 4.8 4.4 4.6 4.4 4.6 3.8 5.2 3.8 5.2 4.1 4.6

42 -42 -60 60 60 -60 32 -32 32 -32 -30 27

0 0 0 0 0 0 3 -3 3 -3 0 -3

0.5 0.5 1.5 1.5 1.5 1.5 0.0 0.0 0.0 0.0 1.1 1.0

0.5 0.5 0.0 0.0 0.0 0.0 1.3 1.3 1.3 1.3 0.8 0.8

2.8 2.8 3.0 3.0 3.0 3.0 2.6 2.6 2.6 2.6 4.5 4.7

IN ST IN ST IN ST

8.6 6.3 2.0

6.8 6.0 4.4

-32 -33 -9

11 7 -32

0.8 1.0 1.0

1.0 0.5 1.1

6.8 7.6 2.1

14.1 19.4 21.3

4.4 4.4 4.1

62 31 -27

1 17 -7

0.4 0.8 1.2

1.1 0.8 0.6

4.6 4.8 4.8

7.1 7.8

5.2 5.8

-23 -27

11 -7

1.0 0.7

0.5 1.1

5.3 7.5

19.9 14.5

4.3 4.2

21 43

-3 11

1.4 0.3

0.4 1.2

5.4 4.2

6.7

5.0

-33

-3

0.8

0.9

4.5

19.5

3.8

25

-17

1.1

0.6

4.3

Table 6 Shape comparison of the vortex generators applied by previous researches Code Researches

Re

Tube arrangement

A

Torii [9]

250/650

IN/ST

B

Jang [10]

250/650

IN/ST

C D

Lemouedda [8] CFD Lemouedda [8] CFU

250/650

IN/ST

E

Salviano [7]

250/650

IN/ST

Vortex generator 1 and 2

Page 27 of 30

F

Present work

250/650

IN/ST

According to Fig. 7 and Fig. 8

Table 7 Comparison of temperature field with iso-value lines in different positions along of the fin. JF, Eq. (29)

JF⅓, Eq.

(30)

Re = 650

Staggered

Re = 250

Re = 650

Inline

Re = 250

Fin-tube

Page 28 of 30

Table 8 Velocity field and streamlines in three different positions for plane-z in inline and staggered tube arrangements for Re = 250. (z/Fp) = 0.25

(z/Fp) = 0.50

(z/Fp) = 0.75

Staggered Inline Staggered Inline Staggered

Present work, JF⅓

Present work, JF

Fin-tube

Inline

Re = 250

Page 29 of 30

Table 9 Velocity field and streamlines in three different positions for plane-z in inline and staggered tube arrangements for Re = 650. (z/Fp) = 0.25

(z/Fp) = 0.50

(z/Fp) = 0.75

Staggered Inline Staggered Inline Staggered

Present work, JF⅓

Present work, JF

Fin-tube

Inline

Re = 650

Page 30 of 30