Effects of in-line deflectors on the overall performance of a channel heat exchanger

ABSTRACT The turbulent convective thermal transfer in channel heat exchangers (CHEs) is studied numerically via the CFD (Computational Fluid Dynamics) method. Deflectors are inserted on the hot bottom walls of the heat channel to enhance the hydrothermal characteristics. Various shapes of in-line deflectors are considered, namely: rectangular (a/b = 0.00), cascaded rectangular-triangular (a/b = 0.25, 0.50, and 0.75), and triangular (a/b = 1.00) shapes. From the obtained results, the inclusion of in-line deflectors with a/b = 0.75 has given the most significant thermal enhancement factor, which was higher than that for a/b = 0.00, 0.25, 0.50, and 1.00 by about 5.36, 5.06, 67.27, and 3.88%, respectively. Also, the in-line cascaded deflector’ case (a/b = 0.75) shows an increase in the enhancement factor (η) from 4 to 15.44% over the cases of one deflector (corrugated, rectangular, triangular, trapezoidal, arc, (+), S, 45° V, 45° W, T, Γ, and ε-shaped) or two deflectors (staggered corrugated). This highlights the effectiveness of in-line cascaded rectangular-triangular deflectors with a/b = 0.75 in improving the performance of the proposed exchanger for the conditions adopted.


Introduction
The continuous increase of the global energy demand caused awareness about climate change and energy shortages. The renewable and environmentally sustainable energies, such as solar energy, have become an efficient alternative in various sectors, i.e. industrial, domestic, rural, etc.
Pipes and channels with attached and detached baffles, fins and ribs are the focus of many research works. Among them, Demartini et al. (2004) examined the details of the turbulent airflow in a duct equipped with two baffles. For a baffled duct heat exchanger, Ary et al. (2012) reported that two baffles yield greater thermal exchange rates than a single baffle and the number of holes has a considerable influence on the flow behavior. The study performed by Khan et al. (2002) revealed that the baffles create larger flow disturbance than that of the ribs. Promvonge and Thianpong (2008) carried out experiments to determine the hydrothermal characteristics of air through an exchanger provided with different shaped ribs (wedge, triangular, and rectangular) in in-line and staggered arrays. Hwang et al. (1999)   predicted the effect of rib length on the turbulent flow characteristics inside a duct. Zhao et al. (2016) carried out experiments on the influence of pin density and shapes (triangle, circular, elliptical, and square) on the flow in a rectangular duct. Wang et al. (2012) inspected Table 1. Structural parameters of the physical domain under study (Demartini et al., 2004).  Table 3. Necessary equations (Menni et al., 2020b;Yang & Hwang, 2003).
the flow and thermal exchange through a duct having various staggered pin fins, (drop, circular, and elliptical). Liu and Wang (2011) found in their study a significant enhancement in the local thermal exchange when using semi-attached ribs in channels. Ameur (2020) presented an investigation of the hydrothermal characteristics of a channel equipped with wavy baffles. Other researchers were interested in the investigation of the effect of baffle Hydro-dynamic diameter of the exchanger, Dh Nu0(for Re ≥ 10,000) (Dittus & Boelter, 1930) Nu 0 = 0.023Re 0.8 Pr 0.4 f 0 (for 3,000 ≤ Re ≤ 5 × 10 6 ) (Petukhov, 1970) where, ε M is the diffusivity for momentum transfer, ε H is the diffusivity for heat transfer, P is the pressure drop, p subscript is the 1st inner-node from the solid-wall, T p is the temperature at the cell adjacent to the wall, y p is the distance from point p to the wall, and Nu 0 and f 0 are the Nusselt number and friction factor for the smooth exchanger. inclination on the flow and thermal exchange behaviors (Eiamsa-ard and Chuwattanakul, 2020). Yongsiri et al. (2014) reported that, at high Reynolds number, the 60°a nd 120°inclination angles of ribs generate comparable thermal exchange rates and heat performance factors that are the most significant than those of the other cases. Promvonge et al. (2015) determined the influence of inclined horseshoes baffles on the overall efficiency of a tubular heat exchanger. For a duct fitted with inclined perforated vortex generators (VGs), Dutta and Hossain (2005) reported that the local thermal exchange is dependent on the design, orientation, and location of the 2ndVG. For an air heater channel, Bopche and Tandale (2009) determined the impact of U-shaped VGs inserted on the absorber surface. The U-and V-shaped baffles have also been studied by many researchers (Misra et al., 2020;Promvonge & Skullong, 2020). The porous baffles were also the subject of some investigations (Ameur & Menni, 2019;Mesgarpour et al., 2018). Detailed reviews on the various techniques to enhance the performance of CHEs (Channel heat-exchangers) may be found in the literature (Alam & Kim, 2017;Kabeel et al., 2017).
According to both the numerical and experimental models, it is well evident that finning technology has a sensitive effect on reinforcing the flow field and improving the heat transfer within CHEs. Several authors and engineers have studied and investigated different CHEs with very complex deflectors in an overlapping arrangement in most of their studies. In our proposed research, new models of in-line regular-deflectors are highlighted to increase the efficiency of CHEs. In the present analysis, steady-state, Newtonian, and incompressible flows of air over in-line deflectors in channels with in-line rectangular (a/b = 0.00), cascaded rectangular-triangular (a/b = 0.25, 0.50, and 0.75), and triangular (a/b = 1.00) shaped deflectors fitted to the hot bottom walls of CHEs are simulated and analyzed numerically. These deflectors direct the fluid flow to circulate in a very small section in the channel duct, to lengthen the trajectory of the fluid, to enhance the heat exchange surface, to force the recirculation zones, to create the turbulence and therefore a successful heat transfer. On the other hand, a comparison of the obtained optimum deflector performance with other deflectors from the literature (Menni et al., 2018a(Menni et al., , 2018b) is shown to highlight the importance of the finning technique for such exchangers.

Channel heat exchanger under study
The hydrothermal characteristics of a CHE are determined in this paper. The computational domain of the problem studied is given in Figure 1. It is a channel fitted with in-line transverse, vertical, and solid deflectors on its bottom wall. The channel length-to-hydraulic diameter, width-to-height aspect ratio of channel, deflector height-to-channel height, and deflector spacingto-channel height ratio are fixed at, L/D h = 3.317, W/H = 1.321, h/H = 0.547, and Pi/H = 0.972, respectively. These values were considered basing on the experiments of Demartini et al. (2004).

Vortex generators under investigation
Effects of the deflector shape were examined by realizing three geometrical cases, namely: flat rectangular (a/b = 0 or case A in Figure 2a), cascaded rectangular-triangular (a/b = 0.25, 0.50, and 0.75 or case B in Figure 2b), and triangular (a/b = 1.00 or case C in Figure 2c).

Used geometrical dimensions and materials' thermal-physical properties
The required details on geometrical parameters are summarized in Table 1. The thermal-physical properties of the working fluid (air) are summarized in Table 2.

Assumptions and hydrothermal boundary conditions
The study concerns turbulent flow and forced-convection of an incompressible Newtonian fluid (Pr = 0.71) with constant thermal physical properties and flowing inside a channel. The 2D steady-state with neglected radiation thermal exchange mode are treated. Air enters the exchanger at 300 K with zero pressure (P in = 0) (Nasiruddin & Siddiqui, 2007). A uniform 1D velocity profile (u = U in , v = 0) is set at the exchanger inlet (Demartini et al., 2004). A temperature of 375 K is set on the entire walls of the exchanger (Nasiruddin & Siddiqui, 2007). The walls of the exchanger, as well as the deflectors, are considered impermeable and no-slip (Demartini et al., 2004). At the exchanger outlet, the atmospheric pressure is set P ex = P atm (Demartini et al., 2004). Further details on the theoretical and mathematical tools are provided in (Menni et al., 2020b;Yang & Hwang, 2003).

Governing equations and parameters
For these considerations, the governing-equations are given in the common form as follows (Menni et al., 2020c(Menni et al., , 2021Yang & Hwang, 2003): where φ is a vector composed of the scalars u, v, T, k, and ε. φ is the turbulent diffusion coefficient, and S φ is the φ source term associated (Table 3). The turbulent-air characteristics are modeled by the standard-type k-ε model (Launder & Spalding, 1974). All the parameters characterizing fluid transport and heat transfer are detailed in the following table (Table 4).

Numerical schemes
The Quick(Quadratic Upstream Interpolation for Convective Kinetics) scheme (Leonard & Mokhtari, 1990) and the SIMPLE(Semi Implicit Method for Pressure Linked Equations algorithm) algorithm (Patankar, 1980)  were employed to achieve computations with the computer software 'ANSYS Fluent 12.0 (2012)'. The probed exchanger is simulated with a two-dimensional, non-uniform, structured, and quadrilateral mesh (Figure 3), as it is very concentrated near the solid boundary.

Mesh density tests
Several grids have been adopted with different (n x × n y ) nodes, starting from 95 to 370 according to the horizontal axis, while from 35 to 145 according to the vertical axis. The performance values (η) were checked according to the hot top wall (0 ≤ x ≤ L, y = H/2) for Re = 3.2 × 10 4 and for all deflector models considered (Table 5). Comparing the different-cell results shows that it is not necessary to raise the number of cells from (245 × 95) to (370 × 145) since the relative error does not exceed 1 percent in the case of a/b = 0.00, 0.25 and 0.50 models, while it does not exceed 1.5 percent in the case of the remaining models (a/b = 0.75 and 1.00). Then, the incoming mesh of (245 × 95) nodes is built for all investigated models.

Turbulence model tests
Five different models are considered, namely (i) Standard -type k-ε, (ii) RNG-k-ε, (iii) Realizable-k-ε, (iv) Standardtype k-ω, and (v) SST-k-ω model. Beginning by the determination of the Nusselt number (Nu 0 ), where Nu 0 values are provided in Figure 4(a) in the case of 1.2 × 10 4 ≤ Re ≤ 3.2 × 10 4 . The results obtained by the five various two-equation models are verified against those calculated by the empirical correlation of Dittus and Boelter (1930). We note that these values are given for a smooth exchanger. Firstly, an increase in Nu 0 is noted with the rise of Re for all turbulence cases. The comparison plots between the different models revealed that the standard-k-ε type is able to provide satisfactory Nu 0 values over the range of Reynolds number used here. The standard-k-ω and SST-k-ω models gave excellent predictions at low Re, but the deviation from those of the correlation increased at high Re. However, it seems that the RNG-k-ε and realizable-k-ε model cases perform better at high Re (as observed for Re ≥ 2.2 × 10 4 ).
In the same smooth exchanger, the variation in the f 0 vs. Re is plotted in Figure 4(b) for the five types of two-equation models. Verification and comparison of these plots is made against the experimental data of Petukhov (1970). From this figure, it seems that the standard k-ε model remains the most efficient for all values of Re. However, a significant deviation is observed for the other models, where the realizable-k-ε case gave the lowest amounts of f 0 and the SST-k-ω yielded the greatest values of f 0 at high Re, especially at Re = 3.2 × 10 4 .

Results and discussion
The overall performances of a heat exchanger depend on the working fluid, as well as the geometrical parameters (Abadi et al., 2020;Ramezanizadeh et al., 2019). The turbulent characteristics of a CHE with two in-line deflectors are evaluated in the present paper. Five different cases (a/b = 0.00, 0.25, 0.5, 0.75, and 1.00) are considered concerning the shape of deflectors. The results are reported for various Reynolds numbers and for the five mentioned cases in the form of streamlines and contours of velocity, dynamic pressure, turbulent kinetic energy and isotherms. The variations of friction factor, Nusselt number, and hydrothermal enhancement factor are also analyzed.
As observed, the streamlines are disturbed near the deflectors on their left and right sides for all studied cases. In the region before the first deflector, the airflow is disturbed near its left side. The presence of a deflector on the lower section of the exchanger allows the current to be directed towards the top region while forming a small recirculation loop at the bottom of the deflector. The current moves over the first deflector, splitting at the front of the sharp edge of the same deflector with the formation of an intense recycling cell on its right side. These cells extend to the left wall of the second deflector. In addition, a large recycling cell is generated behind the second deflector in the upper part of the exchanger. In this figure, the formation of vortices within the baffled channel is similar in all cases, but with different intensities.
At Re = 12,000, the velocity fields of air are plotted in Figure 6 for five different sizes of deflectors, namely: a/b = 0.00, 0.25, 0.50, 0.75, and 1.00. In the following areas, the air velocities are low next to the right sides of both deflectors due to the existence of the recirculation loops in these regions. In the area between the deflector tip and the upper surface of the exchanger, an increase in the axial velocity is observed due to the limitation of the flow passage that is resulted by the presence of deflectors.
At the same Re number, the change in the dynamic pressure (P) is plotted in Figure 7 for three types of deflectors (flat, cascaded, and triangular) in an in-line arrangement. As expected, the most significant amount of pressure (P max ) is obtained at the upper part of the channel, through the gaps, next to its top wall, due to the intense velocity in these regions. While the P min value is observed in the recirculation regions that are located in the lower part, next to the right and left walls of both deflectors.
The distribution of turbulence kinetic energy (k) is highlighted in Figure 8 for the five considered values of a/b. In this figure, the Reynolds number is fixed at Re = 12,000. For all cases studied, the k max is found near the front top sharp edges of the first and second deflectors. However, the k min value is located in the upstream and downstream regions of the deflector, for all cases.
The isotherms are shown in Figure 9 for the five different a/b ratios. For all cases under investigation, small values of the temperature are remarked in the spaces between the tip of each deflector and the upper wall of channel. Moreover, the isotherms show that the vortex fluid temperature in the followed regions of the deflector is higher than that obtained in the same region of the smooth channel. respectively. At the axial location x = 0.159 m, i.e. before the first deflector, the airflow is highly intensified in the top region channel, approaching 114.61-141.12% of the inlet velocity (Figure 10a). However, and in the bottom part of channel, the velocity drops as the flow approaches the first deflector. Figure 10b addresses the variation of axial velocity(u) in the cross-section starting from the upper wall of the channel to the tip of the first deflector, at x = 0.223 m. A strong relationship between the fluid velocity and the ratio a/b is observed in this figure. In terms of improved velocities, the inclusion of in-line deflectors in its triangular geometry (a/b = 1.00) performs better than that of the other deflector cases. The u value in the case a/b = 1.00 is found to be higher by about 18. 89, 9.39, 6.71, and 4.03% over the air baffled channel with a/b = 0.00, 0.25, 0.5, 0.75, and 1.00, respectively. After the first obstacle, at x = 0.218, 0.285, and 0.315 m from the channel inlet, there is appearance of recycling cells in the bottom channel area with low negative values of velocity. These vortices are extended with the changes of the obstacle geometry from the flat rectangular form (a/b = 0.00) to the triangular from (a/b = 1.00). In the upper part of the exchanger, the air current starts to accelerate toward the gap formed between the tip of the second deflector and the opposite wall (Figure 10c and d).

Axial velocity profiles in various stations
In addition, the cascaded rectangular-triangular geometry of the obstacle in the case a/b = 0.75 shows most    and 3.89-9.37%, respectively, compared to the triangular obstacle (a/b = 1.00).

Heat transfer characteristics
Effect of the variation of Reynolds number and obstacle size (a/b = 0.00, 0.25, 0.5, 0.75 and 1.00) on Nu is plotted and reported in Figure 11. As expected, the value of Nu/Nu 0 augments with the raise of Re value in all cases studied. The Nu max is obtained for a/b = 0.75 at Re = 32,000, while the Nu min is reached at a/b = 0.50 for all Re values. At Re = 32,000, the improvements in Nu values for a/b = 0. 00, 0.25, 0.5, 0.75 and 1.00, respectively, are about 320.73, 315.73, 118.79, 355.43, and 323.33% over the smooth channel. The inclusion of obstacles with a/b = 0.75 at Re = 32,000 gives high Nu values than that with a/b = 0.00, 0.25, 0.50, and 1.00 around 9. 76, 11.16, 66.57, and 9.03%, respectively.

Friction loss
The impact of the deflector geometry and Reynolds number on the normalized average skin friction coefficient (f/f 0 ) is highlighted in Figure 12. An enhancement in f values ranging from 3.75 to 24.31 times over the smooth channel is obtained. The amount of f at Re = 32,000 and a/b = 0.50 is found to be around 18. 96, 23.31, 7.31, and 20.26% higher than that with a/b = 0.00, 0.25, 0.75, and 1.00, respectively. Figure 13 reports the changes in the thermal enhancement factor (η)vs Re for all geometrical configurations under study. H augments with the raise of Re. At Re = 32,000, the insertion of deflectors with a/b = 0.75 yields the most significant η. However, the deflectors with a/b = 0.50 yield the lowest η for all Re values considered here. At Re = 32,000, the most significant values of η is reached with a/b = 0.75, which is higher by about 5.36, 5.06, 67.27, and 3.88%, compared to a/b = 0.00, 0.25, 0.50, and 1.00, respectively.

Effect of shape on performance
Moreover, the optimum fin case, obtained with a/b = 0.75, is compared with various deflector models ( Figure 14) from the literature (Menni et al., 2018a(Menni et al., , 2018b. Under the same simulated conditions, the 0.75cascaded model shows an improvement in the performance over all other deflector cases. At Re = 32,000, the cascaded rectangular-triangular deflector case (a/b = 0.75) shows an increase in the enhancement factor (η) by about 10.795, 8.274, 7.486, 7.880, 11.189, 14.814, 11.662, 6.146, 15.445, 14.972, 12.371, 4.018 and 10.795% compared to the cases of one deflector (corrugated, rectangular, triangular, trapezoidal, arc, (+), S, 45°V, 45°W, T, , and ε-shaped) or two deflectors (staggered corrugated), respectively. This highlights the effectiveness of in-line cascaded rectangular-triangular deflectors with a/b = 0.75 in improving the performance of the proposed exchanger for the conditions adopted.

Conclusion
The hydrothermal characteristics of turbulent air flow inside a channel heat exchanger were determined with the CFD computations. The exchanger was equipped with deflectors to enhance the overall performances. Various geometrical configurations concerning the inserted deflectors, as well as various Reynolds numbers were considered in an attempt to augment the thermal enhancement factor. From all calculations, the main findings may be summarized as follows: • The distribution of streamlines within the channel was similar in all cases through the formation of vortices, but with various intensities. • The P max value was observed at the upper part of the channel, through the spaces, next to the top wall due to the intense flow velocity in these regions. However, the P min value was found in the recirculation loops that are formed in the lower part of channel, next to the right and left walls of both deflectors. • The k max was found near the tip of each deflector for all cases studied. However, the k min value was observed in the upstream, and downstream regions of deflectors for all cases studied. • The isotherms showed that the air temperature in the area behind the deflectors was higher than that obtained in the same region for the smooth channel. • At Re = 32,000, the inclusion of deflectors with a/b = 0.75 has given higher Nu values than those with a/b = 0.00, 0.25, 0.50, and 1.00 around 9.76, 11.16, 66.57, and 9.03%, respectively. • The f value at Re = 32,000 and a/b = 0.50 was found to be higher by about 18.96, 23.31, 7.30, and 20.26% than those with a/b = 0.00, 0.25, 0.75, and 1.00, respectively. • The most significant TEF was reached at Re = 32,000 and with a/b = 0.75, which was higher by about 5.36, 5.06, 67.27, and 3.88% compared to a/b = 0.00, 0.25, 0.50 and, 1.00, respectively. • At Re = 32,000, the cascaded rectangular-triangular deflector case (a/b = 0.75) showed an increase in the enhancement factor (η) by about 10. 795, 8.274, 7.486, 7.880, 11.189,14.814, 11.662, 6.146, 15.445, 14.972, 12.371, 4.018 and 10.795% compared to the cases of one deflector (corrugated, rectangular, triangular, trapezoidal, arc, (+), S, 45°V, 45°W, T, , and ε-shaped) or two deflectors (staggered corrugated), respectively. • This improvement in the thermal enhancement factor highlights the effectiveness of in-line cascaded rectangular-triangular deflectors with a/b = 0.75 in improving the performance of the proposed exchanger for the conditions adopted.
This study may be extended by examining the effect of thermo physical properties of the working fluids, as well as the dimensions, inclination, and arrangement (inline/staggered) of porous cascaded fins on the overall performances. The combination of fin, nanofluid, and porous media techniques may be also an interesting subject for future works on the enhancement of the performance of channel heat exchangers.

Disclosure statement
The authors declare that they have no conflict of interest.

Nomenclature b
Deflector height, m C p Specific heat capacity, J kg −1 K