Effect of Flow Attack Angle of V-Ribs Vortex Generators in a Square Duct on Flow Structure , Heat Transfer , and Performance Improvement

A numerical investigation has been carried out to examine the periodic laminar flow and heat transfer characteristics in a threedimensional isothermal wall square duct with 20 inline V-ribs. The computations are based on the finite volume method, and the SIMPLE algorithmhas been implemented.Thefluid flow andheat transfer characteristics are presented for Reynolds numbers based on the hydraulic diameter of the square duct ranging from 100 to 2000. To generatemain streamwise vortex flows through the tested section, V-ribs with an attack angle of 20 are mounted in tandemwith inline arrangement, pointing downstream (V-Downstream) and pointing upstream (V-Upstream) placed on both the upper and lower walls. Effects of different blockage ratio (b/H, BR) with a single pitch ratio (P/H, PR) of 1 on heat transfer, pressure loss, and performance in the ribbed tube are studied. Apparently in each of the main vortex flows, streamwise twisted vortex flows can induce impinging flows on the walls of the interbaffle cavity leading to drastic increase in heat transfer rate over the square duct. In addition, the rise in the V-baffle height results in the increase in the Nusselt number and friction factor values. The computational results show that the optimum thermal enhancement factor is about 4.2 at BR = 0.20 and 0.15 for the V-Downstream and V-Upstream, respectively.


Introduction
The increasing necessity for saving energy and material imposed by the diminishing world resources and environmental concerns have prompted the development of more effective heat transfer equipment with improved heat transfer rates. In many industrial systems, heat must be transferred either to input energy into the system or to remove the energy produced in the system. Considering the rapid increase in energy demand world-wide, both reducing energy lost due to ineffective use and enhancement of the energy transfer in the form of heat has become an increasingly important task for the design and operation engineers for such systems.
From the reasons above, the performance improvement investigation into the heat exchanger system has been interested. Laminar flow behaviors in a channel fitted with 90 ∘ transverse baffles mounted on two opposite walls with a staggered array were studied by Berner et al. [1] who found that the flow is free of vortex shedding at a Reynolds number below 600. Webb and Ramadhyani [2] numerically investigated the fluid flow and heat transfer characteristics in a smooth channel attached with staggered baffles. Kelkar and Patankar [3] reported that the heat transfer in a channel with staggered baffle increases with the rise in baffle height and with the decrease in baffle spacing. Lopez et al. [4] carried out a numerical investigation on laminar forced convection in a three-dimensional channel with baffles for periodically fully developed flow and with a uniform heat flux at the top and bottom walls. The effect of a single baffle in the entrance region on thermal behaviors in a channel was studied by Guo and Anand [5]. Ko and Anand [6] experimentally studied on turbulent channel flow with porous baffles. They found that the porous baffles give a higher level of turbulent flow than solid baffles. Mousavi and Hooman [7] numerically examined the heat transfer behavior in the entrance region of a laminar channel flow over staggered baffles and reported that the Prandtl number affects the precise location of the periodically fully developed region.
Promvonge et al. [8] presented a numerical investigation on laminar flow and heat transfer characteristics in a threedimensional isothermal wall square-channel fitted with inline 45 ∘ V-shaped baffles on two opposite walls. Apparently the longitudinal counter-rotating vortex flows created by the Vbaffle can induce impingement/attachment flows over the walls resulting in a greater increase in heat transfer over the test channel. They found that the V-baffle with blockage ratio BR = 0.2 and pitch ratio PR = 1.5 yields the maximum thermal enhancement factor (TEF) about 3.8, whereas the Nu/Nu 0 is around 14 times above the smooth channel at higher Re.
Promvonge and Kwankaomeng [9] also numerically studied periodic laminar flow and heat transfer characteristics in a three-dimensional isothermal wall channel of aspect ratio, AR = 2 with 45 ∘ staggered V-baffle. They reported that the optimum thermal enhancement factor is around 2.6 at the baffle height of 0.15 times of the channel height for the Vbaffle pointing upstream while it is about 2.75 at the baffle height of 0.2 times for the V-baffle pointing downstream.
In [8,9], the heat transfer augmentation leads to the increase in enlarging pressure. Therefore, the improvement of vortex generators is to reduce the pressure loss that is done by study parameters of the vortex generators, especially, flow attack angle of the vortex generators.
The main aim of the present research is to study numerically the 20 ∘ V-ribs configuration effect on heat transfer, pressure loss, and thermal performance in the square duct is presented. This was decided after the literature search that has revealed that no work has been reported on the numerical computation of the flow in 20 ∘ V-ribbed square duct using the full form of the Navier-Stokes equations. Regarding the above literature reviews, the flow attack angle of 20 ∘ V-ribs leads to significant influence on heat transfer characteristic and helps to reduce the pressure loss. The main objective of the present investigation is to study the influence of the 20 ∘ V-rib with rib height ratio BR = / = 0.1-0.3 at single pitch spacing ratio PR = 1 for pointing downstream (V-Downstream) and pointing upstream (V-Upstream) on the flow field, temperature field, heat transfer rate, friction characteristics, and thermal performance.

Flow Description
2.1. Rib Geometry and Arrangement. The system of interest is a square duct with a 20 ∘ V-rib pair placed on both the upper and lower walls in tandem for inline arrangement and pointing on two different positions, V pointing downstream (V-Downstream) and V pointing upstream (V-Upstream) as shown in Figure 1. The flow under consideration is expected to attain a periodic flow condition in which the velocity field repeats itself from one cell to another. The concept of periodically fully developed flow and its solution procedure has been described in [10]. The air enters the square duct at an inlet temperature, in , and flows over a 20 ∘ inline V-rib pair, where is the rib height, set to 0.05 m is the square duct hydraulic diameter, and / is known as the blockage ratio, BR = 0.1-0.3. The axial pitch, , or distance between the rib cell is set to = in which / is defined as the pitch spacing ratio, PR = 1.

Boundary Conditions.
Periodic boundaries are used for the inlet and outlet of the flow domain. The constant mass flow rate of air with 300 K (Pr = 0.7) is assumed in the inlet flow direction rather than constant pressure drop due to periodic flow conditions. The inlet and outlet profiles for the velocities must be identical. The physical properties of the air have been assumed to remain constant at average bulk temperature. Impermeable boundary and no-slip wall conditions have been implemented over the duct walls as well as the rib. The constant temperature of all the duct walls is maintained at 310 K while the rib plate is assumed at adiabatic wall conditions.

Grid Independent.
A grid independence procedure was implemented using the Richardson extrapolation technique over grids with different number of cells. The characteristics of three grids, 87,320, 126,000, and 186,000 cells, are used in the simulation. The variation in Nu and values for 20 ∘ Vribs at PR = 1, Re = 800, and BR = 0.20 is less than 0.25% when increasing the number of cells from 126,000 to 186,000; hence there is no advantage in increasing the number of cells beyond this value. Considering both convergent time and solution precision, the grid system of 126,000 cells was adopted for the current computational model.

Mathematical Foundation
The numerical model for fluid flow and heat transfer in a square duct was developed under the following assumptions. Based on the above assumptions, the tube flow is governed by the continuity, the Navier-Stokes equations, and the energy equation. In the Cartesian tensor system these equations can be written as follows: Continuity equation: Momentum equation: Energy equation: where Γ is the thermal diffusivity and is given by Apart from the energy equation discretized by the QUICK scheme, the governing equations were discretized by the second order upwind scheme, decoupling with the SIM-PLE algorithm and solved using a finite volume approach [11]. The solutions were considered to be converged when the normalized residual values were less than 10 −5 for all variables but less than 10 −9 only for the energy equation.
Four parameters of interest in the present work are the Reynolds number, friction factor, Nusselt number, and thermal enhancement factor. The Reynolds number is defined as The friction factor, , is computed by pressure drop, Δ across the length of the periodic tube, as follows: The heat transfer is measured by the local Nusselt number which can be written as follows: The average Nusselt number can be obtained by The thermal enhancement factor (TEF) is defined as the ratio of the heat transfer coefficient of an augmented surface, ℎ to that of a smooth surface, ℎ 0 , at an equal pumping power and is given by [12] where, Nu 0 and 0 stand for Nusselt number and friction factor for the smooth duct, respectively.

Validations Test.
In this chapter, the verification is performed to ensure that the computation is reliable. Verification of the heat transfer and friction factor of the square duct without V-rib is performed by comparison with the previous values under similar operating condition as shown in  Figure 3(b) but different in rotating direction. They concluded that the appearance of the vortex flows can help to increase higher heat transfer in the square duct because of highly transporting the fluid from the central core to the near wall regimes.
The plots of streamlines impingement flows on the lower for the V-Downstream and V-Upstream cases are shown in Figures 5(a) and 5(b), respectively. In the figures, it can be noted that impinging jets occur periodically in a region on the lower wall in the rib cavity (as well on the upper wall due to symmetry). A close look shows that the impinging jet on the wall comes from the helical flows rolling up at the side. The helical vortex flow moves along the rib cavity to the RTE (rib trailing edge) side and rolls up to impinge on the wall. After impingement, the jet splits over the wall and recombines into two helical streams at the nearby rib end to create vortex flows again. The helical pitch length of the main vortex flow is about 5 before impingement and becomes shorter (about 3 ) after impingement. This means that the helical vortex flow passes five rib modules from a RTE side to the other RLE (rib leading edge) side before impingement. This behavior is identical on both the upper and lower parts so two streamwise vortices with nonuniform helical pitch are formed throughout the square duct. It can be concluded that vortex flows with nonuniform helical pitches can induce two impingement flows leads to the heat transfer augmentation.

Heat Transfer Characteristics. Figures 6(a) and 6(b)
display the contour plots of temperature field in transverse planes for the V-Downstream and V-Upstream ribs at Re = 800 and BR = 0.2, respectively. The figures show that there is a major change in the temperature field over the duct for both baffle cases. This means that the vortex flows provide a significant influence on the temperature field, because it can induce better fluid mixing between the wall and the core flow regions, leading to a high temperature gradient over the heating wall. The higher temperature gradient can be observed where the flow impinges the walls, while the lower one is found at the RTE sidewall area for V-Downstream cases and on the upper and lower walls for V-Upstream case where the temperature in this region is somewhat high. The results show similar trends with 45 ∘ V-ribs [8] cases as presented in Figure 7.
Local Nu contours of the square duct walls with the Vrib at Re = 800, PR = 1, and BR = 0.20 are presented in Figures 8(a) and 8 respectively. In these figures, it appears that the V-Downstream rib shows higher Nu values over the square duct walls especially on two sidewalls, while the V-Upstream baffle presents higher Nu values on both the upper and lower walls of square duct. This indicates a merit of employing the V-discrete baffle over the smooth tube for enhancing heat transfer. For 45 ∘ V-Downstream rib, as shown in Figure 9, the higher heat transfer area is found to be at the RTE sidewall similar to 20 ∘ V-rib case but of different values of heat transfer rate.
The variation of the average Nu/Nu 0 ratio with BRs at different Reynolds number values is depicted in Figures 10(a) and 10(b) for V-Downstream and V-Upstream, respectively. It is worth noting that the Nu/Nu 0 value tends to increase with the rise of Reynolds number for all cases. The higher BR value results in the increase in the Nu/Nu 0 value. The maximum heat transfer rate is found to be about 13 and 12 times higher than smooth square duct with no ribs for V-Downstream and V-Upstream cases, respectively. Thus, the generation of vortex flows from using the V-rib as well as the role of better fluid mixing and the impingement is the main reason for the augmentation in heat transfer of the square duct. The use of the V-rib with range studied yields heat transfer rate of about 1-13 times higher than the smooth square duct with no rib, while the 45 ∘ V-rib gives the highest heat transfer rate around 1-20 times, reported by Promvonge et al. [8]. values. The / 0 value for both V-ribs is found to be about 1-52 times over the smooth square, duct while the 45 ∘ V-rib gives very enlarged pressure loss of about 1.1-225 times higher than smooth duct [8]. This means that the use of lower flow attack angle helps to reduce the pressure of the system. Figure 12 exhibits the variation of thermal enhancement factor (TEF) for air flowing in the ribbed square duct. In the figure, the enhancement factor of both V-discrete baffles tends to increase with the rise of Re values. The maximum TEF of 20 ∘ V-rib is about 4.2 at BR = 0.20 and 0.15 for the V-Downstream and V-Upstream, respectively. In comparison with 45 ∘ V-rib, on both 20 ∘ and 45 ∘ V-rib gives nearly value of the maximum TEF although the 20 ∘ Vrib provides lower heat transfer rate.

Conclusions
Laminar periodic flow configurations and heat transfer characteristics in a square duct fitted with 20 ∘ V-Downstream and V-Upstream rib elements in tandem, inline arrangements placed on both the upper and lower walls of the tested duct have been investigated numerically.  transfer in the ribbed duct on both V-Downstream and V-Upstream cases. (ii) The order of heat transfer enhancement is 1-13 time higher than smooth duct for using the V-ribs at BR = 0.10-0.20 with single pitch ratio, PR = 1.
(iii) The pressure loss in the range studied is ranging from 1 to 52 times above the smooth plain duct that lower than 45 ∘ flow attack angle of the V-rib. (iv) Thermal enhancement factors for both V-ribs are found to be in a range of 1.00-4.20 and the maximum TEF found at BR = 0.20 and 0.15 for the V-Downstream and V-Upstream, respectively, at the highest Reynolds number. (v) It is noted that the use of 20 ∘ V-rib gives lower heat transfer rate than 45 ∘ V-rib but can help to reduce the pressure loss of the system.

Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.