Laminar Forced Convection and Heat Transfer Characteristics in a Square Channel Equipped with V-Wavy Surface

Corresponding Author: Withada Jedsadaratanachai Department of Mechanical Engineering, Faculty of Engineering, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand Email: kjwithad@kmitl.ac.th Abstract: The numerical investigations on flow pattern and heat transfer characteristic in a heat exchanger channel equipped with Vwavy surface are examined. The influences of the flow attack angles (α = 15°, 20°, 25°, 30°, 35°, 40°, 45°, 50°, 55° and 60°) and wavy surface arrangements (V-tip pointing downstream called “V-Downstream”, Vtip pointing upstream called “V-Upstream) are investigated for laminar regime, Re = 100-1200. The finite volume method with SIMPLE algorithm is used to solve the current research. The numerical result shows that the addition of the wavy surface in the heat exchanger channel can help to develop the heat transfer rate and thermal performance. The wavy surface generates the vortex flow through the test section that disturbs the thermal boundary layer on the heat transfer surface. The insertion of the wavy surface in the heating channel gives higher heat transfer rate around 1.5-10 times above the smooth channel depended on the flow attack angle, Reynolds number and wavy surface arrangement. In addition, the optimum thermal performance is around 3.0 at Re = 1200, α = 40° and V-Upstream.


Introduction
The effort to improve the thermal performance in various types of the heat exchangers is always seen in many industries. The thermal development can help to save production cost, thermal energy and also reduce the size of the heating system. The thermal development in the heat exchanger is divided into two methods; active and passive methods. The active method needs the additional power to enhance heat transfer rate and thermal performance, while the passive method is to generate vortex flow, swirling flow and to disturb the thermal boundary layer by turbulator or vortex generators. The passive method is more popular than the active method.
Many researchers attempt to create the appropriate vortex generator for various engineering works. For examples, the wavy surface (Duan et al., 2016;Khoshvaght-Aliabadi et al., 2016;Lotfi et al., 2016;Lu et al., 2017;Ranganayakulu et al., 2017;Xiao et al., 2017;Xu et al., 2015) is widely selected to augment heat transfer rate and performance in fin-and-tube heat exchanger. The wavy surface performs a better fluid mixing in the heating section that the reason for heat transfer augmentation. The amplitude, profile, flow attack angle, etc., of the wavy surface are important parameters that effect for heat transfer rate and thermal efficiency in the heat exchanger. The distinctive points of the wavy surface are as follows; easy for manufacture and maintenance, low pressure loss.
The V-rib/baffle (Abraham and Vedula, 2016;Deo et al., 2016;Fang et al., 2015;Jin et al., 2017;Kumar and Kim, 2016;Maithani and Saini, 2016;Promthaisong et al., 2016;Ravi and Saini, 2016;Singh and Ekkad, 2017) is another type of the vortex generators, which always use to increase heat transfer rate and thermal performance in tube/channel heat exchanger. The V-rib gives high heat transfer rate and thermal performance, but also performs enlarge pressure loss.
The new design of the vortex generators is the combination between wavy surface and V-rib called "Vwavy surface". The main aim of the new design is to support the generators production, maintenance and to remain the performance as the V-rib.
In the current research, the V-wavy surface is inserted in the middle of the heat exchanger channel to improve heat transfer rate and thermal efficiency. The influences of the flow attack angle (α = 15°, 20°, 25°, 30°, 35°, 40°, 45°, 50° and 60°), V-wavy surface arrangement (V-Downstream and V-Upstream) and Reynolds number (laminar flow, Re = 100-1200) on heat transfer, friction loss and thermal performance are investigated numerically. The numerical method can help to describe the flow and heat transfer mechanisms in the test section that is a way to improve the thermal system. The numerical results are reported in terms of flow visualizations, heat transfer characteristics and performance assessments. Figure 1 presents a heat exchanger channel inserted with V-wavy surface. The square channel height, H, is equal to 0.05 m. The square profile of the wavy surface is fixed at 0.2H×0.2H. The flow attack angle of the wavy surface is varied; α = 15°, 20°, 25°, 30°, 35°, 40°, 45°, 50°, 55° and 60°. The arrangement of the V-wavy surface is divided into two methods; Vtip pointing upstream called "V-Upstream" and V-tip pointing downstream called "V-Downstream". The periodic module of the present investigation is set around H. The laminar flow, Re = 100-1200, is considered for the present study.

Assumption and Boundary Condition
The current work is investigated under following assumptions: • The thermal properties of the test fluid (air) stay constant at average bulk mean temperature. • The Prandtl number of the test fluid is around 0.707.
• The flow and heat transfer are steady in three dimensions. • The laminar flow with incompressible condition is considered for the present study. • The force convection is regarded.
• The body force, viscous dissipation, radiation heat transfer and natural convection are unconsidered.
The computational domain is created with the boundary conditions as follows: • The inlet and outlet of the domain are produced with periodic boundary. • The channel walls are set with constant temperature around 310 K. • No slip wall condition is used for all surfaces of the computational domain. • The V-wavy surface is assumed as adiabatic wall (insulator).

Mathematical Foundation
The heating channel is governed by the continuity, the Navier-Stokes equations and the energy equation as Equation 1-3, respectively.
Continuity equation: Energy equation: Γ is the thermal diffusivity, which equal to: The energy equation and governing equations are discretized by the QUICK scheme and power law scheme, respectively. The present investigation is answered by finite volume method. The SIMPLE algorithm is selected for the test. The solutions are measured to be converged when the normalized residual values are less than 10 −5 for all variables, but less than 10 −9 only for the energy equation. The important variables are Reynolds number, friction factor, local Nusselt number, average Nusselt number and thermal enhancement factor.
The Reynolds number is calculated by: The friction factor, f, is determined by pressure drop, ∆p, through the periodic module, L: The local heat transfer is written as: The average Nusselt number can be obtained by: The Thermal Enhancement Factor (TEF) is considered by the augmentations on both heat transfer and friction factor at similar pumping power: The Nusselt number and friction factor for the smooth square channel are represented by Nu 0 and f 0 , respectively. Figure 2 shows the verification of the smooth heat exchanger channel on heat transfer and friction loss. The comparison of the numerical scheme with the value from the correlation is also reported. As the figure, it is found that the difference of the numerical scheme has no effect for the numerical result on both flow and heat transfer. The deviations on the Nusselt number and friction loss are less than ±0.5%.  Figure 3a and b plot the grid independence on heat transfer and friction loss for the computational domain of the heat exchanger channel inserted with V-wavy surface, respectively. The five different grid cells; 80000, 120000, 240000, 360000 and 420000, are compared. The numerical result shows that the increasing grid cell from 240000 to 360000 has no effect for friction factor and Nusselt number. Therefore, the grid around 240000 is applied for all cases of the present investigation. The optimum number of grid cell can help to save time for investigation and computer resource. Figure 4a and b illustrate the tangential velocity vector in transverse planes (x/H = 0.5, 1.75, 3, 4.25 and 5.5) for the heat exchanger channel inserted with V-Downstream and V-Upstream wavy surfaces, respectively, at Re = 600 and α = 45°. The insertion of the wavy surface on both arrangements changes the flow structure in the square channel. The wavy surface can create the vortex flow through the test section. The vortex flow disturbs the thermal boundary layer on the heat transfer surface that helps to improve the heat transfer rate and thermal performance. The vortex flow in the channel heat exchanger is detected in all cases. Figure 5a and b report the temperature distribution in transverse planes for the heat exchanger channel inserted with the V-Downstream and V-Upstream wavy surface, respectively, at Re = 600 and α = 45°. The installation of the wavy surface in the heat exchanger channel performs a better mixing of the air flow. The cold fluid distributes from the center of the channel, while the hot fluid near the channel wall performs thinner. The improvement of the heat transfer rate and thermal performance is due to the better fluid mixing.

Performance Analysis
The relations of the Nu/Nu 0 with Re at various flow attack angles for the heat exchanger channel installed with V-Downstream and V-Upstream wavy surfaces are plotted as Figure 6a and b, respectively. In general, the Nu/Nu 0 increases when enhancing the Reynolds number for both arrangements. The installation of the wavy surface in the heat exchanger channel provides higher heat transfer rate than the smooth square channel (Nu/Nu 0 > 1) in all cases. The heat transfer rate is around 2-9 and 1.5-10 times above the smooth channel for the wavy surface with V-Downstream and V-Upstream, respectively. The variations of the f/f 0 with the Reynolds number at various flow attack angles for the square channel inserted with V-Downstream and V-Upstream wavy surfaces are depicted as Figure 7a and b, respectively. The present of the wavy surface in the channel heat exchanger not only increases in heat transfer rate, but also enhances the pressure loss. The insertion of the wavy surface provides higher friction loss than the smooth channel for both arrangements (f/f 0 > 1). The f/f 0 is around 6-45 and 6-43 for the wavy surface with V-Downstream and V-Upstream, respectively. Figure 8a and b present the relations of the thermal enhancement factor with the Reynolds number at various flow attack angles for the square channel inserted with V-Downstream and V-Upstream wavy surfaces, respectively. The TEF increases when augmenting the Reynolds number, except from α = 55° and 60° of the V-Downstream wavy surface. Almost cases, the insertion of the wavy surface gives higher thermal performance than the smooth channel (TEF > 1). In the range investigates, the TEF is around 1.05-2.6 and 0.8-3.0 for V-Downstream and V-Upstream wavy surfaces, respectively.  Figure 9a and b report the variations of the Nu/Nu 0 with the flow attack angle at various Reynolds numbers for V-Downstream and V-Upstream wavy surfaces, respectively. In range 15° ≤ α ≤ 40°, the heat transfer rate increases when increasing the flow attack angle. The heat transfer rate decreases when α > 40°. The peak of heat transfer rate is found at the flow attack angle of 40° on both arrangements. Figure 10a and b present the relations of the f/f 0 with the flow attack angle at various Reynolds number for V-Downstream and V-Upstream, respectively. In range 15° ≤ α ≤ 45° for V-Downstream, the f/f 0 increases when increasing the flow attack angle. The friction factor decreases when α > 45° for V-Downstream. For V-Upstream in range 15° ≤ α ≤ 40°, the f/f 0 increases when enhancing the flow attack angle. The friction loss reduces when α > 40° for V-Upstream. In conclusion, the peak of the friction factor in the heat exchanger channel inserted with V-Downstream and V-Upstream wavy surfaces is detected at 45° and 40°, respectively. Figure 11a and b report the relations of the TEF with the flow attack angle for the heat exchanger channel inserted with V-Downstream and V-Upstream wavy surfaces, respectively. The optimum TEF is found at the flow attack angle of 30° and 40° for V-Downstream and V-Upstream wavy surfaces, respectively.

Conclusion
Numerical investigations on flow and heat transfer in a square channel inserted with V-wavy surface are reported. The influences of flow attack angle, V-wavy surface arrangement and Reynolds number on heat transfer and friction factor are studied. The major findings are concluded as follows; The addition of the wavy surface can improve the heat transfer rate and thermal performance due to the wavy surface generates the vortex flow that disturbs the thermal boundary layer on the heat transfer surface.
The heat transfer rate enhances around 1.5-10 times higher than the smooth channel when inserting the wavy surface in the heating system. The insertion of the wavy surface gives the pressure loss around 6-45 times over the plain channel.
The peak of heat transfer rate is found at the flow attack angle of 40° for both arrangements, while the peak of the friction loss is detected at the flow attack angle of 40° and 50°, respectively, for V-Downstream and V-Upstream.
The optimum TEF is around 2.6 and 3.0 for 30° V-Downstream and 40° V-Upstream, respectively, at Re = 1200.