Laminar convective heat transfer of shear-thinning liquids in rectangular channels with longitudinal vortex generators

Heat and fluid flow in a rectangular channel heat sink equipped with longitudinal vortex generators have been numerically investigated in the range of Reynolds numbers between 25 and 200. Aqueous solutions of carboxymethyl cellulose (CMC) with different concentrations (200-2000 ppm), which are shear-thinning non-Newtonian liquids, have been utilised as working fluid. Three-dimensional simulations have been performed on a plain channel and a channel with five pairs of vortex generators. The channels have a hydraulic diameter of 8 mm and are heated by constant wall temperature. The vortex generators have been mounted at different angles of attack and locations inside the channel. The shear-thinning liquid flow in rectangular channels with longitudinal vortex generators are described and the mechanisms of heat transfer enhancement are discussed. The results demonstrate a heat transfer enhancement of 39-188% using CMC aqueous solutions in rectangular channels with LVGs with respect to a Newtonian liquid flow (i.e. water). Additionally, it is shown that equipping rectangular channels with LVGs results in an enhancement of 24-135% in heat transfer performance vis-\`a-vis plain channel. However, this heat transfer enhancement is associated with larger pressure losses. For the range of parameters studied in this paper, increasing the CMC concentration, the angle of attack of vortex generators and their lateral distances leads to an increase in heat transfer performance. Additionally, heat transfer performance of rectangular channels with longitudinal vortex generators enhances with increasing the Reynolds number in the laminar flow regime.


Introduction
Enhancing the thermal efficiency of heat exchangers is a challenging task to meet the heat removal capability needed for development of new devices with better performances. A number of designs and approaches have been proposed to passively enhance the heat transfer performance of cooling devices [1][2][3][4][5][6][7][8][9]. Equipping rectangular channels with vortex generators (VGs) has been demonstrated to be a promising method to passively augment the heat transfer Velocity gradients and therefore shear stresses are larger in channels with LVGs in comparison with plain channels [14,36]. It is known that shear-thinning behaviour influences the structure of vortices generated by VGs [29,33,37]. Therefore, the shear-thinning behaviour can result in a change in heat transfer performance of a heat exchanger [16,26].
According to the best of the author's knowledge and to the reviewed literature, the influence of the non-Newtonian fluid flow behaviour on the thermo-hydraulic performance of rectangular channels equipped with LVGs is not addressed yet. More studies are essential to attain an insight into the effects of non-Newtonian fluid flow behaviour in channels equipped with LVGs and consequently on the heat transfer performance of channels. The primary aim Page 4 of 40 of this study is to understand the influence of shear-thinning behaviour on fluid flow structure and heat transfer characteristics in rectangular channels equipped with LVGs. In the present study, three-dimensional numerical simulations have been performed to explore the shearthinning power-law fluid flow structure and heat transfer in a rectangular channel with LVGs.
To highlight the effect of shear-thinning behaviour on the heat transfer performance of a rectangular channel equipped with LVGs, CMC aqueous solutions with different CMC concentrations are scrutinised. Additionally, the influence of the shear rate on fluid flow structure and heat transfer performance is investigated by changing the angle of attack and the lateral distance of the LVGs. The non-Newtonian fluid flow structure and heat transfer characteristics are compared with a Newtonian fluid (i.e. water). The thermo-hydraulic performances of the channels with LVGs are also compared with a plain channel. The results presented in this paper may introduce new perspectives towards novel approaches to heat transfer enhancement in heat exchangers.

Computational domain
A rectangular channel with five pairs of LVGs was considered in the present study. A schematic diagram of the channel is shown in Figure 1. Six different configurations with VGs mounted at different angles of attack (i.e. α= 30°, 45° and 60°) and lateral distances (i.e. dt= 5, 2.5 and 1.25 mm) were designed. The thickness of the VGs was idealised and supposed to be zero [38]. The geometrical parameters for different configurations are reported in Table 1. The heat and fluid flow were described in a three-dimensional Cartesian coordinate system, in which the mainstream was in the z-axis direction. The computational domain consisted of the Page 5 of 40 inlet, main, and outlet zones. The inlet zone with adiabatic walls and the length of Lin was defined to ensure the flow uniformity at the main zone entrance. The main zone encompasses five pairs of LVGs that were equally spaced in the mainstream direction. The solid walls were kept at constant temperature of 320 K in the main zone. The main zone was extended by adiabatic walls with the length of Lout (i.e. outlet zone) to avoid any flow reversal at the outlet boundary.

Physical model
To highlight the differences between non-Newtonian and Newtonian coolants, aqueous solutions of carboxymethyl cellulose (CMC) were compared with a Newtonian liquid (i.e. water). Thermophysical properties of water and CMC aqueous solutions are presented in Table 2. Because of small temperature variations along the channel (less than 22 K), the material properties were assumed to be independent of temperature. To develop the mathematical model, the generation of longitudinal vortices was assumed to be quasi-steady [12] and the single-phase fluid flow to be laminar due to the low flow Reynolds number.
Furthermore, it was assumed that the effects of body forces, radiation, and compressibility are negligible. Based on these assumptions, the continuum heat and fluid flow in the channel were modelled using the equations of mass, momentum, and energy conservation that are introduced as follows: where V is the fluid velocity vector, ρ density [kg m -3 Table 3.
The boundary conditions at the channel inlet, outlet, solid walls and symmetry plane are mathematically introduced as follows: Inlet boundary: Outlet boundary: Symmetry plane:

Data reduction
The following dimensionless numbers are used to construct a framework of result presentation. The Reynolds number (Re) based on the hydraulic diameter of the channel (Dh) is defined as follows: Page 8 of 40 It should be noted that n is equal to one for Newtonian fluids and therefore K represents the dynamic viscosity of the Newtonian liquid.
The Nusselt number (Nu), Prandtl number (Pr), required pumping power (Ppump), and Fanning friction factor (f) can be calculated using the following equations.

Grid independence test
Five meshes with a different number of cells were generated to investigate the sensitivity of the results to the cell size and to determine the minimum number of required cells to obtain reasonable results. The case of water flow in the channel with α=45° and Re=200 was considered for the grid independence test. The results of the grid independence test are compared with the results obtained from the largest grid (i.e. the mesh with 2.5×10 6 cells) and are presented in Table 4. Based on the results presented in Table 4, a mesh with 2×10 6 cells was selected for simulations considering both accuracy and computational costs.

Solver verification
Page 10 of 40 The ability of the solver to reliably predict Newtonian fluid flow and heat transfer in rectangular channels with VGs was assessed in previous studies conducted by the authors [14, 16, 21]. The solver has been extended to model non-Newtonian fluid flow and heat transfer.
Laminar convective heat transfer in a straight duct heated by constant wall temperature was considered to verify the solver. The duct has a length of 1.2 m and a square cross-section with the side length of 0.01 m. The fluid enters the duct with a constant temperature and velocity.
The validity of the model was examined for both water and CMC aqueous solutions. In Table   5, the results of the present solver for the Reynolds number of 1000 were compared with the numerical results reported by Kurnia et al. [30]. Details assigned to the benchmark case can be found in [30]. The maximum deviation of the present numerical results from the reference data is 0.96 and 0.72% in the prediction of heat transfer rate and pressure drop, respectively, which demonstrates a reasonable agreement. This deviation from the referenced data can be attributed to differences in the grid size, numerical schemes and convergence criteria.  Increasing the angle of attack causes higher velocity gradients and strain rates in the channels.

Results and discussion
Higher strain rates result in a reduction in effective viscosity of shear-thinning fluids, which can lead to an augmentation of the fluid mixing in the channel and therefore the heat transfer performance of the channel. An increase in Num is observed with increasing dt (see Figure   3(c)). Decreasing the lateral distance of VGs leads to lower fluid velocities in the central region of the channel and higher velocities in the outer region. It causes a reduction in the size of the recirculation zone and the strength of the vortices and weakens the secondary flow.
Additionally, due to the lower fluid velocities downstream of the VGs, the contribution of the convective heat transfer in the total heat transfer decreases by reducing the lateral distance of the VGs.
Page 12 of 40 The results presented in Figure 3(d) indicate that, for the range of parameters studied in this paper, increasing the CMC concentration results in higher Nusselt numbers. The viscosity of coolants with higher CMC concentrations is more affected by the velocity gradients and therefore the heat and fluid flow are expected to be more influenced by the shear stresses generated by the VGs. The thermal conductivity of CMC aqueous solutions is higher than water, which enhances the heat absorption from the hot walls. CMC aqueous solutions with higher CMC concentrations have higher Prandtl numbers. Therefore, the momentum diffusivity is the dominant factor that governs the flow behaviour in comparison with the thermal diffusivity at higher CMC concentrations. It means that for the coolants with higher CMC concentrations, convection dominates the energy transportation in the channel compared with the conduction.
For the range of parameters studied here, an enhancement of 38.52-188.43% in the mean Nusselt number is recorded for the CMC aqueous solutions with respect to water. Figure 4 shows the pumping power required to drive the fluid flow through the channels. Employing

Conclusions
Three-dimensional simulations were conducted to investigate laminar convective heat transfer