Cu-Al2O3 Water Hybrid Nanofluid Transport in a Periodic Structure

The present work is a computational investigation of nanofluid and hybrid nanofluid transport in a periodic structure. The governing equations for this work along with the appropriate boundary conditions are solved using the finite-volume method. The simulations are carried out using five wavy amplitudes of the channel shape for a range of Reynolds numbers from 102 to103. It is found that increasing the amplitude and increasing the nanoparticle volume fraction achieve enhancement of the heat transfer at the cost of increased pumping power. Correlations for the friction factor and the Nusselt number for both fluid types are provided.


Introduction
In several engineering applications, microelectronics and compact heat exchangers need significant enhancement in heat transfer. The use of fins, cavities, wavy surfaces, channels or nanofluids can provide substantial heat transfer enhancement in these applications. In case of heat exchangers, nanofluids in the wavy channels offer a much better heat transfer rate. In their review, Hussien et al. [1] summarized some brief survey methods to prepare hybrid nanofluids along with their superior thermal properties compared to base fluids. Several numerical studies, related to the fully developed flows in periodic geometries, exist in the literature [2][3][4][5][6][7][8]. Sarkar et al. [9] considered a furrowed wavy channel at various Reynolds numbers and noticed that critical Reynolds number decreases with an increase in the amplitude and wavelength. They demonstrated that with increasing Reynolds number, Nusselt number, friction, area-goodness, and thermal-performance factors remain unchanged in the steady regime and grow almost linearly in the unsteady regime. Bahaidarah et al. [10][11][12] considered wavy channels with sinusoidal and arc-shaped configurations and compared their numerical results with a parallel-plate channel. They found a negligible increase in heat transfer at low Reynolds number for the two selected configurations as compared to the straight channel. However, at higher Reynolds number, up to 80%, high heat transfer rates were observed. Xie et al. [13] considered a wavy channel and studied the effects of amplitude, lengths, channel widths, and longitudinal pitches on the fluid flow and heat transfer in the channel. They demonstrated that the friction factors and the overall Nusselt numbers rise with an increase in the channel width and Reynold numbers.
Besides these numerical studies, there are also several experimental studies on the fluid flow and heat transfer in periodic channels. A few of them will be summarized here. Rush et al. [14] investigated the flow and local heat transfer rates experimentally for laminar and transitional regimes in wavy passages. They found a substantial increase in the local heat transfer with an increase in the Reynolds where A and o are the amplitude and length of the wave. Here, five-wave amplitudes are considered as shown in Figure 1a; the base case wavy channel amplitude is given by A2, whereas A1, A3, A4, and A5 are scaled from A2 by incrementing the amplitude by 125%, 75%, 50%, and 25%, respectively. Various inflow rates are considered in the ranges of the laminar regime, Reynolds number 10 2 -10 3 ; and the working fluid is water Al2O3-Cu/water hybrid nanofluid with constant properties.
The number of meshes with increasing sizes, (40 × 20), (64 × 34) and (102 × 54) are tested to determine an optimal mesh size. The refinement effect on the average Nusselt number and friction factor is within 2%. A non-uniform, quad-mapped mesh consisting of 5400 elements is adequate for all computations.
Physical properties of nanofluids including the effective density and specific heat are evaluated using the classical mixture formula as follows [42,43]. Density: Specific heat: where hnp = Al 2 O 3 + Cu Since the volume fraction considered here is less than 10%, the fluid mixture is considered a Newtonian fluid. The viscosity and thermal conductivity are given as [44].
Dynamic viscosity: Thermal conductivity: It has been discussed by Chamkha et al. [45] and Mehryan et al. [46] that when considering the thermal conductivity of the Al2O3-Cu/water hybrid nanofluid, either the Levin and Miller [47] or Bruggman [48] are commonly used. Lundgren [49], Brinkman [50], or Batchelor [51] models are employed in the determination of viscosity of the Al2O3-Cu/water hybrid nanofluid. It is interesting to note that at a volume fraction of 2%, the values of conductivity and viscosity models deviate from the experimental results [52] by as much as 6% and 90%, respectively. Therefore, in this study and following the studies [45] and [52], the experimentally determined values of the conductivity and Various inflow rates are considered in the ranges of the laminar regime, Reynolds number 10 2 -10 3 ; and the working fluid is water Al 2 O 3 -Cu/water hybrid nanofluid with constant properties.
The number of meshes with increasing sizes, (40 × 20), (64 × 34) and (102 × 54) are tested to determine an optimal mesh size. The refinement effect on the average Nusselt number and friction factor is within 2%. A non-uniform, quad-mapped mesh consisting of 5400 elements is adequate for all computations.
Physical properties of nanofluids including the effective density and specific heat are evaluated using the classical mixture formula as follows [42,43]. Density: Specific heat: where ϕ hnp = ϕ Al 2 O 3 + ϕ Cu Since the volume fraction considered here is less than 10%, the fluid mixture is considered a Newtonian fluid. The viscosity and thermal conductivity are given as [44].
Dynamic viscosity: Thermal conductivity: It has been discussed by Chamkha et al. [45] and Mehryan et al. [46] that when considering the thermal conductivity of the Al 2 O 3 -Cu/water hybrid nanofluid, either the Levin and Miller [47] or Bruggman [48] are commonly used. Lundgren [49], Brinkman [50], or Batchelor [51] models are employed in the determination of viscosity of the Al 2 O 3 -Cu/water hybrid nanofluid. It is interesting to note that at a volume fraction of 2%, the values of conductivity and viscosity models deviate from the experimental results [52] by as much as 6% and 90%, respectively. Therefore, in this study and following the studies [45,52], the experimentally determined values of the conductivity and viscosity of the hybrid nanofluid should be utilized. The nanofluid properties are obtained using the above equations as given by [28] and summarized in Table 1. It is supposed that the flow is steady and periodically fully developed hydrodynamically and thermally. The governing equations of continuity, momentum, and energy for the fluid flow and heat transfer in the periodic domain are as follows: Continuity: x-direction conservation of momentum: y-direction conservation of momentum: The energy equation: The upstream and downstream ends of the solution domain are subjected to the periodicity conditions [53,54], given as: P(0, y) = P(L, y) The boundary conditions at the solid surface are provided by the no-slip requirement, and the thermal boundary condition employed here is a uniform wall temperature. The Reynolds number and friction factor are defined as The dimensionless temperature θ is defined by where the bulk temperature is given by The average Nusselt number is defined as where the heat transfer coefficient is given as Q is the rate of heat transfer over the periodic module. Ls represents the heated surface area of the channel, and the log-mean temperature difference (LMTD) is defined as The local heat transfer coefficient over the channel surface is given by where q is the local heat flux. The governing equations in terms of primitive variables along with the cyclic boundary conditions Equations (6)-(13) are solved using the Finite Volume Method (FVM). The computational domain consists of a periodic segment as specified by Equation (1). The convection-diffusion was handled using the upwind scheme, while the SIMPLE algorithm [31] was employed to decouple the pressure and velocity. Nanofluid Al 2 O 3 and hybrid nanofluid Cu-Al 2 O 3 with a concentration of 1% and 2% are used as the working fluid and compared to pure water. The Reynolds number range was between 10 2 and 10 3 . The solutions are deemed converged when the residuals of the mass conservation and velocity fall below 10 −5 and the energy residuals are less than 10 −8 . Experimental data on nanofluids flowing inside a wavy channel are lacking. However, the accuracy of the computational method has been validated for a circular tube against the numerical and experimental data of [39] for Al 2 O 3 /water nanofluid flow inside a circular tube of 12.4 mm in diameter, φ = 1% and dp = 30 nm using the Dirichlet thermal boundary condition. The enhancement in the average heat transfer, h r , was validated for the Reynolds number in the laminar range. It is shown in Figure 2 that generally excellent agreement is found with the computational model and good agreement with the experimental results.

Results
This section highlights the results of the conducted numerical experiments to show the effect of using hybrid nanofluid in a wavy channel. We study the influence of changing the nanoparticles volume fraction in the base fluid, the varying amplitude of the channel, and the effect of Reynolds number on the thermal performance. We maintained a constant wave footprint length for the spatially periodic module as shown in Figure 1a.
The non-dimension local wall shear stress distributions (WSS) are shown for the top surface with amplitude A2 (Figure 3a) and amplitude A4 (Figure 3b).

Results
This section highlights the results of the conducted numerical experiments to show the effect of using hybrid nanofluid in a wavy channel. We study the influence of changing the nanoparticles volume fraction in the base fluid, the varying amplitude of the channel, and the effect of Reynolds number on the thermal performance. We maintained a constant wave footprint length for the spatially periodic module as shown in Figure 1a.
The non-dimension local wall shear stress distributions (WSS) are shown for the top surface with amplitude A2 (Figure 3a) and amplitude A4 (Figure 3b).

Results
This section highlights the results of the conducted numerical experiments to show the effect of using hybrid nanofluid in a wavy channel. We study the influence of changing the nanoparticles volume fraction in the base fluid, the varying amplitude of the channel, and the effect of Reynolds number on the thermal performance. We maintained a constant wave footprint length for the spatially periodic module as shown in Figure 1a.
The non-dimension local wall shear stress distributions (WSS) are shown for the top surface with amplitude A2 (Figure 3a) and amplitude A4 (Figure 3b).      For nanoparticle volume fraction ϕ = 2%, doubling the wave amplitude (Figure 3a,b) increases the peak local shear by a factor of 2.5, this is because of increased losses in the expanding recirculating fluid zones of the curved regions of the channel. On the other hand, for a fixed amplitude A2, i.e., Figure 3a, increasing the nanoparticle concentration from ϕ = 1% to ϕ = 2% increases the peak of the local shear stress by a factor of 1.5, which is directly related to the increase in the effective fluid viscosity with increasing nanoparticle volume fraction. Figure 3c,d shows similar trends of the local shear stress at the bottom channel surface, where the location of the peak is shifted due to the asymmetry of the channel configuration. Figure 4a explains the pressure gradient required to drive the flow through the periodic module. We note a significant increase with greater amplitudes and larger volume fractions for both nanofluid and hybrid nanofluid. The hybrid nanofluid increases the pressure gradient requirement by as much as 3.5-fold in comparison to the nanofluid. This is due to the increased resistance with the growing recirculating fluid zones and the higher viscosity of hybrid nanofluid in contrast to the nanofluid. Another way to examine this effect is by studying the friction factor results for various amplitudes at 1% and 2% nanoparticle volume fractions as depicted in Figure 4b. For nanoparticle volume fraction φ = 2%, doubling the wave amplitude (Figure 3a,b) increases the peak local shear by a factor of 2.5, this is because of increased losses in the expanding recirculating fluid zones of the curved regions of the channel. On the other hand, for a fixed amplitude A2, i.e., Figure 3a, increasing the nanoparticle concentration from φ = 1% to φ = 2% increases the peak of the local shear stress by a factor of 1.5, which is directly related to the increase in the effective fluid viscosity with increasing nanoparticle volume fraction. Figure 3c,d shows similar trends of the local shear stress at the bottom channel surface, where the location of the peak is shifted due to the asymmetry of the channel configuration. Figure 4a explains the pressure gradient required to drive the flow through the periodic module. We note a significant increase with greater amplitudes and larger volume fractions for both nanofluid and hybrid nanofluid. The hybrid nanofluid increases the pressure gradient requirement by as much as 3.5-fold in comparison to the nanofluid. This is due to the increased resistance with the growing recirculating fluid zones and the higher viscosity of hybrid nanofluid in contrast to the nanofluid. Another way to examine this effect is by studying the friction factor results for various amplitudes at 1% and 2% nanoparticle volume fractions as depicted in Figure 4b.      in the peak heat transfer coefficient. At a fixed amplitude i.e., A2, and particle concentration of 1%, a significant increase in heat transfer of nearly 14% is noticed when hybrid nanofluid is utilized in comparison to a marginal increase for the case of nanofluid. A less pronounced increase of 3% in the peak heat transfer coefficient is due to the presence of 2% volume fraction nanoparticles in the base fluid in contrast to 1%. This change can be attributed to the improved thermal conductivity with the addition of nanoparticles. Figure 5c,d shows the local heat transfer coefficient at the channel bottom surface. The improvement in the heat transfer is similar to the channel top surface, and the shift in the location of the peak is attributed to the asymmetry of the channel. Moreover, Figure 5c,d shows the effect of changing Reynolds number from 500 to 300 while having the same wave amplitude. This results in decreasing the Nusselt number by nearly 20%.
Processes 2020, 8, x FOR PEER REVIEW 9 of 14 the peak heat transfer coefficient. At a fixed amplitude i.e., A2, and particle concentration of 1%, a significant increase in heat transfer of nearly 14% is noticed when hybrid nanofluid is utilized in comparison to a marginal increase for the case of nanofluid. A less pronounced increase of 3% in the peak heat transfer coefficient is due to the presence of 2% volume fraction nanoparticles in the base fluid in contrast to 1%. This change can be attributed to the improved thermal conductivity with the addition of nanoparticles. Figure 5c,d shows the local heat transfer coefficient at the channel bottom surface. The improvement in the heat transfer is similar to the channel top surface, and the shift in the location of the peak is attributed to the asymmetry of the channel. Moreover, Figure 5c,d shows the effect of changing Reynolds number from 500 to 300 while having the same wave amplitude. This results in decreasing the Nusselt number by nearly 20%.  Figure 6 illustrates the geometric effect, the nanoparticle composition (Al 2 O 3 vs. Cu-Al 2 O 3 ) and the inertial influence on the dimensionless heat transfer or the Nusselt number. We observe that the average Nusselt number is increased by 25% when the channel amplitude is doubled due to increased advection effect driven by vigorous mixing in an expanded recirculating fluid zone. A 2% hybrid nanoparticles addition to the base fluid results in even greater enhancement in the heat transfer by as much as 46%; this increase is directly related to the increase in fluid conductivity. The enhancement of heat transfer with increased waviness happens at the cost of additional frictional effects.  Figure 6 illustrates the geometric effect, the nanoparticle composition (Al 2 O 3 vs. Cu-Al 2 O 3 ) and the inertial influence on the dimensionless heat transfer or the Nusselt number. We observe that the average Nusselt number is increased by 25% when the channel amplitude is doubled due to increased advection effect driven by vigorous mixing in an expanded recirculating fluid zone. A 2% hybrid nanoparticles addition to the base fluid results in even greater enhancement in the heat transfer by as much as 46%; this increase is directly related to the increase in fluid conductivity. The enhancement of heat transfer with increased waviness happens at the cost of additional frictional effects.

Discussion
The average Nusselt number and the friction factor are correlated to Reynolds number and other parameters as follows: where the constant 1 , 2 , 1 , 2 , 1 , 2 , 1 , 2 for the case of nanofluid and hybrid nanofluid are given in Table 2. The above correlations are valid for φ > 0, where A and λ0 are the amplitude and length of the wave. The R 2 values for correlations (24) and (25) are 0.992 and 0.933 for the nanofluid case while these values are 0.992 and 0.932, respectively for the hybrid nanofluid. The expected average error for either correlation is less than 1%.

Discussion
The average Nusselt number and the friction factor are correlated to Reynolds number and other parameters as follows: where the constant C 1 , C 2 , n 1 , n 2 , m 1 , m 2 , s 1 , s 2 for the case of nanofluid and hybrid nanofluid are given in Table 2. The above correlations are valid for ϕ > 0, where A and λ 0 are the amplitude and length of the wave. The R 2 values for correlations (24) and (25) are 0.992 and 0.933 for the nanofluid case while these values are 0.992 and 0.932, respectively for the hybrid nanofluid. The expected average error for either correlation is less than 1%.
In summary, increasing the wave amplitude results in enhancing the local and average heat transfer due to increased mixing in the vicinity of the curved segments of the top and bottom channel surfaces. It is also observed that as Reynolds number increases, the increasing inertial effect intensifies the recirculation zones even more, resulting in its growth. Moreover, it increases the thermal transport exchange between these zones and the fluid in the core region of the channel. In addition, the presence of the hybrid nanoparticles effectiveness in heat transfer enhancement can be attributed to the increase in the thermal conductivity of the base fluid. However, the augmentation of heat transfer with increased waviness and nanoparticle concentration takes place at the cost of increased frictional losses.

Conclusions
A steady two-dimensional analysis of a periodically fully developed flow through a wavy channel module has been carried out. It is found that increasing the volume fraction of nanoparticles increases both the heat transfer rate and frictional effects. The increase in heat transfer is marginal with nanoparticle ϕ = 1% and 2%, and corresponds to an increase of the Nusselt number of 1% and 3%. However, the addition of hybrid nanoparticle has a significant impact with an increase of 16% and 22% in the Nusselt number with the addition of 1% and 2% volume fraction. This is realized at the cost of increasing the friction factor by 1.34 and 2.33-fold for the Cu-Al 2 O 3 fluid volume fraction of 1% and 2% respectively. This is in contrast to a moderate increase of 11% and 27% for the similar volume fractions of Al 2 O 3 nanoparticles. In addition, the increase in the channel wave amplitude by 25% enhances the heat transfer by nearly 12% at the cost of an increase of 15% in frictional losses.