The effect of ribs spacing on heat transfer in rectangular channels under the effect of different types of heat flux in the Presence of a nanofluids

In this study, numerical computations of the influence of adding ribs in a rectangular channel on the forced convection heat transfer and laminar fluid flow characteristics has been carried out. The analysis was carried out by using the finite element method to solve the dimensionless governing equations for two-dimensional channel with 80 mm height and 2000 mm length at the Reynolds number of (10, 100, and 500), rib height e=8mm with different aspect ratios (AR =2.5, 3.125, 3.75, 4.375, and 5). Also, the study compared two cases of investigations with and without nanofluid (Water/ TiO2) at the volume fractions of nanoparticles of 0, 2 and 4%. The results concluded that, for a certain arrangements, the use of extended surfaces within a rectangular channel can significantly enhance the rate of heat transfer and when the aspect ratios decreases, the Nusselt number increased. However, the existence of ribs within channel in case of constant heat flux can cause a significant improvement of heat transfer compared to that in the corresponding channel under the variable heat flux.


Introduction
The use of artificial roughness on a surface in the form of repeated ribs is one of the passive ways to improve the rate of heat transfer. Heat transfer applications have recently become increasingly essential in industrial engineering fields as heat exchangers, power generation, chemical processes, medical applications, air conditioning, electronic devices and solar collectors, etc. [1][2][3][4].In past studies, there were many geometric shapes of the ribs which have been studied [5][6][7][8]. Among these studies, when using Reynolds number 31170 show the influence of the ribs to improving the heat transfer rate about 213% compared with the smooth surface by Al-taie et al. [9]. Also, the forced convection in the grooved channel has been taken the researcher's attention lately [10][11][12]. On the other hand nanotechnology with dispersing solid particles in a base fluid has been serving as coolants with a view to increases the thermal conductivity of fluid [13,14]. The thermal conductivity of Al2O3 and SiO2-water nanofluids was measured by Salman et al. [15].The investigation results showed 22% enhancement of heat transfer when using nanofluids. On the other hand, many researchers have presented several reviews in order to show improved viewpoints and states for improving equipment heat transfer [16][17][18]. Also, another researchers focused on using both method of heat transfer (rough surfaces and nanofluids) [19]. Andreozzi et al. [20] numerically investigated the enhancement of heat transfer of nanofluid flow in a heated channel using various types of ribs. liu et al. [21] performed an experimental and computational study of the thermal performance enhancement in a rectangular passage with perforated ribs. They observed the local heat transfer enhanced due to perforated ribs around 12-24%, and the overall heat transfer increased by 4-8% when compared with the normal ribs.
Chtourou et al. [22] numerically studied the influence of the obstacles on the thermal efficiency inside narrow channels of a plate heat exchangers. With a Reynolds number range 200-800, they investigated and analyzed the impact of the shapes and arrangements of the ribs on hydrothermal behavior. Compared to the smooth duct, the thermal performances are enhanced. They found due to the presence of the obstacles; the heat transfer rate improves from 1.44 to 2.6 times when compared to the smooth case. In the other study, the design of pin-fin heat sink was studied numerically and experimentally by Rezaee et al. [23] Different geometrical parameters of pin-fin were tested in the heat sink. They concluded for all arrangements of pin-fin heat sink that the overall hydrothermal performance significantly increases compared with the smooth case. Wang et al. [24] numerically studied the impact of truncated ribs on the heat transfer and laminar flow characteristics in a ribbed microchannel. The effects of various geometrical parameters (rib width, rib arrangement, and truncation gap height) on hydrothermal performance had been analyzed. The findings showed that the truncated ribs with the same rib arrangement and width can exhibit better thermal behavior than the continuous ribs case, with decreasing the pressure drop in microchannel. Also, a numerical study by Yadav and Bhagoria [25] found that the flow field, average Nusselt number, and overall hydrothermal performance are all highly influenced by the relative roughness height. In this study, the influence of change in aspect ratio of extended surfaces, the volume percentage of nanoparticles and Reynolds number are investigated on the behavior heat transfer and laminar flow in a 2-D rectangular channel under two cases of applied heat flux, the first one constant and in the second case is a variable. For the simulation, Reynolds numbers 10, 100, and 500 were used, with nanoparticle concentrations of 0, 2, and 4%, respectively. Up to best knowledge of the authors, very few previous studies discussed the effect of variable heat flux on the flow characteristics in ribbed channels. Also more attention was paid to microchannels, while this study conducted to ordinary channel. In addition to that the optimum value of the aspect ratios as well as the best scenario of heat flux will be presented in current study using numerical approaches. As a result, more data is presented here with a presence of water-TiO2 nanofluid as a coolant to show the heat transfer characteristics of the extended surfaces with different values of aspect ratios.

Problem description
The two-dimensional ribbed channel with 10 periodic ribs along the lower wall has been considered in this research as shown schematically in Fig.  1. The total length and the height of the horizontal rectangular channel are set to be L=2000 mm and H=80 mm, respectively. The bottom wall with the length of inlet (L1=1260 mm) and the length of exit (L3=200 mm) are insulated, but the middle section with the length of (L2=540mm) is under the effect of heat flux q. The test section has (ten) ribs with height of (e=8), and length of (s=10) for each rib and the space between them varies according to aspect ratio AR. Table 1    The working fluid is water and nano-particles, they are added at volume fraction φ= 0, 0.02, 0.04 to explore the effect of adding them with interaction with different geometries which gives a chance to specify the best and optimum combination. The assumptions regarding the working fluid are as follows: • Incompressible and Newtonian, • Single phase, • The co-existing with nano-particles is in thermal and hydraulic equilibrium, • The physical properties are weak function of temperature and thus approximately constant over the operating range of temperatures' case study.
The applied heat flux is considered at two cases:

Case I: Constant heat flux
In this case the lower wall of channel is exposed to a uniform heat flux condition (qc =1000 W/m2) on the middle section (test section); while the upper and other remaining walls are insulated.

Case II: Variable heat flux
The lower wall of the test section in this case is subjected to a variable heat flux q(x), the heat flux various linearly according to the equation as: q(x)=q0+A.
x Where the constants q0, and A will be chosen in such a way that the total heat flux in both cases (constant heat flux and variable heat flux) will be the same.
At the inlet section, different fluid velocities are set, and they vary to ensure laminar flow with Reynolds numbers of (10, 100, and 500) but the exits were set to pressure outlet. The inlet temperature is introduced Tc=290K. The thermophysical properties of base fluid and the TiO2 nanoparticles are presented in Table 2.

Mathematical model
In this study, the continuity, momentum, and energy equations of the laminar and steady state forced convection of the Newtonian nanofluid in the channel can be written in dimensional form as follows [27]: The following dimensionless parameters can be used to transform the above governing equations into non-dimensional forms [28]: For steady state and laminar flow, the dimensionless governing equations of continuity, momentum, and energy are expressed as follows: [29]: The boundary conditions in dimensionless form are described as follows: U=1, V=0 and = 0 for X=0 and 0≤ Y ≤1 V=0 and = = 0 for X=2.56 and 0≤ Y ≤1 V=0, U=0 and = 0 or = − for Y=0 and 0≤ X ≤ 2.56 (12) V=0, U=0 and = 0 for Y=1 and 0≤ X ≤ 2.56 Nanofluid properties of the nanofluid can be defined based on the properties of the base fluid and nanoparticles [30]: The following formulae are used to calculate the local and average Nusselt number [31]: 1 0 The following relationship can be used to calculate the pressure drop (P) between the inlet and exit sections. [32]:

Numerical procedure
In this study, to acquire the surface heat flux distribution of the test section of cooling channel, a numerical procedure was executed using finite element method. Therefore, the laminar two-dimensional Navier-Stokes equations for fluid flow and heat transfer analysis have been solved and discretized numerically combined with the continuity equation and the energy equation with appropriate boundary conditions. The boundary conditions were illustrated in previous paragraph.

Validation of the numerical model
In order to check the validity and accuracy of procedure for this numerical study. Current results have been validated by comparing them with a previous study. Fig. 2 shows the validation of the present study with Behnampour et al. [22]. Where it can be concluded from Fig. 2-(a) that the values of the local Nusselt number for the current study correspond to the values of the reference at Re=1 and φ=0.04, also Fig. 2-(b) indicates a good matched in the findings for the average Nusselt number.

Gri dindependence test
A grid-independence test is performed to assess the effects of grid sizes on the accuracy and validity of numerical findings. To choose the suitable grid, the average Nusselt number along the lower hot wall of channel for different mesh sizes is estimated. The grid-independence study of the current numerical simulation is performed for ribbed channel with case I (constant heat flux) which is AR=2.5, Re=10 and φ=0.04. As can be shown in Table 3, the optimal mesh for ribbed channel in terms of accuracy and solution time is the Normal grid of (35776). The unstructured triangular meshes for rectangular ribbed channel which used in the present study are shown in Fig. 3.   Fig. 4 indicates the dimensionless isotherms diagrams along the rectangular channel for two cases of heat flux in different aspect ratios with Reynolds number of 500 and a volume fraction (φ) of 4%. As seen from the isotherms lines shown in figures below, more changes are created in the form of non-dimensional temperature contours as the AR ratio is increased, due to more energy transferred across the boundaries. In all rib aspect ratios, using high volume fraction of TiO2 nanoparticles causes improvement of heat transfer since the improved thermal conductivity after addition of nano-material to the base fluid and because of the role of solid particles in transferring and conducting the heat energy. When a fluid with a certain temperature comes into contact with rib-roughened hot channel surfaces, the high surfaces temperature lowers, and heat transfer between the hot surfaces and the flowing fluid occurs which increases the fluid temperature by convection and by conduction at very close fluid layers touching the hot ribbed surface. The heat transfer rate increases as the Reynolds number decreases. The reason for this is because when Re is lower, the fluid stays in touch with the hot surface for longer period of time, so absorbing more heat as it passes through the boundary. Ribs along the test section of channel can generate turbulence and so function as mixer and lower the temperature difference between the surface and the fluid, improving heat transmission. Figure 4: Isotherms contours at Re=500 and φ=0.04 for case I (uniform heat flux), and case II (variable heat flux). Fig. 5 depicts the streamline contours for two heat flux scenarios using Reynolds numbers of 500 and a volume fraction percentage of 4%. The levels of dimensionless velocity for each rib aspect ratio have been compared According to the diagrams below. As the spaces between extended surfaces and Reynolds number increases, multiple circulations occur in a single space, temperature distribution will be improved due to continuous thermal boundary layer towards the channel centerline. The changes in dimensionless velocity contours will be different in all of the pitch-to-height ratios, and by increasing the rib pitch to higher values, the fluid momentum depreciation in the indentation areas will be significant. This means that changing aspect ratios causes changes in the flow's hydrodynamic behavior.  Among the studied aspect ratios, AR =5 has the largest amount of average Nusselt number at Re = 10 and Re = 100, while at Re=500, the largest amount of average Nusselt number is achieved at AR =3.75, while AR = 2.5 has the minimum amount at all Reynolds numbers. Furthermore, Increasing the Reynolds number also causes eddy flows, which improves the convection heat transfer coefficient. As well as when compared behavior, it is observed that nanofluid has a greater value of Nusselt number compared to pure water. This rise is due to the presence of nanoparticles which causes to increased thermal conductivity of ordinary fluid, also due to effect the Brownian motion of nanoparticles on the enhance the thermal conductivity of nanofluid. Nusselt number improves as nanopartical concentration increase.  Also in the second case, the variations of average Nusselt numbers with aspect ratios at different Reynolds number and various fraction factors of nanoparticles are presented in Fig. 8, with the decrease in rib pitch ratios, the average Nu tends to decrease. It can also be seen in this graph that the average Nu tends to rise with the increase in Re, especially when AR=3.75. The flow becomes detached when the distance between two successive ribs increased. Then, it is attached to the walls again. Thus, the rate of heat transfer is increased. On the other hand, the Nusselt number increases with the increasing of aspect ratio. This can be explained by the existence of a ribs turbulator, which causes increase the surface area of channel and lead to rapid mixing between the core and wall flow, leading to high gradients of temperature, especially at lower pitch ratios. Nanofluids are another factor that contributes into increasing the Nusselt number by increases the thermal conductivity of fluid. The thermal behavior is reinforced by increasing the nanoparticles concentration. From the Fig. 9, it can be seen in case (2) that the local Nusselt numbers are increased for all aspect ratios with increased values of Reynolds number, and the values of local Nusselt number along channel in high Reynolds number is larger than that at low Reynolds number. This is due to the rapid rising of velocity at wider spaces between ribs. Ribs play a crucial role in boosting local Nu, causing a significant rise in Nusselt number in the rib-roughened channel surface. The fundamental reason for this rise is that fluid layers between cold and hot places are better mixed. When fluid passes through the ribs, the thermal boundary layers are disturbed and altered, resulting in an increase in heat transfer rate. At mentioned figure, when the nanofluid reaches the left wall of the channel, a fluctuation in the local Nusselt number can be seen in order to recirculate flow in the groove between the ribs, since the graph below has seen a dramatic decrease in the local Nu due to impinging the flow to the rib wall, demonstrating the effect of ribbed channel on heat transfer.

Conclusions
The impacts of rib spacing on the laminar forced convection heat transfer of (Water/ TiO2) nanofluid in a two-dimensional rectangular duct under two situations of heat flux boundary condition were investigated in this study using computational fluid dynamics (CFD). The emphasis was given on the heat transfer enhancement resulting from changing parameters include different aspect ratios of ribs (P/e) changing from 2.5 to 5. The influence of the abovementioned parameters on the average Nusselt number for two cases of heat flux are presented and discussed. As a result, the following findings have been achieved: • The breakdown of the thermal boundary by ribs in a ribbed channel produces mixing enhancement, which results in a rise in the average Nusselt number. Moreover, the average Nusselt number rises with the increase in Reynolds number.

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The heat transfer process in the ribbed channel is affected by the spacing between the extended surfaces. For certain extent, the heat transfer rate increases with the increase in rib pitch.

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Reynolds number of 500 has the largest rate of heat transfer in all instances, whereas Reynolds number of 10 has the lowest rate of heat transmission, which affirms the direct proportional relationship between Nu and Re numbers.

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The Nusselt number was found to be increased as the spacing increased. For this parameter, the case I show better results than case II, because the uniformness and constant distribution of heat transfer rate in all range of Reynolds Number and this result leads to increase of thermal performance factor. • The maximum value of average Nusselt number is found to be 10.76 for the ratio of pitch to height of rib is 3.75 and volume fraction of 0.04 at a higher Reynolds number 500, in case of constant heat flux (Case I).