Pulsating Flow of CNT–Water Nanofluid Mixed Convection in a Vented Trapezoidal Cavity with an Inner Conductive T-Shaped Object and Magnetic Field Effects

: Mixed convection of carbon-nanotube/water nanoﬂuid in a vented cavity with an inner conductive T-shaped object was examined under pulsating ﬂow conditions under magnetic ﬁeld effects with ﬁnite element method. Effects of different parameters such as Richardson number (between 0.05 and 50), Hartmann number (between 0 and 30), cavity wall inclination (between 0 ◦ and 10 ◦ ), size (between 0.1 H and 0.4 H ) and orientation (between − 90 ◦ and 90 ◦ ) of the T-shaped object, and amplitude (between 0.5 and 0.9) and frequency (Strouhal number between 0.25 and 5) of pulsating ﬂow on the convective ﬂow features were studied. It was observed that the average Nusselt number enhanced with the rise of strength of magnetic ﬁeld, solid nanoparticle volume fraction, and amplitude of the pulsation, while the effect was opposite for higher values of Ri number and cavity wall inclination angle. The presence of the T-shaped object and adjusting its size and orientation had signiﬁcant impact on the main ﬂow stream from inlet to outlet and recirculations around the T-shaped object and in the vicinity of hot wall of the cavity along with the magnetic ﬁeld strength. Pulsating ﬂow resulted in heat transfer enhancement as compared to steady ﬂow case for all conﬁgurations. However, the amount of increment was different depending on the variation of the parameters of interest. Heat transfer enhancements were 41.85% and 20.81% when the size of the T-shaped object was increased from 0.1 H to 0.4 H . The T-shaped object can be utilized in the vented cavity as an excellent tool for convective heat transfer control. As highly conductive CNT particles were used in water, signiﬁcant enhancements in the average Nusselt number between 97% and 108% were obtained both in steady ﬂow and in pulsating ﬂow cases when magnetic ﬁeld was absent or present.


Introduction
The convection in vented cavities play a vital role in ventilating systems, food processing applications, electronic cooling, and many others. In heat transfer, many active and passive methods are used to control the convection in cavities. In this context, Saeidi and Khodadadi [1] examined the forced convection in a vented cavity with finite volume method. Different locations of the outlet ports were examined for Reynolds number between 10 and 500. Houshang et al. [2] examined the convective heat transfer features Numerical results from this study can be beneficial when designing and optimizing thermal configurations related to vented cavities, which have many heat transfer engineering applications, as discussed above.

Model Description
Nanofluid mixed convection within a trapezoidal shaped vented cavity with an inner conductive T-shaped obstacle was analyzed considering the effects of magnetic field and pulsating flow. Figure 1 shows the physical problem with boundary conditions. The cavity height is H, while the sizes of the inlet and outlet ports are w i = w o = 0.25 H. Nanofluid with cold temperature enters the inlet of the vented cavity with velocity of u = u 0 [1 + A sin(ωt)] and temperature of T = T c . The cavity is trapezoidal with side wall inclination angles of Ω. W 1 (left wall), W 2 (bottom wall), W 3 (right wall), and W 4 (top wall) denote different hot wall segments of the configuration. A T-shaped conductive object was placed in the cavity with thermal conductivity of k p . d, d 2 , and ω represent the half length, height, and inclination of the T-shaped object, respectively. d 2 was fixed to d 2 = 0.075 H during the simulation. The mid point of the bottom wall of the T-shaped object is placed at (0.5 H, 0.5 H) within the cavity, which denotes the object location. Magnetic field is uniform and γ denotes the inclination angle. A single-walled carbon nanotube/water nanofluid was considered. The Prandtl number is 6.9 for the base fluid. The flow is 2D and laminar while Newtonian and incompressible fluid assumptions were utilized. Impacts of induced magnetic field, joule heating effects, and displacement currents were not considered. The density variation in the buoyancy was modeled by using the Boussinesq approximation. Thermal radiation effect with viscous dissipation effects were also considered to be negligible.

•
On the walls of the T-shaped object, U = V = 0.
• At the interface between the fluid and solid domain (T-shaped object), ∂θ In the pulsating flow configuration, Nusselt number is time dependent. Local Nusselt number is obtained as: where s and t are coordinate along the hot wall and time. Heat transfer enhancement (HTE) is defined by using the Nusselt number values in pulsating flow and steady flow cases as: where the sub-indices sm and m denote the spatial average in pulsating flow and mean value in steady flow, respectively.

Nanofluid Property Relations
In the current study, single-walled carbon nanotubes (SWCNT) were used in water. The density, thermal expansion coefficient, and specific heat of the nanofluid are described as: Thermal conductivity of nanofluid is given as [50]: In the above given representation, the space distribution effects of the carbon nanotubes are taken into account. This model gives very accurate results when compared with the experimental data, even at high particle volume fractions. In the work of Jiang et al. [58], a model for the carbon nanotubes aggregate is proposed. The temperature was found to have small impact on the thermal conductivity enhancement while the nonlinear enhancement of thermal conductivity with solid nanoparticles volume fractions was detected.
The Brinkman model was utilized for the viscosity of the nanofluid [59]: This model for the viscosity was used in many studies for CNT-water heat transfer fluid [60][61][62]. In the study of Halelfadl et al. [63], the effects of solid volume fraction of nanoparticles and temperature on the viscosity of carbon nanotubes are considered for particle volume fraction of 0.0055% and 0.55%. No temperature dependence of the viscosity of the nanofluid was detected at high shear rate .

Solver Method and Code Validation
The above given governing equations were solved with Galerkin weighted residual method considering the appropriate boundary and initial conditions. In the method, the weak form of the equations are obtained. Approximations of flow variables with triangular Lagrange finite elements of different orders are performed. Residuals are established by substituting the approximations into the governing equations and weighted average of this residual is set be be zero as: The weight function is denoted by W. Newton-Raphson method was utilized for the solution of the nonlinear set of algebraic equations. For the time dependent part of the problem, variable-order backward differentiation scheme (BDF) was used with order of 1. Convergence criteria of 10 −6 was set for the relative error.
To assure mesh independence, numerical simulation tests with different grid sizes are performed. The grid was very fine towards the walls of the cavity and in the vicinity of interface between the solid and fluid domains while Figure 1b shows the mesh distribution. Table 1 shows the results of grid independence studies for various values of Ri and Ha number combinations considering all hot walls of the cavity for fixed values of Ri = 1, Ha = 10, Ω = 10 • , ω = 45 • and d = 0.15 H. Grid type G4 with 12,814 elements was used. Then, 1/50 of the period of oscillation (Per) was chosen as the time step (∆t) for the unsteady calculations. Validation was performed by using different available studies in the literature. In the first validation study, the results of Sourtiji et al. [64] were utilized where mixed convective heat transfer in a vented cavity with different locations of the outlet port was examined. The companion results of average Nu versus Reynolds number for Ri number of 10 is shown in Figure 2. The difference in percentage in Figure 2b was defined as the discrepancy of the reference average Nu values obtained by Sourtiji et al. [64] as compared to average Nusselt numbers obtained with the present code. In another validation, magnetic field effects for the free convection in a square cavity were used [65] and Table 2 gives the average Nusselt number comparison results considering various values of Rayleigh number when Hartmann number is fixed to 30. The highest deviation was seen for Ra = 10 7 and the value is −1.34%. The present solver is capable of solving problems in vented cavity with magnetic field effects.

146
Pulsating flow in a vented trapezoidal cavity with a T-shaped conductive obstacle is examined 147 under the effects of an uniform magnetic field. The side wall of the vented cavity is also considered to 148 be as inclined. The size and inclination of the object are used to control the flow re-circulation within 149 the cavity and thus heat transfer characteristics will be altered in pulsating flow conditions. Impacts

Results and Discussion
Pulsating flow in a vented trapezoidal cavity with a T-shaped conductive obstacle was examined under the effects of a uniform magnetic field. The side wall of the vented cavity was also considered to be as inclined. The size and inclination of the object were used to control the flow re-circulation within the cavity and thus heat transfer characteristics were altered in pulsating flow conditions. The impacts Energies 2020, 13, 848 8 of 30 of magnetic field were shown to be effective in the vented cavity and in application with separated flow in various studies. The impact of different parameters such as Richardson number (between 0.05 and 50), Hartmann number (between 0 and 30), side wall inclination of the vented cavity (between 0 and 10), size (between 0.1 H and 0.4 H) and orientation (between −90 and 90) of the conductive object, pulsating flow amplitude (between 0.5 and 0.9) and frequency (Strouhal number between 0) on the fluid flow, and convective heat transfer features were examined.

Effects of Richardson Number
Impact of Ri number on the flow and thermal patterns are shown in Figure 3. A lower value Ri number gives a higher velocity of the incoming fluid since the Rayleigh number was fixed to Ra = 10 5 in the simulations. Multiple re-circulations were established in the cavity and near the T-shaped object. As the value of Ri number was increased, vortices near the T-shaped object first disappeared at Ri = 1 and then enlarged in size for Ri = 50. The extent of the vortex below the inlet port increased at Ri = 50, where forced convection effects were weak. Temperature gradients near the hot walls decreased with higher values of Ri number since less fluid flow energy was introduced. This effect was more pronounced near the upper and right hot wall segments of the vented cavity. The average Nusselt number versus Richardson number showed a decreasing trend for individual walls and all walls while different hot wall parts' contributions to the overall Nu were different, as shown in Figures 4 and 5. When cases with the maximum and minimum values of Richardson number were compared, the highest reduction in the contribution was seen for wall W4 (upper hot wall), which is 28%.  In pulsating flow conditions, variation of streamlines and isotherms are shown in Figure 6 for Richardson number of 1 considering various values of time instances within a period of the flow pulsation. Fifty time steps were used for one period of oscillation, ∆t = Per 50 . The time dependent simulations were performed for 15 periods (15 Per) of pulsating flow. Appearance and disappearance of corner vortex below the inlet port, near the upper wall, and in re-circulation zones in the vicinity of the T-shaped object were seen in the cavity for various time instances of pulsating flow. Spatial average Nusselt number considering all of the hot walls showed periodic variation for different Ri values and highest values were attained for higher Richardson number (Figure 7). Strouhal number, which is a non-dimensional frequency based on the velocity of the incoming fluid, was fixed and thus the frequency of oscillation was higher for lower Ri number where velocity was higher. Comparison of the Nusselt numbers in steady flow and pulsating flow (highest value of the space average Nu) for different Ri numbers are shown in Table 3. Heat transfer enhancement with pulsating flow was obtained for all Ri numbers but enhancement was highest for lower Ri value, which is 46.5%.

Magnetic Field Effects
Effects of strength of uniform magnetic field on the variation of streamlines and isotherms in steady flow configuration are shown in Figure 8. In the absence of magnetic field, re-circulations near the left bottom and top right corner were formed in the vented cavity. As the strength of uniform magnetic field was increased, re-circulations in the cavity were suppressed but the fluid motion dampened as well. Clustering of the isotherms with magnetic field, especially for hot walls W1, W3 and W4, were apparent, as shown in Figure 8d-f. Average Nusselt number for those walls enhanced with magnetic field strength, as shown in Figure 9, while a reduction in the average Nusselt number was seen only for hot wall W2. The average heat transfer when taking into account all hot wall parts of the vented cavity enhanced with higher values of Hartmann number. Different hot wall segments contributed differently to the overall heat transfer rate, as shown in Figure 10. The contribution of wall W2 decreased while other wall contributions enhanced for higher Ha values.  Suppression of the vortices were also seen in pulsating flow condition when magnetic field was imposed, as shown in Figure 11. The effect was influential in the cavity, expect for the regions near the T-shaped object. There were some slight variations in the spatial averaged Nusselt numbers for different magnetic field strengths, as shown in Figure 12. Enhancement of average heat transfer was seen with pulsating flow case, as shown in Table 4. There were some slight variations between the average Nu values for different Hartmann numbers in pulsating flow. However, HTE was reduced with higher values of Ha number and this could be attributed to the increment of the average Nu with magnetic field strength in the pulsating flow configuration.
Version January 13, 2020 submitted to Energies 18 of 37

Impact of Side Wall Inclination of the Vented Cavity
As the value of the inclination angle increased, vortices were established near the inclined surfaces below the inlet port and above the outlet port, as shown in Figure 13. Isotherms became less clustered in the regions where elongation of the walls took place and vortices were formed. The average heat transfer rate reduced for all walls except for hot wall W4 (top wall) with higher values of side wall inclination angles. Overall, the average Nu number was reduced for higher values of side wall inclination angle ( Figure 14). Contributions of walls W1, W2 and W3 reduced with higher Ω while the impact was reversed for wall W4 and the increment was the contribution to the overall heat transfer rising by about 13% when minimum and maximum values of Ω were compared, as shown in Figure 15.
In pulsating flow, similar to steady flow configuration, vortices near the inclined walls were established and multiple re-circulation zones were also seen with different time instances within a period of flow pulsation ( Figure 16). Spatial average and total average heat transfer reduced with higher values of Ω ( Figure 17). Similar to previous configurations with flow pulsations, heat transfer enhanced as compared to steady flow configuration and therefore HTE rose from 21 to 48 when Ω was varied from 0 • to 30 • in pulsating flow conditions.

Effects of Conducive T-shaped Obstacle
The use of a T-shaped object can generate additional sites for the formation of the vortices, which has an impact on the main flow stream in the cavity. Size and orientations of the T-shaped object had profound effects on the variations of streamlines and isotherms, as shown in Figure 18. The configuration without conductive object is also shown. In the absence of the object, re-circulations were established in the lower and right part of the main flow stream from inlet to outlet. When the object was installed, the vortex in the upper right part broke into two small vortices. Suppression of the corner vortices was seen when the size of the object was increased. The main flow stream from inlet to outlet deflected d more toward the bottom wall with higher size as well. As the orientation of the T-shaped object changed, the size of the vortices changed but more impact was seen on the deflection of the main flow stream. Temperature gradients became intensified with the greater size of the T-shaped object and this effect was more pronounced with size change as compared to altering the orientation of the object. The average Nusselt number enhanced for all walls when the size of the object increased except wall W3, which was above the outlet port ( Figure 19). For wall W3, it was reduced until size of d = 0.3 H but increased for d = 0.4 H. This was attributed to the fact that most of the main flow stream was deflected toward the bottom wall for higher sizes but the vortex in the upper right corner was suppressed, which resulted in heat transfer enhancement for size d = 0.4 H. The overall Nu value without object remained below the case with object while the discrepancy between the average Nu value increased with greater size of the object. There were both increment and reduction of the average heat transfer rate for different hot wall segments with different orientations of the object compared to location at ω = 0 • (Figure 20). Version January 8, 2020 submitted to Energies 24 of 36  Spatial average Nusselt number increased as the size of T-shaped object increased and the variation was small when different orientations of the object were compared (Figure 21). The overall average Nusselt number increased with greater size of the T-shaped object, similar to the steady flow configuration. However, HTE value was reduced from 41.85 to 20.81 when d was increased from 0.1 H to 0.4 H (Figure 22), which was attributed to the higher Nusselt number enhancement in pulsating flow compared to the steady flow case. The impact of inclination angle of the T-shaped object is shown in Table 5. When pulsating and steady flow configurations were compared, a maximum value of HTE = 31.32 was attained. The impact of ω on the variation of Nu number was slight in both steady and pulsating flow cases.      Figures 23 and 24 show the impact of amplitude and non-dimensional frequency of the pulsating flow on the variation of Nusselt number and heat transfer enhancements. When the amplitude of pulsation rose, heat transfer enhancement was seen for the average Nusselt number. When the amplitude of the pulsation was varied, there was negligible phase shift for the peak Nu value. For higher amplitudes of the pulsation, impact of magnetic field strength on the average Nu number was less effective. HTE rose for higher amplitudes; however, with the rise of Ha number, its value reduced. At A = 0.9, HTE reduced from 33 to 22 when Ha was increased from 0 to 30. This was due to the higher heat transfer enhancement in steady flow with higher Ha numbers as compared to pulsating flow case. When the flow pulsation frequency was varied, the highest HTE was attained at St = 0.5 for Ha = 0 and Ha = 10 while HTE reduced with higher values of Strouhal number at Ha = 30. When the flow pulsation was promoted, additional vortices were established near the walls of the cavity and T-shaped object and thermal gradients near the hot walls were disturbed. This resulted in heat transfer enhancement for higher amplitudes. The flow pulsation effects became reduced when the value of Ha was further increased. Inclusion of highly conductive CNT nanoparticles increased the heat transfer rate significantly, as shown in Table 6 for steady and pulsating flow configurations. When there was no magnetic field in the system, the average Nu number increments were 107.63% and 97.66% for steady and pulsating flow cases, respectively, for the nanofluid (highest particle volume fractions) as compared to water while these values were 105.42% and 103.62% when magnetic field effects were considered for Ha = 30. HTE values reduced with higher φ in the absence of magnetic field; however, at Ha = 30, the variation of HTE was small when values at different φ were compared.

Conclusions
Mixed convection of nanofluid (CNT-water) in a ventilated cavity considering magnetic field effects and an inner T-shaped conductive obstacle was numerically examined. It was observed that the average Nu number reduces with higher values of Ri number and wall inclination angle but the effect becomes reverse for higher values of Ha number and nanoparticle volume fraction. Different hot wall segments contributed differently to the overall Nu number with respect to changes in Richardson number, Hartmann number, and side wall inclination angle. When pulsating flow was promoted, higher heat transfer rates were found as compared to steady flow configuration. Suppression of the recirculation regions within the enclosure and in the vicinity of the T-shaped object were seen when with higher values of Ha number. Heat transfer enhancement was highest for lower value of Richardson number in pulsating flow but the trend was opposite for higher values of Hartmann number. Increment of side wall inclination angle from 0 • to 30 • resulted in rise of HTE from 21 to 48. The inclusion of the conductive T-shaped object and changing it size and orientation had significant impact on the variation of flow and thermal patterns with the cavity. Corner vortices and main flow in the enclosure were influenced by different sizes of the object. Heat transfer enhancement of 41.85% and 20.81% were obtained with flow pulsation when the size was increased from 0.1 H to 0.4 H. When the flow pulsation amplitude rose, heat transfer enhancement was seen but the amount of increment depended on the magnetic field strength. The impact of flow pulsation frequency was not as significant on the heat transfer enhancement as compared to amplitude of pulsation and the highest HTE was attained at Strouhal number of 0.5 for Ha = 0 and Ha = 10. Significant heat transfer enhancements were obtained with the inclusion of highly conductive CNT particles, which were in the range of 97% and 108% for steady and pulsating flow conditions with or without magnetic field effects, respectively.