# Mixed Convection Heat Transfer of a Hybrid Nanofluid in a Trapezoidal Prism with an Adiabatic Circular Cylinder

Authors: Mohammed Hirpho; Wubshet Ibrahim
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This communication presents the mixed convection flow of hybrid nanofluid flow in a trapezoidal prism cavity. The cavity contains an adiabatic circular cylinder filled withAg−MgO water hybrid nanofluid. The top surface of a prism cavity is heated to a uniform temperature of Th, and the bottom surface of the prism is cooled to a temperature of Tc. The other surfaces of a prism are thermally insulated. The equations governing the flow problem are obtained using conservation principles and nondimensionalized with the help of nondimensional variables. Then, with the help of the Galerkin finite element method algorithm embedded in COMSOL Multiphysics® software, the numerical simulation was performed. The streamlines and isotherms on the surfaces of the cavity have been plotted and thoroughly discussed. The results reveal that buoyant convection distorts the isothermal fields as the Richardson number increases, so does fluid flow penetration in the cavity. The results also reveal that as the Richardson number and volume fraction of hybrid nanoparticles increase, the average Nusselt number diminishes.


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## Body

## 1. Introduction

The two-dimensional study of the flow of convectional fluid in a cavity started two decades ago. Following the works of the pioneering researchers, Chamkha [1] studied the influences of magnetic fields on the flow dynamics in a square cavity under the condition of a heat source or heat sink. Also, Chamkha et al. [2] have computed the dynamics of flow in a double lid-driven cavity containing a rotating frustum of a cone with the application of a magnetic field. Furthermore, Chamkha et al. [3], Mondal et al. [4], and Barnoon et al. [5] have analyzed the combined impact of entropy generation and the magnetic field on the flow of lid-driven cavities. Hamzah et al. [6] have examined the magnetic field influence on the conducting cylinder immersed in a cavity filled with nanofluid, and Huang and Lim [7] have presented the numerical simulation of a lid-driven cavity with some internal obstacles in the shape of cylinders. Another study conducted by Hussein et al. [8] examined mixed convection flow in a cavity-shaped trapezoidal region filled with nanofluid and its bottom face influenced by a temperature-varying sinusoidal function.A few years ago, hybrid nanofluids attracted the attention of scholars due to their usefulness in science and technology. A hybrid nanofluid was invented to enhance the thermal characteristics of conventional nanofluids. Due to synergistic effects, hybrid nanofluids comprise two or more nanoparticles with novel chemical and thermophysical properties that have the potential to increase the rate of heat transmission [9]. Multiple researchers have examined different aspects of hybridized nanoparticles’ influence on the thermal efficiency of coolants. For instance, the use of hybridized nanoparticles in identifying cancer cells was described by Sailor and Park [10]. They claimed that using hybrid nanoparticles to deliver drugs to cancer cells was beneficial. Zhang et al. [11] talked about how hybrid nanoparticles are used in medicine and biomedicine. Also, Landfester [12] discussed how hybrid nanoparticles are used in the manufacturing of polymers. Suresh et al. [13] investigated the effects of convective heat transfer through the tube using Al2O3−Cu/H2O hybrid nanoliquid, finding that hybrid nanofluids had a larger Nusselt number than conventional fluids. Recently, numerous publications (for instance, [14–25]) were produced addressing the impact of nanoparticles in fluids adopting different liquid models.The three-dimensional cavity flow study is a recent trend in fluid dynamics. Accordingly, Iwatsu and Hyun [26] and Albensoeder and Kuhlmann [27] are some pioneers in the examination of the three-dimension cavity flow of Newtonian fluid with a temperature gradient. Later on, Romano et al. [28] examined the 3D cavity flow. Furthermore, De et al. [29] have exhibited the numerical simulation for the three-dimensional lid-driven cavity with the help of the Lattice Boltzmann method. The examination of both 2D and 3D cavity flow simulations induced by enclosed spheres and circles had been discussed by Souayeh et al. [30]. The issue of cavity flow stability and instability is a major concern in fluid dynamics problems. Accordingly, the instability caused in a 3D cavity due to two circles was examined by Gonzalez et al. [31]. Al-Kouz et al. [32] investigated the three-dimensional MHD flow of a water-based CNT/ferroparticles hybrid nanofluid through a wavy-walled trapezoidal cavity with natural convection and entropy generation. The entropy production and mixed convection within a trapezoidal nanofluid-filled hollow with a localized solid cylinder are quantitatively investigated by Ishak et al. [33]. Their findings reveal that the size and position of the solid cylinder are important parameters for optimizing heat transport, as is the Bejan number inside the trapezoidal cavity. Alsabery et al. [34] also examined the transient free convection and entropy formation inside a porous cubical container saturated with an alumina-water nanofluid.The objective of the present study is to conduct a computational simulation of mixed convection heat transfer of a hybrid nanofluid in a trapezoidal prism with an adiabatic circular cylinder under the effect of the Richardson number and volume fraction of hybrid nanoparticles. To the knowledge of the authors and according to the studies cited above, there has been no investigation into the three-dimensional mixed convection hybrid nanofluid within a trapezoidal prism containing an inner cylinder. Hence, the authors are motivated by the knowledge gap that exists in the area.


## 2. Mathematical and Physical Formulation

The current analysis takes into account flow in a trapezoidal prism cavity with an adiabatic circular cylinder, as illustrated in Figure1. The laminar mixed convection flow is considered to be heated from the top surface of the cavity at uniform temperatures of Th and cooled from the bottom surface at uniform temperatures of Tc, respectively. The other surfaces of the cavity are supposed to be thermally insulated. The cavity is considered to be impermeable and rigid on all surfaces. The cavity is filled with Ag-MgO water hybrid nanofluids at various volume fractions. The flow is assumed to be steady, incompressible, and Newtonian. The mixed convection heat transfer problem with the considered hybrid nanofluids has the following governing equations [22, 35]:(1)∇⋅u=0,(2)u⋅∇u=−1ρhnf∇p+μhnfρhnf∇2u+gρβhnfρhnfT−Tcn,(3)u⋅∇T=αhnf∇2T,where p,u=u,v,w,T, and n=0,0,1 are the pressure, velocity vector, temperature, and the unit vector in the buoyancy direction, respectively. ρhnf is density, μhnf is dynamic viscosity, βhnf is the thermal expansion coefficient, and αhnf is thermal diffusivity of hybrid nanofluid mixture properties. The nondimensional variables mentioned are used to transform the governing equations into nondimensional equations.(4)U=U,V,W=u,v,wU0,P=pρhnfU02,T−Tc=Th−Tcθ.Figure 1
Schematic diagram.The governing equations (1)–(3) have now been reduced to the following nondimensional form:(5)∇⋅U=0,(6)U⋅∇U=−∇P+1Reμhnfρfρhnfμf∇2U+GrRe2ρβhnfρhnfβfθn,(7)U⋅∇θ=1RePrβhnfβf∇2θ,where Pr=νf/αf,Gr=gβfTh−Tc/νhnf2L3, and Re=LU0/νf are Prantl, Grashof, and Reynolds numbers, respectively. The configuration’s boundary conditions are written in a nondimensional form.For the upper surface:=1,U=1,V=0,W=0.For the lower horizontal surface:=0,U=V=W=0.For the left and right surfaces:U=V=W=0,∂θ/∂n=0.Along the cylindrical surfaces∂θ/∂n=0=U=V=W.The local and average rate of transfer of heat at the heated top surface can be evaluated by the local Nusselt numbers and average Nusselt numbers given, respectively, as follows:(8)Nu=−khnfkf∂θ∂Y,Nuavg=10.4∫0.30.7∫01NudXdY.The following equations are used to calculate the thermophysical properties of hybrid nanofluids:(9)ρhnf=1−ϕhnfρf+ϕAgρAg+ϕMgOρMgO,ρβhnf=1−ϕhnfρβf+ϕAgρβAg+ϕMgOρβMgO,ρCphnf=1−ϕhnfρCpf+ϕAgρCpAg+ϕMgOρCpMgO,αhnf=khnfρCphnf,μhnf=μf1−ϕAg+ϕMgO2.5,khnf=kf,where φhnf=φAg+φMgO is the volume fraction of hybrid nanoparticles, φAg is volume fraction of Ag and φMgO is volume fraction of MgO. It is assumed to be φAg=φMgO. The thermophysical properties of water and various nanoparticles are shown in Table 1.Table 1
Thermo-physical properties of water,Ag, and MgO [36, 37].
PropertiesWaterMgOAgρKg/m3997.1358010500cpJ/KgK4179879235kW/mK0.61330429βk−121×10−533.6×10−65.4×10−5αm2s−114.7×10−695.3×10−7174×10−3μPas8.55×10−4——


## 3. Numerical Procedure

The nondimensional governing equations (5)–(7) with the boundary conditions were simulated using the Galerkin weighted residual finite element approach. It follows that (5) and equation P=−γ∇⋅U are used to remove the pressure term P, whereas (5) is fulfilled for higher values of γ=107. Then, by substituting the pressure term into the equation (6) and its reduced to(10)U⋅∇U=γ∇⋅∇U+1Reμhnfρfρhnfμf∇2U+GrRe2ρβhnfρhnfβfθn.The solution domain is discretized using a finite element mesh with nonuniform triangular elements. The unknown values of the governing equations are approximated using interpolation functions within those finite elements, such as(11)U≈∑k=1NUkΦk,θ≈∑k=1NθkΦk,where Φkk=1N is the basis function and Uk and θk are the values of the velocity vector at the nodes of the element. Then, Lagrange finite elements have been used in the computational domain to approximate the unknown values. Different types of domain elements, boundary elements, and edge elements were achieved to test the grid sensitivity for the current model, as shown in Table 2. The variation of the average Nusselt number at the hot top surface was considered with φhnf=1%,Gr=104, and Re=100. It was tested by evaluating the percentage error in the nondimensional average Nusselt number Nuavg variation of the hot top surface. To achieve the desired result, a mesh with 427370 domain elements, 25174 boundary elements, and 716 edge elements with a percentage error of less than 0.2 is chosen. Figure 2 depicts the suggested grid layout for this study.Table 2
Grid independency study forRe=100,Gr=104, and φhnf=1%.
Mesh sizeDomain elementsBoundary elementsEdge elementsNuavgPercentage error%Courser1293318261961.993Course2692332542581.9442.4Normal4936949783141.9310.6Fine12743999304481.92070.5Finer427370251747161.91670.2Figure 2
Mesh of the three-dimensional trapezoidal prism with a heated top surface.


## 4. Results and Discussion

This section deals with a thorough discussion of the results obtained and provides meaningful physical interpretations via graphs and tables. A detailed analysis of the effect of the Richardson number and volume fraction of hybrid nanoparticles on streamlines and isothermal contours had been carried out in this context. The Prandtl number of the base fluid was taken as 6.9, and the Reynolds number was set at 100. The nature of average and local Nusselt numbers is also taken into account due to the variation of different parameters.Figure3 exhibited significant behavior for various values of the Richardson number on isothermal contours and streamlines with a volume fraction of hybrid nanoparticles of φhnf=1%. The Richardson number indicates the significance of free convection compared to forced convection effects. The isothermal contours are shown in Figure 3(a) for Ri=0.1 demonstrate that the large temperature zone is symmetrically concentrated and stratified around the hot surface. Indeed, the patterns clearly show that for Ri=0.1, the mechanically driven forced convection outperforms buoyancy-driven convection, suggesting that forced convection is mostly caused by lid movement. This phenomenon is driven by the minor impact of buoyancy force for small values of the Richardson number. When the free convection impact is almost equal to the forced convection effect (i.e., if Ri=1), the pattern of isotherm is identical to that seen in the prior situation (i.e., the case where Ri=0.1). The isothermal fields are distorted by buoyant convection when Ri increases to Ri=10, as illustrated in Figure 3(e). This is due to the buoyancy force’s strong influence. This is a sensible fact because the free convection influence becomes powerful at peak values of the Richardson number. The presence of the wall movement clearly affects isotherms, because the isotherms have distinct patterns from one another. The definition of the Richardson number relates to this behavior since a low Richardson number indicates that forced convection dominates the cavity’s heat transfer mechanism to the greatest extent. As a result, the isotherms are significantly impacted by the orientation of the moving wall. That’s because the impact of free convection is modest in this situation. Further, the isosurfaces corresponding to the middle and lower temperatures are parallel to one another, indicating thermal convection heat transmission occurs. Figure 3(b) shows a streamlined plot for a low Richardson number Ri=0.1, indicating that forced convection is dominating the flow. In this situation, the flow inside the cavity is made up of clockwise spinning convection vortices. When the top wall of the hollow moves, the flow vortices begin to travel in the same direction, and as a result, all portions of the cavity fill uniformly. It is also seen that the velocity of the upper cavity’s moving surface is higher and that natural convection effects are diminished. More fluid flow vortices in the cavity are noticed, as shown in Figure 3(f), when the Richardson number increases to Ri=10 owing to the sliding top surface.Figure 3
Plots showing the effect of Richardson number on isothermal contours and streamlines with volume fraction of hybrid nanoparticlesφhnf=1%. (a) Ri=0.1. (b) Ri=0.1. (c) Ri=1. (d) Ri=1. (e) Ri=10. (f) Ri=10.
(a)(b)(c)(d)(e)(f)Figure4 depicts the distribution of the local Nusselt number along the top horizontal hot surface for various values of Richardson numbers with volume fraction of hybrid nanoparticles φhnf=1%. The results show that the highest values of the local transfer of heat are distributed throughout the lower part of the heated surface, and there are minor differences in the local heat transfer with varied Richardson numbers. It can also be shown that when the Richardson number increases, the value of the local Nusselt number decreases due to an increase in heat distribution resistance.Figure 4
Plots showing the effect of Richardson number on local Nusselt number with volume fraction of hybrid nanoparticlesφhnf=1%. (a) Ri=0.1. (b) Ri=1. (c) Ri=10.
(a)(b)(c)The effects of the volume fraction of hybrid nanoparticles on isothermal contours and streamlines with Richardson numberRi=1 are presented in Figure 5. When nanoparticles are introduced to conventional fluids, even in modest amounts, increased heat transfer rates are seen due to thermal conductivity enhancement. The graphs of the isosurfaces show that there is a slight difference in the values of the stream function between the volume fraction of hybrid nanoparticles. Furthermore, each volume fraction of hybrid nanoparticles has a comparable flow field pattern comprised of clockwise rotating convection vortices. It is worth noting that the use of a hybrid nanofluid boosts fluid flow strength. This is due to the fact that as thermal energy increases, so does the velocity of the fluid and stream function.Figure 5
Plots showing the effects of the volume fraction of hybrid nanoparticles on isothermal contours and streamlines with Richardson numberRi=1. (a) φhnf=0%. (b) φhnf=0%. (c) φhnf=1%. (d) φhnf=1%. (e) φhnf=2%. (f) φhnf=2%.
(a)(b)(c)(d)(e)(f)Figure6 and Table 3 depict the heat transfer rate in terms of the average Nusselt number versus the Richardson number for various values of the volume fraction of a hybrid nanoparticle. The result shows that the average Nusselt number decreases with increasing the Richardson number. This is because natural convection strengthens at a higher Richardson number, and thus heat transfer decreases.Figure 6
Plots showing heat transfer rate in terms of average Nusselt number.Table 3
Table of values showing the average Nusselt number with Richardson numberRi on the heated top surface for different values φhnf.
φhnf (%)RiNuavg00.14.37432014.284480103.3155110.14.19976114.123311103.2971720.14.05995214.000442103.34167


## 5. Conclusion

The mixed convection heat transfer of a hybrid nanofluid in a trapezoidal prism with an adiabatic circular cylinder under the effect of the Richardson number and volume fraction of hybrid nanoparticles has been investigated numerically. The nanoliquid under consideration consists of nanosizedAg−MgO particles suspended in water. A numerical solution using the finite element method has been elaborated for a system of partial differential equations with boundary conditions governing the flow dynamics. The grid independence test has been performed for convergence of the solution. The impact of various parameters on the performance of the transfer of heat and the structure of the flow has been investigated. It has been found that the isothermal fields are distorted by buoyant convection, and more fluid flow penetration in the cavity is noticed as the values of the Richardson number increase. It is also observed that the usage of hybrid nanofluids increases the strength of the fluid flow. Furthermore, as the values of the Richardson number and hybrid nanoparticle percentage increase, the total transfer of heat in terms of the average Nusselt number decreases. In the future, the investigation could be broadened to include entropy generation analysis by considering other nanoparticles with various base fluids.



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