# Study of Heat Transfer in Magnesium Zinc ZirconiumMgZn6Zr Alloy Suspended in Engine Oil

Authors: Saima Noor; Safyan Mukhtar
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, alloy nanoparticleMgZn6Zr will be analyzed for the first time in the flow of ferromagnetic nanofluid. The combination of two or more metallic substances is termed as alloy. In other words, alloys are a solid solution or mixture of more than one metallic substance. Thus, alloys have always distinct melting points. This means that alloys have melting ranges instead of melting points. Due to these characteristics, alloy nanoparticles are efficient for heat transfer in liquid flows. Thus, we will consider magnesium alloy MgZn6Zr which is suspended in the base fluid engine oil C8H18. The suspension will be analyzed along a dipole; thus, the fluid is known as ferromagnetic MgZn6Zr−C8H18 nanofluid. Solutions will be obtained numerically through the RK-method (shooting method). Our motivation is to optimize the heat transfer through the parametric study. Further, velocity of the fluid is observed to be decreasing with increasing the volume fraction of nanoparticles while temperature profile gets a rise with increase in volume fraction. Moreover, the presence of alloys in any viscous base fluid can decline the fraction between certain fluid layers which results in fast velocity field for the proposed fluid.


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## Body

## 1. Introduction

Nanofluids are widely used in many engineering applications ranging from the use in the mechanical industry to the medical science [1]. Usually, 90% of alloy consists of the metals present in the market place. The alloy is a metal-metal mixture or solid solution. In everyday life, bicycles, aircraft, and cooking pots are typically made up of many kinds of alloys. Solder metal, brass, sterling silver, pewter, and magnesium MgZn6ZrR are the popular and widely used alloys. Having a solid solution or combining or mixing different metals with nonmetals contributes to many benefits. Thus, combination or solid solution of metallic substances can reduce the melting points, increase hardness, and make tensile strength better. The thermal properties, i.e., heat power, density, thermal expansion, and thermal conductivity of the alloys can also be influenced by this combination or solid solution of the metallic material. This means that, due to the metallic substance mixture, the thermophysical characteristics of metals differ from the thermophysical properties of the resulting alloy. This results in the alloy having stronger thermophysical characteristics than the pure metallic material. This phenomenon therefore makes alloys effective for heat flow efficiency [2–8].Pure metals have the characteristics of high melting point; thus, they are too soft for many uses. For instance, gold can be bend easily even at low heat. This is why gold jewelry is usually an alloy. This means that if the Curie temperature of the metals of ferrites is smaller, then the mixture of these ferrites will definitely disturb the resultant alloy. It seems from the properties of the resulting alloys that for pure metals and after taking the form of alloys, the melting point, hardness, and tensile strength vary. Similarly, along with the density, thermal expansion, thermal conductivity, and heat power of the solid solution of metal substances or alloys, the Curie temperature will also be increased. This will help the alloys in reducing friction drag and enhancing the transfer of heat. These types of characteristics of nanoparticles and the like are appropriate and applicable to all devices used for cooling or heating purposes [9–13]. Normal nanoparticles, i.e., gold, copper, aluminum, and titanium, and ferrite nanoparticles, i.e., nickel zinc ferrite, manganese zinc ferrite, and magnetite ferrite, were used by scientists and mathematicians in the history [14–17]. Due to the use of alloys for heat transfer and friction drag, this article distinguishes from all others. If the alloys of nanosized particles and base fluids are in isothermal balance, the presence of these alloys in base fluid develops the thermal properties.As the alloys or ferrites are in isothermal equilibrium with the base fluids, thus, these alloys or ferrites are also very reactive. For examples, due larger melting point, iron is very hard and strong but its surface reacts with air moisture and easily rusts. To prevent iron from rusting, it is sufficient to cast iron as alloy, which enhances its inertness. This is why if the alloy nanosized particle MgZn6Zr is used with engine oil will definitely alter the properties of the resulting ferromagnetic nanofluids. The resultant fluid is still called ferromagnetic nanofluid since alloy particles and ferrite particles have same properties, i.e., alloy particles like ferrite particles have the property of Curie temperature, which ensures that the alloy particles at some temperatures become nonmagnetic [18–21]. Alloys are multiple metals, so these particles can do best in heat transfer in moving fluids, so the article focuses on these particles.In most of the engineering problems, the differential equations are inherently nonlinear in nature. The exact solution for these partial differential equations is often difficult to obtain. To overcome this issue, researchers developed some novelties that can handle these nonlinear partial differential equations or convert them to more solvable form. For the said purpose, various techniques are employed. For instance, the reduction of partial differential equations to ordinary ones by using the similarity transformations is one of those. This method provides a suitable transformation that without disturbing the physics of problem gives a more reliable form of differential equations that can be handled either numerically or analytically.The researchers identified various nanoparticles like magnetic particles as nonmagnetic particles and investigated their effect on surface friction and cooling rate or heating rate. Yet very rare literature on the study of alloy nanoparticles in moving fluids exists. In particular, the authors who studied alloys treat these particles as natural particles, which means that they treat them as nonmagnetic particles. Since alloys fulfill the properties of ferrites, the nanosized particle magnesium alloy MgZn6ZrZn is therefore studied in this article. Magnesium zinc zirconium MgZn6Zr is used because of the lower density and higher thermal conductivity. Because of the higher thermal conductivity, it can provide a very useful source for the heat transfer and temperature control in many industrial as well as physical situations. In the base fluid engine oil, C8H18, the magnesium alloy MgZn6Zr is suspended. In the presence of a dipole, the study is carried out to establish the moving fluid as a magnetic fluid. The graphical findings are discussed, and the effect of alloys on the transmission of heat is studied.


## 2. Mathematical Modeling

Consider a steady two-dimensional flow of an incompressible, viscous, and electrically nonconducting fluid driven by an impermeable sheet in the horizontal direction shown in Figure1. By applying two equal and opposite forces along the horizontal direction which is taken as the x-axis, with the y-axis in a direction normal to the flow, the sheet is stretched with a velocity which is proportional to the distance from the origin. A magnetic dipole is located with its center on the y-axis at a distance S from the sheet. The engine oil C8H18 and magnesium alloy MgZn6Zr are in thermal equilibrium. The external dipole is examined in the flow problem. The flow problem and its mathematical molding, i.e., the thermo-mechanical coupling are given in equations (1)–(5). The equations of interest are as follows:(1)∂u∂x+∂v∂y,(2)ρnfu∂u∂x+v∂u∂y=−∂P∂x+μnf∂2u∂y2+μ0M∂H∂x,(3)ρcpnfu∂T∂x+v∂T∂y+ρcpnfμ0T∂M∂Tu∂H∂x+v∂H∂y=knf∂2T∂y2.Figure 1
Geometry of the MgZn6Zr-C8H18 nanofluid flow.Equation (1) is the continuity equation. Equation (2) stands for the momentum equation, and Equation (3) gives the description for law of conservation of energy and provides the mathematical description for the temperature profile.Boundary conditions are as follows:(4)u|y=0=Sx,v|y=0=0,∂T∂y|y=0=−hcT,(5)u|y⟶∞⟶0,T|y⟶∞⟶Tc.The respective thermophysical properties and expression of engine oilC8H18 and magnesium alloy MgZn6Zr are given in Tables 1 and 2.Table 1
Thermophysical properties of engine oilC8H18 and magnesium alloy MgZn6Zr.
ρ (kg/m3)Cp (J/kgK)k (W/mK)PrC8H1889018680.14512900MgZn6Zr2.00960.0120---Table 2
Thermophysical expressions of engine oilC8H18 and magnesium alloy MgZn6Zr.
PropertiesMgZn6Zr−C8H18Viscosityμμnf=μf1−Φ−25/10Densityρρnf=1−Φρf+ΦρsThermal conductivitykknf/kf=2kf+ks−2Φkf−ks/2kf+ks+Φkf−ksHeat capacityρCpρcpnf=1−Φρcpf+Φρcps


### 2.1. Magnetic Dipole

The presence of alloy particle in any base fluid needs a magnet to give magnetization to the internal alloy particles in the fluid. In this direction, for the flow of ferromagneticMgZn6Zr−C8H18 nanofluid, a dipole is taken at a distance l. The center of dipole is taken along vertical axis at distance l away from horizontal axis. The dipole attracts the alloys particles when the temperature is below the Curie temperature, and for higher temperature, the alloy particles lose their magnetization. ς is given as follows:(6)ς=γ12πxx2+y+l2.H can be defined as(7)H=∂ς∂y2+∂ς∂x2.Moreover, the magnetization is taken here as follows:(8)M=KmT−T∞.Figure1 portrays the physical interpretation.


### 2.2. Similarity Analysis

Similarity variables are given as follows for the present problem:(9)ψη,ξ=ημfρffξ,θη,ξ≡Tc−TTc=θ1ξ+η2θ2ξ.The velocity components and dimensionless coordinates are(10)η=xρfSμf1/2,u=∂ψ∂r=Sxf′ξ,ξ=yρfSμf1/2,v=−∂ψ∂x=−Sνf1/2fξ.These dimensionless variables transform equations (1)–(5) to equations as follows:(11)11−Φ25/10A1f‴−f′2+ff″−2βθ1A1ξ+γ4=0,(12)knfkfA2θ1″+Prfθ1′−2f′θ1+2λβfθ1−εA2ξ+γ3−4λA2f′2=0,(13)knfkfA2θ2″−Pr4f′θ2−fθ2′+2λβfθ2A2ξ+γ3−λβθ1−εξ+γ32f′ξ+γ4+4fξ+γ5−λA2f″2=0,(14)fξ=0,f′ξ=1,θ1′ξ=−λ11+θ1ξ,θ2ξ=0,atξ=0,f′ξ⟶0,θ1ξ⟶0,θ2ξ⟶0,atξ⟶∞.A1 and A2 are(15)A1=1−Φ+Φρsρf,A2=1−Φ+Φρcpsρcpf.The parameters and dimensionless number areε (dimensionless Curie temperature), λ (viscous dissipation), Pr (Prandtl number), λ1 (conjugate parameter), and β (ferrohydrodynamic interaction). The mathematical representation is as follows:(16)ε=T∞Tc,γ=Sρfl2μf,β=γ12πμ0KmTcρfμf2,λ=Sμf2ρfKmTc,λ1=hcμfρfS,Pr=νfαf.The surface friction and heat transfer rate of engineering interest are as follows:(17)Cf=τw1/2ρnfUw2,τw=μnf∂u∂y|y=0,Nux=xknfkfTc−Tw∂T∂y|y=0.The dimensionless forms are as follows:(18)12Rex1/2Cf=11−Φ25/10f″0,Rex−1/2Nux=−λ1knfkf1+1θ10+η2θ20.Here,Rex denotes the Reynolds number that is the ratio of inertial to viscous forces; mathematically, one can write Rex=xUw/νf.


## 2.1. Magnetic Dipole

The presence of alloy particle in any base fluid needs a magnet to give magnetization to the internal alloy particles in the fluid. In this direction, for the flow of ferromagneticMgZn6Zr−C8H18 nanofluid, a dipole is taken at a distance l. The center of dipole is taken along vertical axis at distance l away from horizontal axis. The dipole attracts the alloys particles when the temperature is below the Curie temperature, and for higher temperature, the alloy particles lose their magnetization. ς is given as follows:(6)ς=γ12πxx2+y+l2.H can be defined as(7)H=∂ς∂y2+∂ς∂x2.Moreover, the magnetization is taken here as follows:(8)M=KmT−T∞.Figure1 portrays the physical interpretation.


## 2.2. Similarity Analysis

Similarity variables are given as follows for the present problem:(9)ψη,ξ=ημfρffξ,θη,ξ≡Tc−TTc=θ1ξ+η2θ2ξ.The velocity components and dimensionless coordinates are(10)η=xρfSμf1/2,u=∂ψ∂r=Sxf′ξ,ξ=yρfSμf1/2,v=−∂ψ∂x=−Sνf1/2fξ.These dimensionless variables transform equations (1)–(5) to equations as follows:(11)11−Φ25/10A1f‴−f′2+ff″−2βθ1A1ξ+γ4=0,(12)knfkfA2θ1″+Prfθ1′−2f′θ1+2λβfθ1−εA2ξ+γ3−4λA2f′2=0,(13)knfkfA2θ2″−Pr4f′θ2−fθ2′+2λβfθ2A2ξ+γ3−λβθ1−εξ+γ32f′ξ+γ4+4fξ+γ5−λA2f″2=0,(14)fξ=0,f′ξ=1,θ1′ξ=−λ11+θ1ξ,θ2ξ=0,atξ=0,f′ξ⟶0,θ1ξ⟶0,θ2ξ⟶0,atξ⟶∞.A1 and A2 are(15)A1=1−Φ+Φρsρf,A2=1−Φ+Φρcpsρcpf.The parameters and dimensionless number areε (dimensionless Curie temperature), λ (viscous dissipation), Pr (Prandtl number), λ1 (conjugate parameter), and β (ferrohydrodynamic interaction). The mathematical representation is as follows:(16)ε=T∞Tc,γ=Sρfl2μf,β=γ12πμ0KmTcρfμf2,λ=Sμf2ρfKmTc,λ1=hcμfρfS,Pr=νfαf.The surface friction and heat transfer rate of engineering interest are as follows:(17)Cf=τw1/2ρnfUw2,τw=μnf∂u∂y|y=0,Nux=xknfkfTc−Tw∂T∂y|y=0.The dimensionless forms are as follows:(18)12Rex1/2Cf=11−Φ25/10f″0,Rex−1/2Nux=−λ1knfkf1+1θ10+η2θ20.Here,Rex denotes the Reynolds number that is the ratio of inertial to viscous forces; mathematically, one can write Rex=xUw/νf.


## 3. Numerical Simulation

The problem of heat transfer is simulated numerically in Matlab [22, 23]. Equations (11)–(14) are transformed to first-order differential equations and then entertained via shooting technique in Matlab. The discussion section contains its simulation and physical explanation. The procedure and its reduction to first-order equations are as follows:(19)z1=fξ,z2=f′ξ,z3=f″ξ,z3′=f‴ξ,z4=θ1ξ,z5=θ1′ξ,z5′=θ1″ξ,z6=θ2ξ,z7=θ2′ξ,z7′=θ2″ξ.Equation (19) reduces equations (11)–(14) to the following form:(20)z3′=1−Φ25/10A1z2∗z2−z1∗z3+2βz4A1ξ+γ4,z5′=−kfA2knfPrz1z5−2z2z4+2λβz1z4−εA2ξ+γ3−4λA2z22,z7′=kfA2knfPr4z2z6−z1z7−2λβz1z6A2ξ+γ3+λA2z22+λβz4−εξ+γ32z2ξ+γ4+4z1ξ+γ5.The boundary conditions given in equation (14) take the form as follows:(21)z1=0,z2=1,z5=−λ11+z4,z6=0,atξ=0,z2⟶0,z4⟶0,z6⟶0,atξ⟶∞.


## 4. Discussion

The ferromagnetic nanofluid consists of alloyMgZn6Zr with C8H18 base fluid. The alloy nanoparticle in the flow analysis is taken to address what happens exactly to heat transfer when an external field is placed or removed. This analysis is incorporated only to know whether ferromagnetic MgZn6Zr−C8H18 nanofluid is efficient for flow of heat; if the ferromagnetic MgZn6Zr−C8H18 nanofluid performs, then the ferromagnetic MgZn6Zr−C8H18 nanofluid must attract the industrial users and engineers from the fields indulging all those equipments and devices which transfer heat. The Newtonian heating is defined at the surface; this leads to help the alloy nanoparticles in transferring heat from internal side to outside the fluid. Thus, this model can be more suitable for transferring heat as compared with other studied nanoparticles or ferrite nanoparticles.In this direction, Figures2 and 3 are sketched to check out the impacts of alloy MgZn6Zr on velocity and temperature fields. Enhancement in axial velocity for ferromagnetic MgZn6Zr−C8H18 nanofluid is examined. The ferromagnetic MgZn6Zr−C8H18 nanofluid contains alloy MgZn6Zr nanoparticle as the alloy MgZn6Zr has the property of Curie temperature. Further, the base fluid and alloy have different thermophysical properties; thus, they disturb the velocity and temperature field. Thus, the variation or abrupt change in axial velocity of ferromagnetic MgZn6Zr−C8H18 nanofluid is valid because of the inclusion of alloy MgZn6Zr nanoparticle; the reduction is observed for the respective cases, i.e., (i) Φ≠0, β≠0, and λ1≠0. On the other hand, the low velocity in Figure 2 is evident for the ferromagnetic MgZn6Zr−C8H18 nanofluid when (ii) Φ=0, β≠0, and λ1≠0, (iii) Φ≠0, β=0, and λ1≠0, and (iv) Φ=0, β=0, and λ1≠0. This means that the presence of alloys in any viscous base fluid can decline the fraction between certain fluid layers, and as a result, one can get fast velocity field for the fluid under study. Now, the behavior of ferromagnetic MgZn6Zr−C8H18 nanofluid describes that if alloy MgZn6Zr is suspended in a single base fluid, it has higher energy of transferring heat as compared with the fluids having no alloy MgZn6Zr nanoparticles. Thus, the absence of alloy MgZn6Zr leads to more resistance in the base fluids; hence, velocity declines when alloy MgZn6Zr is removed. On the other hand, temperature field in Figure 3 shows variations in ferromagnetic MgZn6Zr−C8H18 nanofluid. Basically, the base fluid and alloy nanoparticle and there interaction arise the resistance; thus, the respective resistance is responsible for enhancement in temperature field. Further, the thermphysical properties of base fluid and alloy also enlarge the temperature of the flowing fluid. Here, alloy MgZn6Zr in a base fluid C8H18 is suspended. This means that MgZn6Zr alloy and C8H18 base fluid have different thermophysical properties in all aspects. Therefore, resulting ferromagnetic MgZn6Zr−C8H18 nanofluid has different thermophysical properties and Curie temperature. Thus, the temperature field declines in the ferromagnetic MgZn6Zr−C8H18 nanofluid in such a way that the highest temperature is depicted for Φ=0, β=0, and λ1≠0 and decline for Φ≠0, β=0, and λ1≠0; Φ=0, β≠0, and λ1≠0; and Φ≠0, β≠0, and λ1≠0.Figure 2
Comparision of velocity field inMgZn6Zr−C8H18 for ferrohydrodynamic parameter, Newtonian heating, and solid volume fraction.Figure 3
Comparision of temperature distribution inMgZn6Zr−C8H18 for ferrohydrodynamic parameter and solid volume fraction.The ferromagneticMgZn6Zr−C8H18 nanofluid is significant to compute their results and demonstrate its impacts on heat flow. The presence of alloy MgZn6Zr in a ferrofluid is analyzed for the first time. Thus, it is important to plot comparative results with a simple base fluid. In this way, the graphical results, i.e., Figures 4 and 5, delineate the hydrodynamic interaction parameter β on the fields of axial velocity and temperature. The influence of hydrodynamic interaction β on the axial velocity of ferromagnetic MgZn6Zr−C8H18 nanofluid is inspected in Figure 4. The simulation of alloy MgZn6Zr in the base fluid C8H18 shows that enhancement in axial velocity takes place when hydrodynamic interaction β parameter arises. Maximum enhancement is noticed for the MgZn6Zr−C8H18 nanofluid, where the minimum change takes place in MgZn6Zr−C8H18, when Φ=0. Giving variation to hydrodynamic interaction parameter β leads to enhanced velocity of ferromagnetic MgZn6Zr−C8H18 nanofluid compared with the ferromagnetic fluid, i.e., Φ=0. This happens because of the presence of alloy MgZn6Zr in the ferromagnetic MgZn6Zr−C8H18 nanofluid; thus, alloy MgZn6Zr gives more attraction to the dipole. Maximum decrease in temperature distribution is described for the ferromagnetic fluid, i.e., Φ=0, as presented in Figure 5. The simulation of hydrodynamic interaction parameter means that the dipole is essential for the flow of heat if the cooling or heating agent MgZn6Zr−C8H18 is used. Thus, magnetic dipole and alloy nanoparticles are significant for each other as well.Figure 4
Axial velocity and its comparison in the flow ofMgZn6Zr−C8H18 and C8H18 via β.Figure 5
Temperature field in the flow ofMgZn6Zr−C8H18 and C8H18 via β.Fraction drag and heat transfer analysis is significant parameter for the analysis. Friction drag tells us what kind of resistance faces the flowing fluid in the analysis under discussion whereas the rate of heat flux between two points in a material is proportional to the difference in temperature between the points and the ratio of the distance between two points. Fourier’s law helps in simulation of heat flow. This section includes the analysis of heat transfer flow and skin friction for the present problem. Figure6 examines friction drag in the ferromagnetic MgZn6Zr−C8H18 nanofluid. The comparison of skin friction coefficient is made for different cases, i.e., maximum resistance is described for Φ=0, β=0, and λ1≠0 and decline for Φ≠0, β=0, and λ1≠0; Φ=0, β≠0, and λ1≠0; and lowest resistance is observed for Φ≠0, β≠0, and λ1≠0. Maximum reduction in wall shear stress is demonstrated for ferromagnetic MgZn6Zr−C8H18 nanofluid when Φ=0, β=0, and λ1≠0. Figure 7 reveals that the Nusselt number arises in the flow of ferromagnetic MgZn6Zr−C8H18 nanofluid, when Φ≠0, β≠0, and λ1≠0; thus, the ferromagnetic MgZn6Zr−C8H18 nanofluid is efficient for the heat transfer whereas the minimum heat transfer is examined for the case when Φ=0, β=0, and λ1≠0. Physically, this means that the presence of alloy MgZn6Zr particles in the base C8H18 ferrofluid really enhances the rate of heat transfer.Figure 6
Different cases comparison of friction drag in the flow ofMgZn6Zr−C8H18 and C8H18.Figure 7
Different cases comparison of heat transfer rate in the flow ofMgZn6Zr−C8H18 and C8H18.


## 5. Concluding Remarks

The article discuses heat transfer for the first time in the alloy nanoparticles suspended in a ferrofluid theoretically. The analysis described in the article further studied friction drag in alloy nanoparticles. Heat transfer is needed in fluid flow that helps to reduce heat in mechanical devices. Usually, scientists use nonmagnetized particles or particles that have no Curie temperature, but when heat transfer involves a magnetic fluid, those particles become useless. The nonmagnetized particles are also useless for heat transfer.NiZnFe3O4, MnZnFe3O4, and so many others are particles satisfy the magnetization property. These mentioned particles are available in the literature. The alloys are rarely used in this direction for heat transfer; thus, the analysis is made. It is observed that alloys in any viscous base fluid might reduce the fraction between particular fluid layers, resulting in a fast velocity field for the proposed fluid. Hence, it is concluded that for heat transfer instead of viscous ferrofluid described in the results section, the presence of alloy is effective. This study can be further extended by considering variable thermophysical properties.



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