# CuO–Water Nanofluid Magnetohydrodynamic Natural Convection inside a Sinusoidal Annulus in Presence of Melting Heat Transfer

Authors: M. Sheikholeslami; R. Ellahi; C. Fetecau
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

Impact of nanofluid natural convection due to magnetic field in existence of melting heat transfer is simulated using CVFEM in this research. KKL model is taken into account to obtain properties of CuO–H2O nanofluid. Roles of melting parameter (δ), CuO–H2O volume fraction (ϕ), Hartmann number (Ha), and Rayleigh (Ra) number are depicted in outputs. Results depict that temperature gradient improves with rise of Rayleigh number and melting parameter. Nusselt number detracts with rise of Ha. At the end, a comparison as a limiting case of the considered problem with the existing studies is made and found in good agreement.


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## Body

## 1. Introduction

Melting process has various uses such as application like in thermocouple, heat exchangers, and heat engines. Chamkha and Ismael [1] examined the Cu-H2O nanofluid mixed convection in an enclosure with lid wall. Also, Ellahi et al. [2] reported the similar study for nanofluid over permeable wedge in mixed convection. Harikrishnan et al. [3] investigated melting behavior of composite PCM for heating application. Nanofluid melting in a pipe with external fins has been studied by Mat et al. [4]. Manikandan and Rajan [5] utilized hybrid nanofluid for heat transfer augmentation. Koca et al. [6] utilized Ag-H2O nanofluid for natural circulation loops. Shyam and Tiwari [7] investigated the coiled heat exchanger by means of nanofluid. Das [8] reported the radiative flow in existence of melting heat.Sheremet et al. [9] illustrated the free convective flow of ferrofluid in a rotating enclosure. Wavy duct in existence of Brownian forces has been examined by Shehzad et al. [10]. Sheikholeslami et al. [11] illustrated different uses of nanotechnology in their article paper. Abbas et al. [12] demonstrated the nanofluid flow through a horizontal Riga plate. Chamkha and Rashad [13] reported the nanoparticle migration on porous cone. Ellahi et al. [14] analyzed particle shape effects on Marangoni convection boundary layer flow of a nanofluid. They considered the Lorentz forces impact in governing equations. Malvandi et al. [15] reported nanofluid flow inside a channel in existence of Lorentz forces. Garoosi et al. [16] investigated performance of heat exchanger via nanofluid. They indicated that optimum volume fraction of nanoparticle exists to obtain maximum Nusselt number. Mesoscopic method has been utilized by Sheikholeslami and Ellahi [17] for a three-dimensional problem.This research intends to present the impact of melting heat transfer on free convection of ferrofluid in the presence of Lorentz forces. CVFEM is selected to find the outputs. Roles of melting parameter, CuO–water volume fraction, and Hartmann and Rayleigh numbers are presented.


## 2. Problem Statement

Figure1 depicts the geometry, boundary condition, and sample element. The inner wall is hot wall T=Th and the outer one is melting surface T=Tm. Other walls are adiabatic. Horizontal magnetic field has been applied. The enclosure is field with nanofluid.Figure 1
(a) Geometry; (b) sample element.
(a)
(b)


## 3. Governing Equation and Simulation

### 3.1. Governing Formulation

2D steady convective nanofluid flow is considered in existence of constant magnetic field. The fluid flow is laminar and incompressible. The flow is steady and Newtonian. The viscous dissipation is negligible in this study. The effects of Brownian force and thermophoretic force are not taken into condition. The flow also follows the Boussinesq assumption. The prevailing equations under these constraints can be written as [20](1)∂v∂y+∂u∂x=0,ρnfv∂u∂y+u∂u∂x=ByσnfvBx-By2σnfu+μnf∂2u∂y2+∂2u∂x2-∂P∂x.ρnf∂v∂yv+∂v∂xu=uByσnfBx+T-Tcβnfgρnf-vBxσnfBx+∂2v∂x2+∂2v∂y2μnf-∂P∂y,Bx=Bocos⁡λ,By=Bosin⁡λ,ρCpnfv∂T∂y+u∂T∂x=knf∂2T∂x2+∂2T∂y2,ρCpnf,ρnf, ρβnf, and σnf are calculated as [17, 18](2)ρCpnf=ϕρCps+1-ϕρCpf,ρnf=ρf1-ϕ+ϕρs,ρβnf=ϕρβs+1-ϕρβf,σnf=σf+2+σs/σf-ϕ-1+σs/σf3ϕ-1+σs/σf-1+1.knf, μnfare calculated via KKL model [19]:(3)knfkf=1+3kp/kf-1ϕkp/kf+2-kp/kf-1ϕ+5×104g′ϕ,T,dpϕρfcp,fκbTdpρp,Rf=4×10-8km2/W,Rf=-dp1/kp-1/kp,eff,g′ϕ,dp,T=ln⁡Ta1+a3ln⁡ϕ+a2ln⁡dp+a5ln⁡dp2+a4ln⁡dpln⁡ϕ+a6+a8ln⁡ϕ+a7ln⁡dp+a10ln⁡dp2+a9ln⁡ϕln⁡dp,μnf=μf1-ϕ2.5+kBrowniankf×μfPr.Properties and needed parameters are provided in Tables 1 and 2 [19].Table 1
Coefficient values of CuO–H2O [19].
Coefficient values
CuO–water
a
1
-
26.5933108
a
2
-
0.403818333
a
3
-
33.3516805
a
4
-
1.915825591
a
5
6.421858
E
-
02
a
6
48.40336955
a
7
-
9.787756683
a
8
190.245610009
a
9
10.9285386565
a
10
-
0.72009983664Table 2
Properties of H2O and CuO [19].
ρ (kg/m3)
C
p (j/kgk)
k (W/m⋅k)
β
×
1
0
5
(
K
-
1
)
d
p (nm)
σ
Ω
⋅
m
-
1
Water
997.1
4179
0.613
21
—
0.05
C
u
O
6500
540
18
29
45
10−10Vorticity and stream function should be used to eliminate pressure source terms:(4)ω+∂u∂y-∂v∂x=0,∂ψ∂x=-v,∂ψ∂y=u.Dimensionless quantities are introduced as follows:(5)P=pρnfαnf/L2,U=uLαnf,V=vLαnf,θ=T-TmTh-Tm,X,Y=x,yL,Ψ=ψαnf,Ω=ωL2αnf.The final formulae are(6)Ω+∂2Ψ∂Y2+∂2Ψ∂X2=0,∂Ω∂XU+V∂Ω∂Y=PrA5A1A2A4∂2Ω∂Y2+∂2Ω∂X2+PrHa2A6A1A2A4-∂V∂XBxBx+By∂U∂XBx-∂V∂YByBx+By∂U∂YBy+PrRaA3A22A1A42∂θ∂X,∂θ∂XU+∂θ∂YV=∂2θ∂X2+∂2θ∂Y2.Boundary conditions are(7)∂θ∂n=0.0onotherwalls,θ=0onouterwall,θ=1.0oninnerwall,Ψ=0.0on all walls except melting surfaceand in melting surface, we have(8)∂θ∂n=A2A11δL+csTm-T0∂Ψ∂n,where dimensionless and constants parameters are illustrated as(9)Pr=υfαf,Ra=gβfΔTL3υfαf,Ha=LB0σfμf,δ=ρCpfρfTh-Tm1+csTm-T0,A1=ρnfρf,A5=μnfμf,A2=ρCPnfρCPf,A6=σnfσf,A4=knfkf,A3=ρβnfρβf.It should be mentioned thatδis related to Stefan numbers.Local and average Nusselt over the cold wall can be calculated as follows:(10)Nuloc=knfkf∂θ∂r,Nuave=10.5π∫00.5πNulocdξ.


### 3.2. Numerical Procedure

CVFEM uses both benefits of two common CFD methods. This method uses triangular element (see Figure1(b)). Upwind approach is utilized for advection term. Gauss-Seidel approach is applied to solve the algebraic equations. Further notes can be found in [21].


## 3.1. Governing Formulation

2D steady convective nanofluid flow is considered in existence of constant magnetic field. The fluid flow is laminar and incompressible. The flow is steady and Newtonian. The viscous dissipation is negligible in this study. The effects of Brownian force and thermophoretic force are not taken into condition. The flow also follows the Boussinesq assumption. The prevailing equations under these constraints can be written as [20](1)∂v∂y+∂u∂x=0,ρnfv∂u∂y+u∂u∂x=ByσnfvBx-By2σnfu+μnf∂2u∂y2+∂2u∂x2-∂P∂x.ρnf∂v∂yv+∂v∂xu=uByσnfBx+T-Tcβnfgρnf-vBxσnfBx+∂2v∂x2+∂2v∂y2μnf-∂P∂y,Bx=Bocos⁡λ,By=Bosin⁡λ,ρCpnfv∂T∂y+u∂T∂x=knf∂2T∂x2+∂2T∂y2,ρCpnf,ρnf, ρβnf, and σnf are calculated as [17, 18](2)ρCpnf=ϕρCps+1-ϕρCpf,ρnf=ρf1-ϕ+ϕρs,ρβnf=ϕρβs+1-ϕρβf,σnf=σf+2+σs/σf-ϕ-1+σs/σf3ϕ-1+σs/σf-1+1.knf, μnfare calculated via KKL model [19]:(3)knfkf=1+3kp/kf-1ϕkp/kf+2-kp/kf-1ϕ+5×104g′ϕ,T,dpϕρfcp,fκbTdpρp,Rf=4×10-8km2/W,Rf=-dp1/kp-1/kp,eff,g′ϕ,dp,T=ln⁡Ta1+a3ln⁡ϕ+a2ln⁡dp+a5ln⁡dp2+a4ln⁡dpln⁡ϕ+a6+a8ln⁡ϕ+a7ln⁡dp+a10ln⁡dp2+a9ln⁡ϕln⁡dp,μnf=μf1-ϕ2.5+kBrowniankf×μfPr.Properties and needed parameters are provided in Tables 1 and 2 [19].Table 1
Coefficient values of CuO–H2O [19].
Coefficient values
CuO–water
a
1
-
26.5933108
a
2
-
0.403818333
a
3
-
33.3516805
a
4
-
1.915825591
a
5
6.421858
E
-
02
a
6
48.40336955
a
7
-
9.787756683
a
8
190.245610009
a
9
10.9285386565
a
10
-
0.72009983664Table 2
Properties of H2O and CuO [19].
ρ (kg/m3)
C
p (j/kgk)
k (W/m⋅k)
β
×
1
0
5
(
K
-
1
)
d
p (nm)
σ
Ω
⋅
m
-
1
Water
997.1
4179
0.613
21
—
0.05
C
u
O
6500
540
18
29
45
10−10Vorticity and stream function should be used to eliminate pressure source terms:(4)ω+∂u∂y-∂v∂x=0,∂ψ∂x=-v,∂ψ∂y=u.Dimensionless quantities are introduced as follows:(5)P=pρnfαnf/L2,U=uLαnf,V=vLαnf,θ=T-TmTh-Tm,X,Y=x,yL,Ψ=ψαnf,Ω=ωL2αnf.The final formulae are(6)Ω+∂2Ψ∂Y2+∂2Ψ∂X2=0,∂Ω∂XU+V∂Ω∂Y=PrA5A1A2A4∂2Ω∂Y2+∂2Ω∂X2+PrHa2A6A1A2A4-∂V∂XBxBx+By∂U∂XBx-∂V∂YByBx+By∂U∂YBy+PrRaA3A22A1A42∂θ∂X,∂θ∂XU+∂θ∂YV=∂2θ∂X2+∂2θ∂Y2.Boundary conditions are(7)∂θ∂n=0.0onotherwalls,θ=0onouterwall,θ=1.0oninnerwall,Ψ=0.0on all walls except melting surfaceand in melting surface, we have(8)∂θ∂n=A2A11δL+csTm-T0∂Ψ∂n,where dimensionless and constants parameters are illustrated as(9)Pr=υfαf,Ra=gβfΔTL3υfαf,Ha=LB0σfμf,δ=ρCpfρfTh-Tm1+csTm-T0,A1=ρnfρf,A5=μnfμf,A2=ρCPnfρCPf,A6=σnfσf,A4=knfkf,A3=ρβnfρβf.It should be mentioned thatδis related to Stefan numbers.Local and average Nusselt over the cold wall can be calculated as follows:(10)Nuloc=knfkf∂θ∂r,Nuave=10.5π∫00.5πNulocdξ.


## 3.2. Numerical Procedure

CVFEM uses both benefits of two common CFD methods. This method uses triangular element (see Figure1(b)). Upwind approach is utilized for advection term. Gauss-Seidel approach is applied to solve the algebraic equations. Further notes can be found in [21].


## 4. Grid Independency and Validation

Outputs should not rely on mesh size. Therefore, several grids should be tested. For example, as shown in Table3, a grid size of71×211 can be selected. The correctness of CVFEM code for nanofluid natural convective heat transfer is demonstrated in Figure 2 ([18]). This CVFEM code has good accuracy for magnetohydrodynamic flow as depicted in Table 4 [20].Table 3
Mesh independency analysis whenRa=10000, δ=0.2, Ha=40, and ϕ=0.04.
Mesh size
51
×
151
61
×
181
71
×
211
81
×
241
91
×
271
N
u
a
v
e
0.890241
0.895531
0.907145
0.908743
0.909166Table 4
N
u
a
v
e for various Gr and Ha at Pr = 0.733.
H
a
G
r
=
2
×
1
0
4
G
r
=
2
×
1
0
5
Present work
Rudraiah et al. [20]
Present work
Rudraiah et al. [20]
0
2.5665
2.5188
5.093205
4.9198
10
2.26626
2.2234
4.9047
4.8053
50
1.09954
1.0856
2.67911
2.8442
100
1.02218
1.011
1.46048
1.4317Figure 2
Validation of present code (Khanafer et al. [18]) when Gr=104, ϕ=0.1, and Pr=6.8 (Cu–water).


## 5. Results and Discussion

Nanofluid flow in a half sinusoidal annulus due to magnetic field in presence of melting surface is examined.μnf, knf of CuO–water nanofluid are estimated by means of KKL model. Graphs and tables are depicted for different amounts of CuO–H2O volume fraction (ϕ=0 to 0.04), melting parameter (δ=0 to 0.2), Rayleigh number (Ra=500 to 5000), and Hartmann number (Ha=0 to 40).Impact of adding CuO nanoparticles in water on velocity and temperature contours is depicted in Figure3. Temperature gradient decreases with augment of ϕ.Ψmax augments with adding nanoparticles because of increment in the solid movements. In presence of melting heat transfer and magnetic field, effect of adding nanoparticles on isotherms becomes negligible.Figure 3
Impact of adding CuO in water on streamlines and isotherms (nanofluid (ϕ=0.04) (—) and pure fluid (ϕ=0) (- - -)) when Ra=5000 and δ=0.2.Figures4 and 5 depict the impact of Rayleigh and Hartmann numbers in absence of melting heat transfer. There is only one eddy in streamline. In low Rayleigh number, conduction mechanism is dominant. As Ra increases, the distortion of isotherms is enhanced close to the hot wall. Adding magnetic field makes isotherms become parallel. In presence of melting heat transfer the primary eddy diminishes and increasing Lorentz forces generates three layers for streamline. Increasing melting parameter augments the bottom eddy.Figure 4
Isotherms and streamlines for various Ha, Ra whenδ=0 and ϕ=0.04.Figure 5
Isotherms and streamlines for various Ha, Ra whenδ=0.2 and ϕ=0.04.Figure6 illustrates the impact of δ, Ra, and Ha on Nuave. The formula for Nuave corresponding to active parameters is(11)Nuave=0.799+0.305δ+0.19Ra∗-0.16Ha∗+0.003Ra∗δ-0.0039δHa∗-0.028Ra∗Ha∗+0.058δ2-0.012Ra∗2+0.039Ha∗2,where Ha∗=0.1Ha and Ra∗=0.001Ra. As melting parameter augments, temperature gradient is enhanced and in turn Nusselt number is enhanced. Increasing buoyancy forces leads the thermal boundary layer thickness to reduce. So Nusselt number increases with enhancement ofRa. As Hartmann number augments, isotherms become parallel. Therefore Nusselt number has opposite relationship with Ha.Figure 6
Impacts ofδ, Ha, and Ra on Nuave.


## 6. Conclusions

Nanofluid free convection due to Lorentz force in existence of melting surface is reported. Combination of FEM and FVM is utilized to solve the PDEs. KKL model is considered for nanofluid. Roles of Hartmann number, CuO–water volume fraction, Rayleigh number, and melting parameter are presented. Outputs depict that temperature gradient improves with augment of melting parameter and Rayleigh number. Adding magnetic field makes the temperature gradient reduce due to domination of conduction mechanism in high Hartmann number.



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