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Nonlinear Radiative Heat Transfer of Cu-Water Nanoparticles over an Unsteady Rotating Flow under the Influence of Particle Shape

By K. Ganesh Kumar, B.J. Gireesha and S. Manjunatha

Submitted: November 27th 2017Reviewed: February 2nd 2018Published: September 5th 2018

DOI: 10.5772/intechopen.74807

Downloaded: 206

Abstract

A 3D study on Cu-water-rotating nanofluid over a permeable surface in the presence of nonlinear radiation is presented. Particle shape and thermophysical properties are considered in this study. The governing equations in partial forms are reduced to a system of nonlinear ordinary differential equations using suitable similarity transformations. An effective Runge-Kutta-Fehlberg fourth-fifth order method along with shooting technique is applied to attain the solution. The effects of flow parameters on the flow field and heat transfer characteristics were obtained and are tabulated. Useful discussions were carried out with the help of plotted graphs and tables. It is found that the rate of heat transfer is more enhanced in column-shaped nanoparticles when compared to tetrahedron- and sphere-shaped nanoparticles. Higher values of rotating parameter enhance the velocity profile and corresponding boundary layer thickness. It has quite the opposite behavior in angular velocity profile. Further, unsteady parameter increases the velocity profile and corresponding boundary layer thickness.

Keywords

  • particle shape effect
  • nonlinear radiation
  • Cu-water nanoparticles
  • unsteady rotating flow

1. Introduction

The interaction of thermal radiation has increased greatly during the last decade due to its importance in many practical applications. We know that the radiation effect is important under many isothermal and nonisothermal situations. If the entire system involving the polymer extrusion process is placed in a thermally controlled environment, then radiation could become important. The knowledge of radiation heat transfer in the system can, perhaps, lead to a desired product with a sought characteristic. Magnetohydrodynamic 3D flow of viscoelastic nanofluid in the presence of nonlinear thermal radiation has been examined by Hayat et al. [1]. Shehzad et al. [2] proposed the nonlinear thermal radiation in 3D flow of Jeffrey nanofluid. Refs. [3, 4, 5, 6, 7, 8, 9, 10, 11] are some of the works associated with stretching sheet problem of thermal radiation.

Nanotechnology has been widely utilized in the industries since materials with the size of nanometers possess distinctive physical and chemical properties. Nanofluids are literally a homogeneous mixture of base fluid and the nanoparticles. Common base fluids embody water, organic liquids, oil and lubricants, biofluids, polymeric solution and other common liquids. Nanoparticles are created from totally different materials, like oxides, nitrides, carbide, ceramics metals, carbons in varied (e.g., diamond, graphite, carbon nanotubes, fullerene) and functionalized nanoparticles. Nanofluids have novel properties that are potentially helpful in several applications in heat transfer, as well as microelectronics, pharmaceutical processes, heat exchanger, hybrid-powered engines, domestic refrigerator, fuel cells, cooling/vehicle thermal management, nuclear reactor agent, in grinding, in space technology, ships and in boiler flue gas temperature reduction. Choi [12] was the first who composed the analysis on nanoparticles in 1995. Later, Maiga et al. [13] initiated the heat transfer enhancement by using nanofluids in forced convective flows. The laminar fluid flow which results from the stretching of a flat surface in a nanofluid has been investigated by Khan and Pop [14]. Recently, a number of researchers are concentrating on nanofluid with different geometries; see [15, 16, 17, 18, 19].

Experimental studies have shown that the thermal conductivity of nanofluids is determined by the parameters related to: nanoparticles, concentration, size, spherical and nonspherical shapes, agglomeration (fractal-like shapes), surface charge and thermal conductivity, base fluids (e.g., thermal conductivity and viscosity), nanofluids (e.g., temperature), the interfacial chemical/physical effect or interaction between the particles and base fluid and others. For more details, readers are referred to the studies [20, 21, 22, 23, 24, 25].

Impact of nonlinear thermal radiation on 3D flow and heat transfer of Cu-water nanoliquid over unsteady rotating flow have been considered. The heat transfer characteristics are studied in the presence of different particle shapes, thermophysical properties and nonlinear thermal radiation. The principal equations of continuity, momentum, energy and mass equations are transferred into a set of nonlinear similarity equalities by applying the appropriate transformations. The condensed equalities are solved numerically, and the impacts of relevant parameters are discussed through plotted graphs and tables.

2. Mathematical formulation

An unsteady laminar flow over a permeable surface in a rotating nanofluid is considered in this study. The copper-water motion is 3D due to Coriolis force in the present problem. The Cartesian coordinates are x,yand zwhere the rotation of the nanofluid is at an angular velocity Ω¯tabout the z-axis, and time is denoted as t. Let uwxt=bx1δtand vwxtrepresent the surface velocity in xand ydirections, respectively, and wwxtis the wall mass flux velocity in the z-direction as represented in Figure 1. Under these conditions, the governing equations can be written as:

ux+vy+wz=0,E1
ut+uux+vuy+wuz2Ω¯v=1ρpx+μnfρnf2uz2,E2
vt+uvx+vvy+wvz+2Ω¯v=1ρpy+μnfρnf2vz2,E3
wt+uwx+vwy+wwz=1ρpz+μnfρnf2vz2,E4
Tt+uTx+vTy+wTz=αnf2Tz2+1ρcpnfqrz.E5

Figure 1.

Influence of Ω on f'η.

Boundary conditions for the problem are,

u=uwxt=v=0,w=0,T=Twatz=0,u0,v0,w=0,TTaszE6

where velocity components in x,yand zdirections are u,vand w, constant angular velocity of the Nano fluid is Ω, dynamic viscosity of the Nano fluid is μnf, density of the nanofluid is ρnf, thermal diffusivity of the nanofluid is αnf, Tis temperature of nanofluid and wall temperature is Tw,Tdenotes temperature outside the surface (Table 1).

ρcpk
Copper (Cu)3858933400
Water997.141790.613

Table 1.

Thermophysical properties of water and nanoparticles.

The radiative heat flux expression in Eq. (5) is given by:

qr=16σ3kT3TzE7

where σand kare the Stefan-Boltzmann constant and the mean absorption coefficient, respectively, and in view to Eq. (7), Eq. (4) reduces to:

Tt+uTx+vTy+wTz=yαnf+16σT33ρcpnfkTZTy.E8

Parameters μnf, ρnfand αnfare interrelated with nanoparticle volume fraction; ϕand can be defined as:

ρnf=ρf1ϕ+ϕρsρf,μnf=μf1ϕ2.5,αnf=knfρcpnfρcpnf=ρcpf1ϕ+ϕρcpsρcpf,knfkf=ks+m1kfm1ϕkfksks+m1kf+ϕkfksE9

where volumetric heat capacity of the solid nanoparticles is ρcps, volumetric heat capacity of the base fluid is ρcpf, volumetric heat capacity of the nanofluid is ρcpnf, thermal conductivity of the nanofluid is knf, thermal conductivity of the base fluid is kf, thermal conductivity of the solid nanoparticles is ks,nanoparticle volume fraction is ϕ, density viscosity of the base fluid is ρfand dynamic viscosity of the base fluid is μf(Table 2).

Particle shapesSphereTetrahedronColumn
m34.06136.3698

Table 2.

Values of the empirical shape factor for different particle shapes.

Now, we introduce similarity transformations:

u=bx1δtf'η,v=bx1δtgη,w=bx1δtfη,η=bxυ1δtz,θη=TTTwTE10

with T=T1+θw1θand θw=TwT, θw>1is the temperature ratio parameter.

Using Eqs. (2)(6) and (10), we can have

11ϕ2.51ϕ+ϕρsρff'''f'2ff'2Ωg+λη2f''+f'=0,E11
11ϕ2.51ϕ+ϕρsρfg''f'gfg'2Ωf+λη2g'+g=0,E12
knfkf1ϕ+ϕρcpsρcpf1+R1+θw1θ3θPrλη2θ''=0,E13

The transformed boundary conditions are as follows:

f0=0,f'0=1,g0=0,θ0=1atη=0f'η0,gη0,θη0asηE14

where Ω=ωbis rotation rate, λ=δbis unsteadiness parameter R=16σT33kfkis radiation parameter, Pr=αnfυnfis Prandtl number and primes denote the differentiation with respect to η.

Rosca et al. [11] mentioned that the pressure term pcan be integrated from Eq. (4); thus, we obtain:

p=υρwzρw22+cE15

The physical quantities of interest in this problem are the skin friction coefficients in x and ydirections, Cfxand Cfyas well as the local Nusselt number Nuxwhich are defined as:

Cfx=τwxρuw2xt,Cfy=τwyρuw2yt,Nux=xqwkfTwT,E16

Surface shear stress τwx,τwyand surface heat flux qware defined as:

τwx=μnfuzz=0,τwy=μnfvzz=0andqw=knfTz+qrz=0E17

Using Eqs. (16) and (17), we obtain

RexCfx=11ϕ2.5f''0,RexCfy=11ϕ2.5g'0andNuxRex=knfkf1+Rθw3θ'0,E18

where Rex=uwxυis local Reynolds number.

3. Numerical method

Numerical solutions of nonlinear coupled differential Eqs. (11)(13) subject to the boundary conditions (14) constitute a two-point boundary value problem. Due to coupled and highly nonlinear nature, which are not amenable to closed-form solutions; therefore, we resorted to numerical solutions. In order to solve these equations numerically, we follow most efficient fourth-fifth order Runge-Kutta-Fehlberg integration scheme along with shooting technique. In this method, it is most important to choose the appropriate finite values of η. The asymptotic boundary conditions at ηwere replaced by those at η8in accordance with standard practice in the boundary layer analysis.

4. Result and discussion

To get a clear insight into the physical situation of the present problem, numerical values for velocity and temperature profile are computed for different values of dimensionless parameters using the method described in the previous section. The numerical results for the local Nusselt number are presented for different values of the governing parameters in Table 3.

λΩPrRθwϕ.Nusselt number
m=3m=4.0613m=6.3698
0.20.367220.351440.32983
0.30.296710.292560.28353
0.40.218970.226350.23028
0.010.330690.321120.30611
0.020.290050.287390.27973
0.030.249360.253640.25333
5.7760.358650.342470.32053
6.5870.367220.351510.32990
7.5780.375980.360900.33978
0.50.367220.351440.32983
10.332660.316420.29491
1.50.309570.293320.27217
1.20.367220.351440.32983
1.40.325650.310780.29076
1.60.288070.274160.25572
1%0.436650.424920.40752
2%0.367220.351440.32983
3%0.308410.292120.27084

Table 3.

Numerical values of Nusselt number for different physical parameters.

Figure 1 portrays the effect of Ωon velocity profile f'η. The velocity profile and corresponding thickness of the boundary layer enhance with larger values of Ω. This is because the larger value of Ωparameter leads to higher rotation rate as compared to stretching rate. Therefore, the larger rotation effect enhances velocity field. Figure 2 shows the impact of Ωon angular profile gη. From this figure, one can see that gηreduces for larger values of Ω. Further, it is noticed that rate of heat transfer is larger in column-shaped nanoparticles when compared to tetrahedron- and sphere-shaped nanoparticles.

Figure 2.

Influence of Ω on gη.

Figures 3 and 4 depict the effect of λon the fηand gηprofile. It is clear from both the figures that an increase in λdecreases the momentum boundary layer thickness resulting in velocity decrease. It is also noted that gηdecreases smoothly with the increase in the unsteadiness parameter. This shows an important fact that the rate of cooling is much faster for higher values of λ, whereas it may take a longer time in steady flows.

Figure 3.

Influence of λ on f'η.

Figure 4.

Influence of λ on gη.

Influence of the solid volume fraction parameter ϕon temperature profiles θηcan be visualized in Figure 5. It is observed that the temperature profile increases by increasing values of the solid volume fraction parameter. This is due to the fact that the volume occupied by the dust particles per unit volume of mixture is higher so that it raises the rate of heat transfer. It was noticed that the development in the temperature profiles of column-shaped nanoparticles is high when compared to temperature profiles of sphere- and tetrahedron-shaped nanoparticles due to the increase in volume fraction of nanoparticles.

Figure 5.

Influence of Pr on θη.

The effect of temperature ratio parameter θwon temperature profile is shown in Figure 6. The influence of temperature ratio parameter enriches the temperature profile and corresponding boundary layer thickness. This may happen due to the fact that the fluid temperature is much higher than the ambient temperature for increasing values of θw,which increases the thermal state of the fluid. It is also observed that the rate of heat transfer is higher in the column-shaped nanoparticles than that of tetrahedron- and sphere-shaped nanoparticles.

Figure 6.

Influence of ϕ on θη.

Figure 7 demonstrates the effect of the Prandtl number Pron temperature profiles of θη. The above mentioned graph elucidate that the temperature profile and corresponding thermal boundary layer thickness decrease rapidly with increasing values of Pr. Physically, the Prandtl number is the ratio of momentum diffusivity to thermal diffusivity. In fact, the larger Prandtl number means that the lower thermal diffusivity. A decrease in the thermal diffusivity leads to a decrease in the temperature and its associated boundary layer thickness.

Figure 7.

Influence of θw on θη.

The temperature distribution θηfor various values of radiation parameter Ris shown in Figure 8. This figure reveals that the larger values of radiation parameter increase the temperature profile and thermal boundary layer thickness. Generally, higher values of radiation parameter produce additional heat to the operating fluid that shows associate enhancement within the temperature field. We have noticed an improvement within the temperature profile because of increase in the radiation parameter. Moreover, the rate of heat transfer at the wall is less in case of the sphere-shaped particles when compared to the tetrahedron- and column-shaped nanoparticles.

Figure 8.

Influence of R on θη.

Figure 9 shows the effect of θwand Ron the skin friction coefficient. Here, we observed that the skin friction coefficient decreases for larger values of θwand R. Figure 10 delineates the influence of ϕand Econ Nusselt number. One can observe from the figure that Nusselt number decreases for larger values of ϕand Ec. It is also perceived from these figures that the maximum decrease in the rate of heat transfer of nanofluid is motivated by the column-shaped, followed by tetrahedron- and sphere-shaped nanoparticles, respectively. It is just because of the nanofluid which contains column-shaped nanoparticles having maximum thermal conductivity than nanofluids containing tetrahedron- and sphere-shaped nanoparticles. Table 3 presents the numerical values of Nusselt number for various values physical parameter values. It is observed that Nusselt number increases with increasing Pr. Further, from Table 3, we observe that Nusselt number decreases with increasing values of θw,R,ϕand λ.

Figure 9.

Influence of θw and R on Nusselt number.

Figure 10.

Influence of ϕ and Ω on Nusselt number.

5. Conclusions

In the present analysis, impact nonlinear radiative heat transfer of Cu-water nanoparticles over an unsteady rotating flow under the influence of particle shape is considered. Effects of various parameters are studied graphically. The main points of the present simulations are listed as follows:

  • The highlight of this study is that temperature profile is more enhanced in column-shaped nanoparticles when compared to tetrahedron- and sphere-shaped nanoparticles.

  • Temperature profile and thermal boundary layer thickness increase with increasing values of Rand θw.

  • The thermal boundary layer thickness and temperature profile enhance with increasing values of ϕ.

  • Higher values of rotating parameter enhance the velocity profile and corresponding boundary layer thickness. It has quite opposite behavior in angular velocity profile.

  • Unsteady parameter increases the velocity profile and corresponding boundary layer thickness.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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K. Ganesh Kumar, B.J. Gireesha and S. Manjunatha (September 5th 2018). Nonlinear Radiative Heat Transfer of Cu-Water Nanoparticles over an Unsteady Rotating Flow under the Influence of Particle Shape, Impact of Thermal Conductivity on Energy Technologies, Aamir Shahzad, IntechOpen, DOI: 10.5772/intechopen.74807. Available from:

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