Boundary-Layer Flow in a Porous Medium of a Nanofluid Past a Vertical Cone

1. Introduction

The natural convection flow over a surface embedded in saturated porous media is encountered in many engineering problems such as the design of pebble-bed nuclear reactors, ceramic processing, crude oil drilling, geothermal energy conversion, use of fibrous material in the thermal insulation of buildings, catalytic reactors and compact heat exchangers, heat transfer from storage of agricultural products which generate heat as a result of metabolism, petroleum reservoirs, storage of nuclear wastes, etc.

The derivation of the empirical equations which govern the flow and heat transfer in a porous medium has been discussed in [1-5]. The natural convection on vertical surfaces in porous media has been studied used Darcy’s law by a number of authors [6–20]. Boundary layer analysis of natural convection over a cone has been investigated by Yih [21-24]. Murthy and Singh [25] obtained the similarity solution for non-Darcy mixed convection about an isothermal vertical cone with fixed apex half angle, pointing downwards in a fluid saturated porous medium with uniform free stream velocity, but a semi-similar solution of an unsteady mixed convection flow over a rotating cone in a rotating viscous fluid has been obtained Roy and Anilkumar [26]. The laminar steady nonsimilar natural convection flow of gases over an isothermal vertical cone has been investigated by Takhar et al. [27]. The development of unsteady mixed convection flow of an incompressible laminar viscous fluid over a vertical cone has been investigated by Singh and Roy [28] when the fluid in the external stream is set into motion impulsively, and at the same time the surface temperature is suddenly changed from its ambient temperature. An analysis has been carried out by Kumari and Nath [29] to study the non-Darcy natural convention flow of Newtonian fluids on a vertical cone embedded in a saturated porous medium with power-law variation of the wall temperature/concentration or heat/mass flux and suction/injection. Cheng [30-34] focused on the problem of natural convection from a vertical cone in a porous medium with mixed thermal boundary conditions, Soret and Dufour effects and with variable viscosity.

The conventional heat transfer fluids including oil, water and ethylene glycol etc. are poor heat transfer fluids, since the thermal conductivity of these fluids play an important role on the heat transfer coefficient between the heat transfer medium and the heat transfer surface. An innovative technique for improving heat transfer by using ultra fine solid particles in the fluids has been used extensively during the last several years. Choi [35] introduced the term nanofluid refers to these kinds of fluids by suspending nanoparticles in the base fluid. Khanafer et al. [36] investigated the heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. The convective boundary-layer flow over vertical plate, stretching sheet and moving surface studied by numerous studies and in the review papers Buongiorno [37], Daungthongsuk and Wongwises [38], Oztop [39], Nield and Kuznetsov [40,41], Ahmad and Pop [42], Khan and Pop [43], Kuznetsov and Nield [44,45] and Bachok et al. [46].

From literature survey the base aim of this work is to study the free convection boundary-layer flow past a vertical cone embedded in a porous medium filled with a nanofluid, the basic fluid being a non-Newtonian fluid by using similarity transformations. The reduced coupled ordinary differential equations are solved numerically. The effects of the parameters governing the problem are studied and discussed.

2. Mathematical formulation of the problem

Consider the problem of natural convection about a downward-pointing vertical cone of half angle $$\gamma$$ embedded in a porous medium saturated with a non-Newtonian power-law nanofluid. The origin of the coordinate system is placed at the vertex of the full cone, with $$x$$ being the coordinate along the surface of the cone measured from the origin and $$y$$ being the coordinate perpendicular to the conical surface Fig (1). The temperature of the porous medium on the surface of the cone is kept at constant temperature$$T_w$$, and the ambient porous medium temperature is held at constant temperature$$T_{\infty }$$.

Figure 1.

A schematic diagram of the physical model.

The nanofluid properties are assumed to be constant except for density variations in the buoyancy force term. The thermo physical properties of the nanofluid are given in Table 1 (see Oztop and Abu-Nada [39]). Assuming that the thermal boundary layer is sufficiently thin compared with the local radius, the equations governing the problem of Darcy flow through a homogeneous porous medium saturated with power-law nanofluid near the vertical cone can be written in two-dimensional Cartesian coordinates $$(x,y)$$ as:

$$\frac{\partial (r^{m}u)}{\partial x}+\frac{\partial (r^{m}v)}{\partial y}=0,$$E1

$$\frac{\partial u^n}{\partial y}=\frac{{\left(\rho \beta \right)}_{nf}Kg\cos \gamma }{{\mu }_{nf}}\frac{\partial T}{\partial y}\text{,}$$E2

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}={\alpha }_{nf}\frac{{\partial }^{2}T}{\partial y^2}.$$E3

Where $$u$$ and $$v$$ are the volume-averaged velocity components in the $$x$$ and $$y$$ directions, respectively, $$T$$is the volume-averaged temperature. $$n$$is the power-law viscosity index of the power-law nanofluid and $$g$$ is the gravitational acceleration. $$m=\gamma =0$$corresponds to flow over a vertical flat plate and $$m=1$$ corresponds to flow over a vertical cone. $$n$$is the viscosity index. For the case of$$n=1$$, the base fluid is Newtonian. We note that $$n<1$$ and $$n>1$$ represent pseudo-plastic fluid and dilatant fluid, respectively. Property $${\rho }_{nf}$$ and $${\mu }_{nf}$$ are the density and effective viscosity of the nanofluid, and $$K$$ is the modified permeability of the porous medium. Furthermore, $${\alpha }_{nf}$$and $${\beta }_{nf}$$ are the equivalent thermal diffusivity and the thermal expansion coefficient of the saturated porous medium, which are defined as (see Khanafer et al. [36]):

$$\begin{array}{l}{\rho }_{nf}=\left(1-\varphi \right){\rho }_f+\varphi {\rho }_s,{\mu }_{nf}=\frac{{\mu }_f}{{\left(1-\varphi \right)}^{2.5}},{\alpha }_{nf}=\frac{k_{nf}}{{\left(\rho C_p\right)}_{nf}},\\ {\left(\rho C_p\right)}_{nf}=\left(1-\varphi \right){\left(\rho C_p\right)}_f+\varphi {\left(\rho C_p\right)}_s,\frac{k_{nf}}{k_f}=\frac{\left(k_s+2k_f\right)-2\varphi \left(k_f-k_s\right)}{\left(k_s+2k_f\right)+2\varphi \left(k_f-k_s\right)}.\end{array}$$E4

Here $$\varphi$$ is the solid volume fraction.

The associated boundary conditions of Eqs. (1)-(3) can be written as:

$$\begin{array}{l}v=0;T=T_w\text{at }y=0;\\ u=0;T\to T_{\infty }\text{as }y\to \infty ,\end{array}$$E5

where $${\mu }_f$$ is the viscosity of the basic fluid, $${\rho }_f$$and $${\rho }_s$$ are the densities of the pure fluid and nanoparticle, respectively, $${\left(\rho C_p\right)}_f$$and $${\left(\rho C_p\right)}_s$$ are the specific heat parameters of the base fluid and nanoparticle, respectively, $$k_f$$and $$k_s$$ are the thermal conductivities of the base fluid and nanoparticle, respectively. The local radius to a point in the boundary layer $$r$$ can be represented by the local radius of the vertical cone$$r=x\sin \gamma$$.

	$$\rho (kg/m^3)$$	$$C_p(J/kgK)$$	$$k(W/mK)$$	$$\beta \times {10}^5(K^{-1})$$
Pure water	997.1	4179	0.613	21
Copper (Cu)	8933	385	401	1.67
Silver (Ag)	10500	235	429	1.89
Alumina (Al2O3)	3970	765	40	0.85
Titanium Oxide (TiO2)	4250	686.2	8.9538	0.9

Table 1.

Thermo-physical properties of water and nanoparticles [39].

By introducing the following non-dimensional variables:

$$\begin{array}{l}\eta =\frac{y}{x}R{a}_x{}^{1l2},f(\eta )=\frac{\psi \left(x,y\right)}{{\alpha }_f{r}^{m}R{a}_x{}^{1l2}},\\ \theta (\eta )=\frac{T-T_{\infty }}{T_w-T_{\infty }}.\end{array}$$E6

The continuity equation is automatically satisfied by defining a stream function $$\psi (x,y)$$such that:

$$r^{m}u=\frac{\partial \psi }{\partial y}\text{and }{r}^{m}v=-\frac{\partial \psi }{\partial x}.$$E7

where;

$$R{a}_x=\left(\frac{x}{{\alpha }_f}\right){\left[\frac{Kg{\left(\rho \beta \right)}_f\cos \gamma (T_w-T_{\infty })}{{\mu }_f}\right]}^{1/n}.$$E8

Integration the momentum Eq. (2) we have:

$$\frac{{\mu }_{nf}}{{\mu }_f}{u}^n=\frac{{\left(\rho \beta \right)}_{nf}Kg\cos \gamma }{{\mu }_f}\left(T-T_{\infty }\right)\text{.}$$E9

Substituting variables (6) into Eqs. (1)–(5) with Eq. (9), we obtain the following system of ordinary differential equations:

$$\frac{1}{{(1-\varphi )}^{2.5}}{\left(f^{\prime }\right)}^n=\left[1-\varphi +\varphi \frac{{\left(\rho \beta \right)}_s}{{\left(\rho \beta \right)}_f}\right]\theta ,$$E10

$$\frac{k_{nf}/k_f}{\left[1-\varphi +\varphi \frac{{\left(\rho C_p\right)}_s}{{\left(\rho C_p\right)}_f}\right]}{\theta }^{″}+\left(m+\frac{1}{2}\right)f{\theta }^{\prime }=0,$$E11

along with the boundary conditions:

$$\begin{array}{l}f(0)=0,\theta (0)=1,\\ f^{\prime }(\infty )=0,\theta (\infty )=1.\end{array}$$E12

where primes denote differentiation with respect to$$\eta$$, the quantity of practical interest, in this chapter is the Nusselt number $$N{u}_x$$ which is defined in the form:

$$N{u}_x=\frac{hx}{k_m}=\frac{{-\frac{\partial T}{\partial y}|}_{y=0}x}{T_w-T_{\infty }}=-R{a}_x{}^{1/2}{\theta }^{\prime }(0).$$E13

where $$h$$ denotes the local heat transfer coefficient.

3. Results and discussion

In this study we have presented similarity reductions for the effect of a nanoparticle volume fraction on the free convection flow of nanofluids over a vertical cone via similarity transformations. The numerical solutions of the resulted similarity reductions are obtained for the original variables which are shown in Eqs. (10) and (11) along with the boundary conditions (12) by using the implicit finite-difference method. The physical quantity of interest here is the Nusselt number $$N{u}_x$$ and it is obtained and shown in Eqs. (13) and (14). The distributions of the velocity$$f^{\prime }(\eta )$$, the temperature $$\theta (\eta )$$ from Eqs.(10) and (11) and the Nusselt number in the case of Cu-water and Ag-water are shown in Figs. 2–8. The computations are carried for various values of the nanoparticles volume fraction for different types of nanoparticles, when the base fluid is water. Nanoparticles volume fraction $$\varphi$$ is varied from 0 to 0.3. The nanoparticles used in the study are from Copper (Cu), Silver (Ag), Alumina (Al₂O₃) and Titanium oxide (TiO₂).

In order to verify the accuracy of the present method, we have compared our results with those of Yih [22] for the rate of heat transfer $${\theta }^{\prime }(0)$$ in the absence of the nanoparticles$$\left(\varphi =0\right)$$. The comparisons in all the above cases are found to be in excellent agreement, as shown in Table 2. It is clear that as a geometry shape parameter $$m$$ increases, the local Nusselt number increases. While Table 3 depict the heat transfer rate $${\theta }^{\prime }(0)$$ for various values of nanoparticles volume fraction $$\varphi$$ for different types of nanoparticles when the base fluid is water. Figs. 2 and 3 show the effects of the nanoparticle volume fraction $$\phi$$ on the velocity distribution in the case of Cu-water when$$\varphi =0,0.05,0.1,0.15,0.2,0.3$$. It is noted that the velocity along the cone increases with the nanoparticle volume fraction in both of the two cases (i.e. Cu-water and Ag-water), moreover the velocity distribution in the case of Ag-water is larger than that for Cu-water. We can show that the change of the velocity distribution when we use different types of nanoparticles from Fig. 4, which depict the Ag-nanoparticles are the highest when the base fluid is water and when$$\varphi =0.1$$. Thus the presence of the nanoparticles volume fraction increases the momentum boundary layer thickness.

Figure 2.

Effects of the nanoparticle volume fraction $$\varphi$$ on velocity distribution $$f^{\prime }(\eta )$$ in the case of Cu-Water.

$$n$$	Vertical plate		Vertical cone
Yih [22]	Present method	Yih [22]	Present method
0.5	0.3766	0.3768	0.6522	0.6524
0.8	0.4237	0.4238	0.7339	0.7340
1.0	0.4437	0.4437	0.7686	0.7686
1.5	0.4753	0.4752	0.8233	0.8233
2.0	0.4938	0.4938	0.8552	0.8552

Table 2.

Comparison of results for the reduced Nusselt number $$-{\theta }^{\prime }(0)$$ for vertical plate $$(\lambda =0)$$ and vertical cone $$(\lambda =1)$$ when$$\varphi =0$$.

$$\varphi$$	$$Cu$$	$$Ag$$	$$A{l}_2{O}_3$$	$$Ti{O}_2$$
0.05	0.7423	0.7704	0.6604	0.6725
0.1	0.6931	0.7330	0.5642	0.5852
0.15	0.6301	0.6732	0.4780	0.5057
0.2	0.5591	0.6002	0.4006	0.4331
0.3	0.4052	0.4357	0.2673	0.3062

Table 3.

Values of $$-{\theta }^{\prime }(0)$$ for various values of $$\varphi$$ when$$n=1$$.

Figure 3.

Effects of the nanoparticle volume fraction $$\varphi$$ on velocity distribution $$f^{\prime }(\eta )$$ in the case of Ag-Water.

Figure 4.

Velocity profiles $$f^{\prime }(\eta )$$ for different types of nanofluids when$$\varphi =0.1$$.

Figs. 5 and 6 are presented to show the effect of the volume fraction of nanoparticles Cu and Ag respectively, on temperature distribution. These figures illustrate the streamline for different values of$$\varphi$$, when the volume fraction of the nanoparticles increases from 0 to 0.3, the thermal boundary layer is increased. This agrees with the physical behavior, when the volume of copper and silver nanoparticles increases the thermal conductivity increases, and then the thermal boundary layer thickness increases. Moreover Fig. 7 displays the behavior of the different types of nanoparticles on temperature distribution when$$\varphi =0.1$$. The figure showed that by using different types of nanofluid as the values of the temperature change and the Ag-nanoparticles are the lower distribution.

Figure 5.

Effects of the nanoparticle volume fraction $$\varphi$$ on temperature distribution $$\theta (\eta )$$ in the case of Cu-Water.

Figure 6.

Effects of the nanoparticle volume fraction $$\varphi$$ on temperature distribution $$\theta (\eta )$$ in the case of Ag-Water.

Figure 7.

Temperature profiles $$\theta (\eta )$$ for different types of nanofluids when$$\varphi =0.1$$.

Fig. 8 shows the variation of the reduced Nusselt number with the nanoparticles volume fraction $$\varphi$$ for the selected types of the nanoparticles. It is clear that the heat transfer rates decrease with the increase in the nanoparticles volume fraction$$\varphi$$. The change in the reduced Nusselt number is found to be lower for higher values of the parameter$$\varphi$$. It is observed that the reduced Nusselt number is higher in the case of Ag-nanoparticles and next Cu-nanoparticles, TiO₂-nanoparticles and Al₂O₃-nanoparticles. Also, the Fig. 8 and Table 3 show that the values of $${\theta }^{\prime }(0)$$ change with nanofluid changes, namely we can say that the shear stress and heat transfer rate change by taking different types of nanofluid. Furthermore this depicts that the nanofluids will be very important materials in the heating and cooling processes.

Figure 8.

Effects of the nanoparticle volume fraction $$\varphi$$ on dimensionless heat transfer rates.

4. Conclusions

The problem of the steady free convection boundary layer flow past a vertical cone embedded in porous medium filled with a non-Newtonian nanofluid has been studied and the special case when the base fluid is water has been considered. The effects of the solid volume fraction $$\varphi$$ on the flow and heat transfer characteristics are determined for four types of nanofluids: Copper (Cu), Silver (Ag), Alumina (Al₂O₃) and Titanium oxide (TiO₂). It has been shown, as expected, that increasing of the values of the nanoparticles volume fraction lead to an increase of the velocity and the temperature profiles and to an decrease of the Nusselt number for the values of the parameter$$\varphi$$. It has been found that the Ag-nanoparticles proved to have the highest cooling performance and Alumina-nanoparticles enhanced to have highest heating performance for this problem.

Nomenclature

$$C_p$$ specific heat at constant temperature

$$f$$ dimensionless stream function

$$g$$ acceleration due to gravity

$$h$$ local heat transfer coefficient

$$K$$ permeability coefficient of the porous medium

$$k$$ thermal conductivity

$$m$$ geometry shape parameter

$$N{u}_x$$ reduced Nusselt number

$$n$$ viscosity index,$$n\ge 0$$

$$R{a}_x$$ modified Rayleigh number

$$r$$ local radius of the cone

$$T$$ temperature

$$T_w$$ temperature at the surface of the cone

$$T_{\infty }$$ ambient temperature attained as $$y\to \infty$$

$$u,v$$ Darcian velocity components in $$x$$- and$$y$$ -directions

$$x,y$$ Cartesian coordinates

Greek symbols

$$\alpha$$ thermal diffusivity

$$\beta$$ volumetric expansion coefficient

$$\gamma$$ half angle of the cone

$$\eta$$ similarity variable

$$\theta$$ dimensionless temperature

$$\mu$$ effective viscosity

$${\rho }_f$$ density of the fluid

$${\rho }_{\text{s}}$$ nanoparticles mass density

$${\left(\rho C_p\right)}_{nf}$$ heat capacitance of the nanofluid

$${\left(\rho C_p\right)}_f$$ heat capacity of the fluid

$${\left(\rho C_p\right)}_s$$ effective heat capacity of the nanoparticles material

$$\varphi$$ nanoparticles volume fraction

$$\psi$$ stream function

Subscripts

$$f$$ fluid fraction

$$nf$$ nanofluid fraction

$$s$$ solid fraction

$$w$$ condition at the wall

$$\infty$$ stream function condition at the infinity