2018, Vol. 39
Shanghai University
Article Information
- A.A. SIDDIQUI, M. SHEIKHOLESLAMI
- TiO2-water nanofluid in a porous channel under the effects of an inclined magnetic field and variable thermal conductivity
- Applied Mathematics and Mechanics (English Edition), 2018, 39(8): 1201-1216.
- http://dx.doi.org/10.1007/s10483-018-2359-6
Article History
- Received Dec. 18, 2017
- Revised Feb. 25, 2018
2. Department of Mechanical Engineering, Babol Noshirvani Uiniversity of Technology, Babol, Iran
Usually, the thermal transfer rate of the fluid used in industrial equipments, e.g., radiator, transformer, and heat exchanger, is very low. In order to enhance the thermal transfer rate or the performance of the industrial equipment, tiny nanosized (< 100 nm) solid particles will be added in the fluid. This mixture is called nanofluid, the fluid is called the base fluid, and the tiny solid particles are called nanoparticles. Usually, the nanoparticles are copper, gold, silver, TiO2, etc.[1]. Among all the nanoparticles, the features and characteristics of TiO2 nanoparticles are distinguished. TiO2 nanoparticles have been widely used[2-10], e.g., to cleave protein[2], to detoxify the wastewater[3], to produce Li-ion batteries[4-5] and Na-ion batteries[6].
Tremendous efforts have been made by various researchers for TiO2-based nanofluids[11-38]. Said et al.[12] performed experiment to evaluate the efficiency of the evacuated tube solar collector with the water-TiO2 nanofluid. Dinarvand and Pop[13] used the TiO2 -water nanofluid to enhance the performance of a flat plate solar collector. Fedele et al.[14] studied the magnetohydrodynamics of the TiO2-water nanofluid, and measured the viscosity and thermal conductivity. Kao and Lin[15] used the TiO2-oil nanofluid to reduce the friction between two pieces of cast iron. Sajadi and Kazemi[16] investigated the turbulent convective heat transfer and pressure drop of the TiO2-water nanofluid in a circular tube. Radiom et al.[17] investigated the turbulent thermal transfer of the TiO2-water nanofluid in a circular pipe. Hamid et al.[18] examined the dynamic contact angle of the TiO2 -water nanofluid. Matthias and Buschmann[19] focused on the effect of temperature on the heat transfer behavior of the TiO2-ethylene glycol nanofluid, and determined the heat transfer coefficient. Naphon et al.[20] probed the enhancement of heat pipe thermal efficiency with the TiO2-alcohol nanofluid.
In this paper, we aim to study the steady, viscous, creeping, and laminar flow of the TiO2-water nanofluid in a rectangular channel bounded by two plates. The nanofluid flow is generated by the pressure gradient based on the solid volume fraction of the TiO2 nanoparticles. The nanofluid flows under the influence of an oblique magnetic field. The main objectives of this paper are to find an exact/analytical solution of the BVP by using the homotopy analysis method (HAM); to elucidate the effects of the magnetic field angle and the intensity on the physical field variables such as the nanofluid velocity, the temperature, and the concentration of the nanoparticles; to confirm and capture the effect of the TiO2 nanoparticles in raising the thermal transfer-rate; to highlight the flow/thermal-difference between pure water and the nanofluid; to elucidate the ramification of the volume fraction (%) of the TiO2 nanoparticles; to find the empirical-expression for the friction-factor and to study its variation with respect to the pure water and nanofluid; and to examine the dependence of the dimensionless field parameters such as the Hartmann number and the Nusselt number on the volume fraction (%) of the TiO2 nanoparticles.
The paper is organized as follows. In Section 2, we formulate the aforementioned flow problem mathematically to get a boundary value problem (BVP). In Section 3, we present the detail of the analytical solution of the BVP. In Section 4, the obtained results and its validity are presented. Finally, some conclusions are given in Section 5.
2 Mathematical formulationOthman et al.[39] recorded that TiO2 had the molecular structure as nanoparticles with the radius of about 7.5 mm. Owing to this TiO2-nanoparticles have been used as a network in the dye-sensitized solar cell (DSSC)[40-44].
We consider the steady TiO2-water nanofluid flow in a porous rectangular open channel |y|≤ H under the effect of the oblique magnetic field such that the origin is in coincidence with the centroid of the channel (see Fig. 1).
|
| Fig. 1 Schematic of the fluid flow |
|
|
Furthermore, we have the following assumptions:
(ⅰ) The plates of the channel are stationary;
(ⅱ) The lower plate is permeable, and the nanofluid can pass through the pores at the velocity Vs;
(ⅲ) The temperatures and concentration at the lower and upper plates of the channel are, respectively, Tl, Cl and Tu, Cu so that Tu >Tl and Cu >Cl;
(ⅳ) The electrically conducting flow is driven by the constant pressure gradient analogous to the plane Poiseuille flow;
(ⅴ) The flow is creeping, laminar, and unidirectional;
(ⅵ) The nanoparticles are in the thermally equilibrium state;
(ⅶ) There is no no-slip velocity between the nanoparticles and the molecules of the base fluid.
In the light of the aforementioned assumptions, the non-zero components of the momentum, energy, and concentration equations[45-46] for the mixture of the nanoparticles and the fluid in dimensional form are as follows:
|
(1) |
|
(2) |
|
(3) |
where C', p', T', and u' are the dimensional concentrations of the nanoparticles, pressure, temperature, and velocity of the nanofluid, respectively. B0, cp, D, α, μ, ρ, and σ are the magnetic field intensity, the specific heat coefficient, the mass diffusivity, the angle of inclination of the applied magnetic field, the viscosity coefficient, the density, and the electric conductivity respectively. knf* is the thermal conductivity of the nanofluid, and knf* =knf (1+ε T), where ε is a conductivity parameter[47]. The subscript nf signifies nanofluid.
The empirical relations for the physical quantities { k*, μ, ρ, ρ cp, σ } among the base fluid, nanoparticles, and nanofluid are as follows[48-51]:
|
(4) |
where the subscripts f and n represent for the base fluid and the nanoparticles, respectively, whereas ϕ is called the volume solid fraction[47].
The boundary conditions of the flow problem are
|
(5) |
Now, let us introduce the following dimensionless variables:
|
(6) |
where 
Substituting Eq. (6) in Eqs. (1)-(3), we have
|
(7) |
|
(8) |
|
(9) |
where Ec, Ha, Pr, Re, and Sc are the Eckert number, the Hartmann number, the Prandtl number, the Reynolds number, and the Schmidt number, respectively, which are defined by
|
Note that if ϕ =0 (for the pure water with non-zero conductivity), Eqs. (7)-(8) coincide with those in Ref. [47].
Eventually, the boundary conditions given in Eq. (5) can be rewritten as follows:
|
(10) |
In this section, let us solve the BVP expressed in Eqs. (7)-(10). Note that Eq. (7) is a linear ordinary differential equation. Its solution with respect to the boundary conditions in Eq. (10) is
|
(11) |
where
|
Note that if ϕ =0 (for pure water with non-zero conductivity), solution is in agreement with that in Ref. [47].
Similarly, Eq. (9) will satisfy the following solution:
|
(12) |
However, Eq. (8) is a highly non-linear differential equation, and its solution cannot be obtained analytically conveniently. Usually, such a type of equations is solved numerically. Here, we try to find the analytical solution by the means of the HAM[52].
According to this method, we consider the linear and nonlinear operators Λ (·) and Θ (·), respectively, as follows:
|
(13) |
|
(14) |
where
|
and p0 ∈ (0, 1) is an embedded parameter.
Next, we will consider the zeroth-order deformation equation as follows:
|
(15) |
where ħ is a non-zero arbitrary parameter. Accordingly, the boundary conditions on T will become
|
(16) |
The T0 (y) in Eq. (15) is an initial guessed solution, in the light of the aforementioned boundary conditions. In addition, this T0 (y) can be chosen for the problem in hand, without loss of generality, as follows:
|
(17) |
Obviously, the solution of Eq. (15) with the boundary condition (16) when p0 =0 is
|
(18) |
while the solution when p0 =1 is equal to the exact solution of the boundary value problem expressed in Eqs. (8) and (16), i.e.,
|
(19) |
Therefore, we can say that the variation of p0 from 0 to 1 will reflect the continuous deformation of T(y; p0) from the initial guessed solution T0 (y) to the exact solution T(y) of Eq. (8) with the boundary conditions in Eq. (16). This is, actually, the basic theme of the homotopy, and such types of variations are named as deformation in the topology. Now, if we employ the Taylor theorem to expand T(y; p0) in the power of p0 as follows:
|
(20) |
we will have
|
(21) |
Owing to Eqs. (19) and (20), we have
|
(22) |
Now, let us take n times derivative of Eq. (15) with respect to p0 in the light of the boundary condition (16), divide them by n!, and put p0 =0. Then, we get the nth-order deformation problem as follows:
|
where
|
and
|
(23) |
The boundary conditions in Eq. (16) take the form as follows:
|
(24) |
Next, if we solve Eq. (23) subject to the boundary condition (24) for n=1, 2, 3, ..., according to Eq. (22), we will obtain the first two approximate analytical solutions, in addition to Eq. (17), as follows:
|
(25) |
|
(26) |
where
|
In this section, we shall highlight the magnetic concentration of the nanoparticles and the convective effects on the field variables such as the velocity and internal temperature of the water-TiO2 nanofluid. The physical properties of water and TiO2 are given in Table 1.
Figure 2 illustrates the effects of the solid volume fraction ϕ of the TiO2 nanoparticles, which control the percentage-quantity of TiO2 in the water. It is plotted on the basis of the analytical solution given in Eq. (11). From this figure, we can see that the fluid velocity decreases if the solid volume fraction ϕ increases for all values of the magnetic field intensity and the angle of incident when Re=1. In addition, the fluid velocity also decreases if the magnetic field intensity increases for both the pure water and the nanofluid. Particularly, if ϕ =0, for the pure water, the velocity profiles agree with those in Ref. [47].
|
| Fig. 2 Variations of the velocity u(y) across the channel width y for different values of the solid volume fraction ϕ of the TiO2 particles when B0 =1× 10-3 T, 3× 10-3 T, 6× 10-3 T, and 9× 10-3 T |
|
|
Moreover, the maximum velocity umax occurs at not exactly the middle of the channel. There is a little bit deviation towards the lower plate, which is adsorbing the fluid at a constant velocity. The effects of the Reynolds number on umax is captured in Fig. 3. It is shown that, umax decreases if the Reynolds number increases. Moreover, umax decays when the angle of incidence of the imposed magnetic field α increases within the range from 0 to π/2 while intensifies when α increases within the range from π /2 to π for all values of the solid volume fraction of the TiO2 nanoparticles weather the fluid is pure water or nanofluid. umax decreases when the magnetic field intensity B0 increases for all values of the Reynolds number and all angles of the magnetic field.
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| Fig. 3 Variations of the maximum velocity umax with the angle of the magnetic field α for different values of the solid volume fraction ϕ of the TiO2 particles, where B0 =3× 10-3 T while Re=1.0 and 5.0 |
|
|
As we know, a boundary layer will be formed in the vicinity of the upper wall of the channel when the fluid flows. In addition, the effects of the stresses and vorticity are dominant here. We intend to assess the effect of the concentration of the nanoparticles, which is controlled by ϕ, on the wall-loading (stress on the wall). This effect is captured shortly in Table 2.
|
Table 2 is constructed in the light of the following data:
|
It shows that if the concentration of the nanoparticles in the fluid increases, the Reynolds number, wall vorticity, and wall stress will increase while the Hartmann number will decrease.
The Hartmann number Ha, as mentioned in Section 2, depends upon the magnetic field intensity B0, the angle of the magnetic field α, and the solid volume fraction ϕ. This fact is highlighted in Fig. 4.
|
| Fig. 4 Variations of the maximum velocity umax with the solid volume fraction ϕ for different values of B0 when α =π /4 and π/2 |
|
|
Figure 5 depicts that, unlike umax, Ha decreases when α varies from 0 to π /2 while intensifies when α varies from π /2 to π for all values of the Reynolds number and the solid volume fraction of the TiO2 nanoparticles whether the fluid is pure water or nanofluid.
|
| Fig. 5 Variations of Ha with α for different values of the solid volume fraction ϕ of the TiO2 particles, where B0 =3× 10-3 T, and Re=1.0 and 5.0 |
|
|
Next, the variation of the temperature T(y), based on the analytical solution given in Eq. (22), across the channel for different values of the magnetic field intensity B0 and the solid volume fraction ϕ is captured in Fig. 6. It is shown that the internal temperature T(y) increases from the lower plate to the upper plate of the channel both for the pure water and the nanofluid or when the solid volume fraction ϕ of the TiO2 nanoparticles increases for all values of the magnetic field intensity.
|
| Fig. 6 Variations of T(y) across the channel width y for different values of the concentration of the TiO2 particles, where Re=1, α=π /2, and B0 =1× 10-3 T and 9× 10-3 T |
|
|
From Fig. 7, we can see that the temperature decreases when the Reynolds number increases.
|
| Fig. 7 Variations of the temperature T with y for different values of the concentration of the TiO2 particles ϕ, where B0=9× 10-3 T, α=π /2, and Re=1.1 and 2.1 |
|
|
Next, let us discuss two technically important quantities, i.e., the friction factor f and the local Nusselt number Nu, which are defined as follows[55-56]:
|
(27) |
|
(28) |
The fRe in this case can be expressed explicitly as follows:
|
(29) |
The effects of the magnetic field intensity and the angle on the friction factor and the local Nusselt number for pure water and nanofluid are displayed in Figs. 8 and 9.
|
| Fig. 8 Variations of fRe with the solid volume fraction ϕ for different values of α when B0 =9× 10-3 T and different values of B0 when α =π/4 |
|
|
|
| Fig. 9 Variations of the local Nusselt number Nu on the upper plate with the solid volume fraction ϕ for different values of α when B0 =9× 10-3 T and different values of B0 when α =π /4, where the inset shows the magnified view |
|
|
In Fig. 8, it is observed that the quantity fRe (product of the friction factor and the Reynolds number) decreases with the increase in the solid volume fraction ϕ for all values of the magnetic field angle α and the magnetic field intensity B0. Moreover, fRe also decreases if either α or B0 increases for both pure water and the nanofluid. Unlike fRe, the local Nusselt number Nu, on the upper plate, increases if the solid volume fraction of the TiO2 nanoparticles increases for all values of the magnetic field intensity and the angle (see Fig. 9). In addition, a little bit increase in Nu is observed if we increase either the magnetic field intensity or the angle.
Finally, the variation of the concentration C(y) is analyzed based on the analytical solution given in Eq. (12) (see Fig. 10).
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| Fig. 10 Variations of C(y) with y, C(0) with ϕ, and C(0) with ReSc |
|
|
It shows that the concentration increases from the lower plate to the upper plate of the channel analogous to the temperature. In addition, the concentration depends merely upon the factor ReSc (the product of the Reynolds number and the Schmidt number). However, ReSc =HVs /D. It means that the concentration depends upon the channel-width H, the suction velocity Vs, and the mass diffusivity D. It does not depend upon the solid volume fraction ϕ, the magnetic field intensity B0, the magnetic field angle α, and even the temperature. Moreover, we can conclude that the concentration C is inversely proportional to the mass diffusivity D (see Fig. 10(b)), and the concentration C intensifies with the increase in ReSc (see Fig. 10(c)).
5 ConclusionsIn this paper, the TiO2-water nanofluid flow in a permeable channel with an inclined magnetic field is studied. The flow problem is formulated to get a BVP. The effects of the existence of the TiO2 nanoparticles/nanoparticles are highlighted. From the results and analysis, we can give the following findings:
(Ⅰ) The analytical solution of the BVP can be obtained by using the HAM for the fluid velocity u, the temperature T, and the concentration C.
(Ⅱ) The fluid velocity and temperature depend upon the magnetic field intensity and angle. In contrast, the concentration of the nanoparticles is independent of the magnetic field intensity and the angle.
(Ⅲ) The fluid velocity decays while the temperature intensifies if the volume fraction of the TiO2 nanoparticles increases, which confirms the fact that the occurrence of the TiO2 nanoparticles can enhance the thermal transfer rate.
(Ⅳ) The fluid velocity decreases whereas the temperature increases for both pure water and the nanofluid when the magnetic field intensity and angle increase.
(Ⅴ) The maximum velocity umax does not exist at the center of the symmetric channel, in contrast to the plane Poiseuille flow, while deviates a little bit towards the lower plate, which is absorbing the fluid at a very low velocity. If this suction velocity increases, the temperature in the vicinity of the lower plate will increase.
(Ⅵ) An explicit expression for fRe is developed.
(Ⅶ) The Hartmann number Ha decreases whereas the Nusselt number Nu increases for nanofluids as compared with pure water. In addition, the fall in Ha and the rise in Nu are observed when the volume fraction of the TiO2 nanoparticles increases. However, both Ha and Nu increase with the increase in the magnetic field intensity.
(Ⅷ) The concentration C is inversely proportional to the mass diffusivity D, and intensifies from the lower plate to the upper plate or when ReSc increases.
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