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In this paper, the effect of geometrical parameters of microtube on enhancing the heat transfer has been studied numerically. Single phase model was used to simulate the flow of sio2ethlyen glycol nanofluids inside a microtube at different reynolds numbers ranged from 10 to 160 at constant heat flux boundary condition. The results show that the higher tube diameter and entrance size has the highest Nusselt number and lower pressure drop. Furthermore, no effect of inclination angles was found.
Numerical modelling, Nanofluids, Microtube, Heat transfer enhancement.
In recent years, liquid suspensions containing nanoparticles gained significant attention in science and engineering. Their unique thermophysical properties, such as a drastically reduced melting temperature of the nanoparticles compared to their bulk counterpart [1, 2], allow for novel methods in the manufacturing of nano and microelectronic devices. Many researchers intended to investigate experimentally and numerically the effects of the microtube and microchannel shapes, sizes, heat transfer and the other parameters that can be influence on the performance for the microelectronic devices. However, the recent numerical and experimental researches that related to microtube have been summarized in Table1.
The purpose of this work is to provide a numerical study on the effect of geometrical parameters (entrance size, shape and inclination angle on enhancing the heat transfer at different Reynolds number ranged from 10 to 120 for different inclination angles ranged from 0° to ±90°.
2.1 Geometry and the governing equations
Navier–Stokes and energy equations were used to describe the flow and heat transfer in the microtube. The following assumptions are adopted: (i) The nanofluid is incompressible, the flow is laminar, the radiation heat transfer effects are negligible; (ii) The nanoparticles are assumed to be spherical and single phase model is used; (iii) Constant thermophysical properties are considered for the nanofluid. The governing equations used are as follows:
Continuity equation:
$\frac{1}{R} \frac{\partial}{\partial \theta}\left(\rho_{e f f} U\right)+\frac{1}{R} \frac{\partial}{\partial R}\left(\rho_{e f f} R V\right)+\frac{\partial}{\partial Z}\left(\rho_{e f f} W\right)=0$ (1)
Momentum equation:
$\theta$ Components
$1 \frac{\partial}{\partial \theta}\left(\rho_{e f f} U U\right)+\frac{1}{R} \frac{\partial}{\partial R}\left(\rho_{e f f} R V U\right)+\frac{\partial}{\partial Z}\left(\rho_{e f f} W U\right)+\frac{1}{R}\left(\rho_{e f f} U V\right) =\frac{1}{R^{2}} \frac{\partial P}{\partial \theta}+\frac{1}{R^{2}} \frac{\partial}{\partial \theta}\left(\mu_{e f f} \frac{\partial U}{\partial \theta}\right)+\frac{\partial}{\partial R}\left(\frac{\mu_{e f f} }{R} \frac{\partial(R U)}{\partial R}\right)+\frac{2\mu_{e f f}}{R^{2}} \frac{\partial V}{\partial \theta}$ (2a)
R Components
$\frac{1}{R} \frac{\partial}{\partial \theta}\left(\rho_{e f f} U V\right)+\frac{1}{R} \frac{\partial}{\partial R}\left(\rho_{e f f} R V V\right)+\frac{\partial}{\partial Z}\left(\rho_{e f f} W V\right)\frac{1}{R}\left(\rho_{e f f} U^{2}\right) =\frac{1}{R} \frac{\partial P}{\partial \theta}+\frac{1}{R^{2}} \frac{\partial}{\partial \theta}\left(\mu_{e f f} \frac{\partial V}{\partial \theta}\right)+\frac{\partial}{\partial R}\left(\frac{\mu_{e f f}}{R} \frac{\partial(R V)}{\partial R}\right)\frac{2 \mu_{e f f}}{R^{2}} \frac{\partial U}{\partial \theta}$ (2b)
Z Components
$\frac{1}{R} \frac{\partial}{\partial \theta}\left(\rho_{e f f} U W\right)+\frac{1}{R} \frac{\partial}{\partial R}\left(\rho_{e f f} R V W\right)+\frac{\partial}{\partial Z}\left(\rho_{e f f} W W\right)= =\frac{\partial P}{\partial Z}+\frac{1}{R^{2}} \frac{\partial}{\partial \theta}\left(\mu_{e f f} \frac{\partial W}{\partial \theta}\right)+\frac{1}{R} \frac{\partial}{\partial R}\left(R \mu_{e f f} \frac{\partial W}{\partial \theta}\right)$ (2c)
Energy equation:
$\frac{1}{R} \frac{\partial}{\partial \theta}\left(\rho_{e f f} U \Theta\right)+\frac{1}{R} \frac{\partial}{\partial R}\left(\rho_{e f f} R V \Theta\right)+\frac{\partial}{\partial Z}\left(\rho_{e f f} W \Theta\right) =\frac{1}{R^{2}} \frac{\partial}{\partial \theta}\left(\frac{K e f f}{(C p)_{e f f}} \frac{\partial \Theta}{\partial \theta}\right)+\frac{\partial}{R \partial R}\left(R \frac{K e f f}{(C p)_{e f f}} \frac{\partial \Theta}{\partial R}\right)$ (3)
Where the nondimensional variables are defined as
$R=\frac{r}{D}, \quad Z=\frac{z}{D}, \quad U=\frac{u}{u_{\infty}}, \quad V=\frac{v}{u_{\infty}}, \quad W=\frac{w}{u_{\infty}}, \quad \Theta=\frac{TT}{T_{w}T_{\infty}}, \quad P=\frac{p}{\rho u_{\infty}^{2}}$
The $\rho_{e f f}, \mu_{e f f}, C_{e f f}$ and $k_{e f f}$ are density, viscosity, heat capacity and thermal conductivity of nanfluid, respectively.
2.2 Boundary conditions
At the tube inlet, the inlet temperature was taken as T_{in}=301 K. Noslip conditions and uniform heat flux used was 50000 W/m^{2} to heat up the tube. Different velocities depending on the values of Reynolds number were used. At the tube outlet, the flow and heat transfer are assumed to be fully developed.
2.3 Nanofluids thermophysical properties
The thermophysical properties which are density, heat capacity, dynamic viscosity and thermal conductivity for SiO_{2}–EG nanofluid are given in Table 2. These properties are calculated using the following equations:
Effective thermal conductivity [32]:
$k_{e f f}=k_{\text {static}}+k_{\text {Brownian}}$ (4)
$k_{\text {static}}=k_{f}\left[\frac{\left(k_{n p}+2 k_{f}\right)2 \phi\left(k_{f}k_{n p}\right)}{\left(k_{n p}+2 k_{f}\right)+\phi\left(k_{f}k_{n p}\right)}\right]$ (5)
Where $k_{n p}$ and $k_{f}$ are the thermal conductivity of the solid particles and the base fluid respectively.
Thermal conductivity due to the Brownian motion presented by [33] as:
$k_{\text {Brownian}}=5 \times 10^{4} \beta \phi p_{f} c_{p f} \sqrt{\frac{K T}{2 R_{n p}}} f(T, \phi)$
$f(T, \phi)=(0.028217 \varnothing+0.003917) \frac{T}{T_{0}}+(0.030669 \varnothing0.00391123)$ (6)
Where K is the Boltzman constant, T is the fluid temperature; T_{0} is the reference temperature, the $\beta$ values for SiO_{2} particle expressed in as follow [33]:
$\begin{array}{llll}1.9526(100 \phi)1.4594 & \text { for } & 1 \% \leq \varphi \leq 10 \% & \text { at } 298 K \leq T \leq 363 K\end{array}$ (7)
The effective dynamic viscosity is given as [34]:
$\frac{\mu_{e f f}}{\mu_{f}}=\frac{1}{134.87\left(d_{p} / d_{f}\right)^{0.3} \varnothing^{1.03}}$ (8)
Where $d f=\left[\frac{6 \times M}{N \times \pi \times \rho_{f 0}}\right]^{1 / 3}$ (9)
While $\mu_{e f f}$ and $\mu_{f}$ are the viscosity of nanofluid and base fluid respectively, $d_{p}$ is the nanoparticle diameter, $d_{f}$ is the base fluid equivalent diameter and is the nanoparticle volume fraction. M is the molecular weight of the base fluid and N is the Avogadro number. $\rho_{f_{0}}$ is the mass density of the base fluid calculated at temperature T_{0}=293 K.
The effective density is given as [34]:
$\rho_{e f f}=(1\varnothing) \rho_{f}+\varnothing \rho_{s}$ (10)
Where $\rho_{e f f}$ and $\rho_{f}$ are the nanofluid and base fluid densities respectively, and $\rho_{s}$ is the density of nanoparticle.
The effective specific heat is given as [34]:
$(\mathrm{Cp})_{\mathrm{eff}}=\frac{(1\varnothing)\left(\rho C_{p}\right)_{f}+\varnothing\left(\rho C_{p}\right)_{x}}{(1\varnothing) \rho_{f}+\varnothing \rho_{s}}$ (11)
Where C p_{s} is the heat capacity of the solid particles and c p_{f} is the heat capacity of the base fluid.
Table 2. Thermophysical properties of nanofluids.

Base fluids 
Nanoparticle 
Properties 
EG 
SiO_{2} 
ρ (kg/m^{3}) 
1114.4 
2200 
μ (Nm/s) 
0.0157 
 
k (W/mK) 
0.252 
1.2 
c_{p }(kJ/kgK) 
2415 
703 
2.4 Numerical implementation
The Finite volume approach is used to solve the continuity, momentum and energy equations along with the corresponding boundary conditions. SIMPLE algorithm is used to solve the flow field inside the MT. The diffusion term in the momentum and energy equations is approximated by secondorder central difference which gives a stable solution. The secondorder upwind differencing scheme is considered for the convective terms. Five different sets of the grid sizes were imposed to the geometry and simulated by calculating the Nusselt number along the MT. The five grids sizes ($8 \times 6 \times 200$ ; $10 \times 8 \times 400$ ; $12 \times 10 \times 600$ ;$14 \times 12 \times 800$ and $16 \times 14 \times 1000$ ) show no much difference in the values of Nusselt number. The grid size of $14 \times 12 \times 800$ is selected in this study as it is found to provide a more stable solution. About the code validation part is documented in the Ref. [31].
The simulations are performed for Reynolds number in the range of 10 ≤ Re ≤ 160 and SiO_{2} nanoparticle with pure EG as a base fluid. The nanoparticles volume fraction used was 0.04 with nanoparticles diameter 25nm. The MT used has a 100 mm length with different diameters ranged from 0.5 to 0.9 mm and different entrance sizes ranged from 0.371 to 2.5 mm. The heat flux that used to heat up the microtube was 50000 W/m^{2}. The inclination angles from the horizontal position were 0°, ± 45°, ± 90° are used in this investigation.
Table 1. Summary of numerical and experimental studies for MT
3.1 Effect of tube diameter
The effect of tube geometry on, the axial and wall temperatures, axial velocity along the tube radius, Nusselt number and the pressure drop are presented in this section.
In Figure 1a and 1b temperature profile at z/L=1 along the tube radius and tube axis shows that the tube with 0.9 mm diameter has the highest temperature along the tube radius and tube wall followed by 0.7 mm, 0.5 mm respectively. This is because the velocity proportional inversely with the tube diameter. Smaller diameter increases the velocity which leads to decrease the temperature and vice versa.
(a)
(b)
Figure 1. Profiles of temperature at z/L=1for different tube diameters: (a) Temperature along the tube radius; (b) Temperature along tube axis
In Figure 2 the axial velocity along tube radius profile shows that the tube with 0.5 mm diameter has the highest
axial velocity along the tube radius at z/L=0.2 for Reynolds number Re=80 followed by 0.7 mm, then 0.9 mm. This is because the smaller tube diameter is proportional directly to the velocity.
Figure 3 shows the average Nusselt number for different Reynolds numbers. It can be obtained from this Figure that 0.9 mm has the highest Nusselt number comparing to other tubes. The larger tube diameter has highest Nusselt number. This is because for Nusselt number is proportional directly with the tube diameter; higher tube diameter leads to higher Nusselt number and vice versa. Figure 4 the pressure drop along the tube axis show that the tube with 0.5 mm diameter has the highest pressure drop followed by 0.7 mm and 0.9 mm. This is because the static pressure for the fluid is proportional directly with the velocity. Thus, 0.5 mm tube has the highest velocity which leads to increases the pressure drop.
Figure 2. Profiles of axial velocity for different tube diameters at z =0.1 and Re= Re= 80
Figure 3. Average Nusselt number versus Reynolds number for different tube diameters
Figure 4. Pressure drop along the tube axis for different tube diameters at Re =80
3.2 Effect of entrance size
Three different entrance sizes were investigated in this section. The configuration for different entrance is illustrated in Figure 5. According to the entrance sizes three different diameters which is 0.371 mm, 0.9 mm and 2.5 mm were investigated. The results as shown in Figure 6 show that though the entrance region is just 40% from the tube length but it affects the heat transfer rate through the tube. It is evident that 2.5 mm entrance diameter has the highest Nusselt number followed by 0.9 mm, 0.371 mm respectively. This is because the Nusselt number is proportional directly with the entrance size.
(a)
(b)
(c)
Figure 5. The configuration for different entrance sizes: (a) D= 0.9 mm; (b) D=2.5 mm; (c) D=0.371 mm
Figure 6. Average Nusselt number versus Reynolds number for different entrance sizes
3.3 Effect of inclination angles
Different inclination angles were investigated in this section. The configuration for different inclination angles is shown in Figure 7. The results for different inclination angles show no changes in the Nusselt number values and the pressure drop at different Reynolds numbers as shown in Table 3. This is because no effect of the gravity, the Richardson number which is a function of Grashof number divided on the Reynolds number is less than Ri ˂ 0.1, so the natural convection is neglected and the forced convection is dominated.
Figure 7. Configuration of microtube at different inclination angles
Table 3. The effects of inclination angle on Nusselt number average and pressure drop at different Reynolds number
(a) At Reynolds number 10
Angle 
Nu_{ave} 
Pressure 
0° 
7.362852 
79515.71 
± 45° 
7.362852 
79515.71 
± 90° 
7.362852 
79515.71 
(b) At Reynolds number 40
Angle 
Nu_{ave} 
Pressure 
0° 
10.11316 
317295.8 
± 45° 
10.11316 
317295.8 
± 90° 
10.11316 
317295.8 
(c) At Reynolds number 80
Angle 
Nu_{ave} 
Pressure 
0° 
11.75585 
638552.4 
± 45° 
11.75585 
638552.4 
± 90° 
11.75585 
638552.4 
(d) At Reynolds number 120
Angle 
Nu_{ave} 
Pressure 
0° 
12.86489 
960093.9 
± 45° 
12.86489 
960093.9 
± 90° 
12.86489 
960093.9 
The effect of geometrical parameters of microtube on the heat transfer enhancement was investigated numerically. It is concluded from the results that the heat transfer rate strongly depends on the geometry and the entrance size of the microtube. Higher tube diameter and entrance size have the highest Nusselt number and vice versa. No effects of inclination angle on heat transfer rate were found. The buoyancy force is neglected because high values of Reynolds number was used in this investigation. An investigation for the effects of natural convection is needed for the future studies.
Al_{2}O_{3} 
Aluminum oxide 
C_{p} 
Specific heat of the fluid, J/ kgK 
CuO 
Copper oxide 
d_{f} 
Diameter of base fluid molecule 
d_{p} 
Nanoparticle diameter, m 
g 
Gravitational acceleration, $m / s^{2}$ 
k 
Thermal conductivity, $W / m K$ 
k_{eff } 
Effective thermal conductivity 
L 
Length of the tube, m 
M 
Molecular weight of base fluid 
Avogadro No 
$N=6.022 \times 10^{23} \mathrm{mol}^{1}$ 
Nu 
Nusselt number, 
P 
Pressure of fluid, $P a$ 
Pr 
Prandtl number, $P r=C p \mu_{0} / k$ 
q_{x } 
Heat flux, $W / m^{2}$ 
Re 
Reynolds number, $R e=\rho_{0} V_{0} D / \mu_{0}$ 
R_{np} 
Nanoparticle radius, m 
Ri 
Richardson number, $R i=G r / R e$ 
SiO_{2} 
Silicon oxide 
T 
Temperature of fluid, K 
T 
Bulk temperature, K 
T_{0 } 
Reference temperature 
U´, V´, W´ 
Velocities in x´, y´ and z´ directions, $m / s$ 
U, V, W 
Dimensionless velocities in x, y and z directions 
U_{0 } 
Average jet velocity at the entrance, $m / s$ 
V 
Axial velocity, $m / s$ 
ZnO 
Zinc oxide 
Greek symbols 

μ_{eff } 
Effective viscosity 
α 
Thermal diffusivity, $m^{2} / s$ 
μ 
Dynamic viscosity of fluid, $\mathrm{kg} / \mathrm{m} \mathrm{s}$ 
υ 
Kinematic viscosity of fluid, $m^{2} s$ 
θ 
Inclination of tilted wall 
ρ 
Fluid density, $\mathrm{kg} / \mathrm{m}^{3}$ 
ρ_{1 } 
Nanofluid density $\mathrm{kg} / \mathrm{m}^{3}$ 
φ 
Volume fraction of nanoparticles 
Subscript 

b_{f} 
basefluid 
n_{f} 
nanofluids 
n_{p } 
nanoparticle 
eff 
effective 
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