Conjugate heat transfer of laminar mixed convection of a nanofluid through an inclined tube with circumferentially nonuniform heating
 Shahriar Allahyari^{1},
 Amin Behzadmehr^{1}Email author and
 Seyed Masoud Hosseini Sarvari^{2}
DOI: 10.1186/1556276X6360
© Allahyari et al; licensee Springer. 2011
Received: 29 October 2010
Accepted: 26 April 2011
Published: 26 April 2011
Abstract
Laminar mixed convection of a nanofluid consisting of water and Al_{2}O_{3} in an inclined tube with heating at the top half surface of a copper tube has been studied numerically. The bottom half of the tube wall is assumed to be adiabatic (presenting a tube of a solar collector). Heat conduction mechanism through the tube wall is considered. Threedimensional governing equations with using twophase mixture model have been solved to investigate hydrodynamic and thermal behaviours of the nanofluid over wide range of nanoparticle volume fractions. For a given nanoparticle mean diameter the effects of nanoparticle volume fractions on the hydrodynamics and thermal parameters are presented and discussed at different Richardson numbers and different tube inclinations. Significant augmentation on the heat transfer coefficient as well as on the wall shear stress is seen.
Introduction
Many different industries such as electronic, automotive and aerospace have been facing heat transfer limitation for improving performance of their thermal systems. Heat transfer enhancement has been considered as one of the key parameter for developing more efficient and effective thermal devices. Thus this issue has been studied extensively. Different active and passive methods have been considered for the heat transfer augmentation. Improving the thermophysical properties of the working fluids such as water, oil and ethylene glycol mixture is one of the possible methods. Therefore, there has been a strong motivation to develop a new heat transfer fluids with substantially higher thermal conductivity. Choi [1] presented a new generation of solidliquid mixtures that is called nanofluid. It demonstrates significant improvement over the thermal characteristics of the base fluids. Various nanofluids with different nanoparticle and base fluid materials have been prepared and their thermofluid characteristics have been investigated by many researchers.
Among them, experimental studies of [2–5] on confined geometries could be cited. In general they found that the Nusselt number increases with the nanoparticle concentrations and significant heat transfer enhancement has been achieved. Many works have been dedicated to determine and model the effective physical properties of different nanofluid. For instance, investigations of Refs. [6–13] on the effective thermal conductivity or the works that have been done by [14, 15] on the nanofluid effective viscosity could be mentioned.
Convective heat transfer with nanofluids can be modelled using the twophase or singlephase approach. The first provides the possibility of understanding the function of both the fluid phase and the solid particles in the heat transfer mechanisms. The second assumes that the fluid phase and particle are in thermal and hydrodynamic equilibrium. This approach is simpler and requires less computational time. Thus it has been used in several theoretical studies of convective heat transfer with nanofluids [16–18]. However, the concerns in singlephase modelling consist in selecting the proper effective properties for nanofluids and taking into account the chaotic movement of ultra fine particle. To partially overcome this difficulty, some researches [19–21] used the dispersion model which takes into account the improvement of heat transfer due to the random movement of particles in the main flow. In addition several factors such as gravity, friction between the fluid and solid particles and Brownian forces, the phenomena of Brownian diffusion, sedimentation and dispersion may coexist in the main flow of a nanofluid. This means that the slip velocity between the fluid and particle may not be zero [22]. Therefore, it seems that the twophase approach could better model nanofluid behaviours. Behzadmehr et al. [23] studied the turbulent forced convection of a nanofluid in a circular tube by using a twophase approach. They implemented the twophase mixture model for the first time to study nanofluid. Their comparison with the experimental results showed that the twophase mixture model is more precise than the singlephase model. Mirmasoumi and Behzadmehr [24] studied the laminar mixed convection of a nanofluid in a horizontal tube using twophase mixture model. They showed that the twophase mixture model could better simulate the experimental results than the singlephase model. Recently, Lotfi et al. [25] studied twophase Eulerian model that has been implemented to investigate such a flow field. Their comparison of calculated results with experimental values shows that the mixture model is more precise than the twophase Eulerian model.
This work intends to investigate conjugate mixed convectionconduction of a nanofluid though an inclined tube. The tube is subjected to a uniform heat flux on its top surface; it is insulated on its bottom surface. Therefore, the effects of tube inclinations and particle volume fractions on the hydrodynamic and thermal parameters have been presented over a wide range of ReGr combinations.
Mathematical formulation
are the mean axial velocity and shear stress, respectively, and ϕ is the volume fraction of phase k.
The physical properties in the above equations are:
Boundary condition
This set of nonlinear elliptical governing equations has been solved subject to the following boundary conditions:
At the tube outlet: atmospheric static pressure is assumed.
Numerical method and validation
It should be mentioned that our numerical results were obtained using the twophase mixture model and considering a very small volume fraction for the solid particles. Therefore, the numerical procedure is reliable and can well predict developing mixed convection flow in a tube.
Results and discussions
Calculations have been performed over wide range of ReGr combinations and nanoparticle concentrations. The Grashof number (or Richardson number) has been limited in order to respect the validity of the Boussinesq's approximation for the fluid density variation. The results presented here are for different Richardson numbers and three nanoparticle volume fractions.
Conclusion
Conjugate laminar mixed convection of water/Al_{2}O_{3} nanofluid in an inclined copper tube has been investigated numerically by using twophase mixture model. The top half of tube wall is heated while the other half of tube is considered to be adiabatic. Copper is a good conductive material and transfers the heating energy from the upper part of tube to the lower half of tube by heat conduction mechanism. This could also increase the fluid temperature at this region. The latter could generate the secondary flow for which its strength depends on the nanoparticle volume fraction, the Richardson number and tube inclination angle. The buoyancy induced secondary flow augments with the nanoparticle volume fraction and the Richardson number. However, by tube inclination the axial component of the buoyancy forces increases and so the strength of secondary flow decreases. Nanoparticle concentration does not have significant effect on the axial velocity profile. However, at the high value of the Richardson number for which the effect of thermal energy is become more important than the hydrodynamic energy, nanoparticle concentration could affect the axial velocity profiles. Heat transfer coefficient is augmented with the nanoparticle volume fraction as well as the Richardson number. Combinations of the axial and radial component of the buoyancy forces could determine the inclination angle for which the maximum heat transfer enhancement occurs. However, the wall shear stress is significantly increased with the nanoparticle volume fraction. It is also augmented with the tube inclination because of increasing the axial component of the buoyancy forces.
Abbreviations
 List of symbols :

C_{p}: Specific heat
 D :

Tube diameter (m)
 d _{ f } :

Molecular diameter of base fluid
 d _{p} :

Nanoparticle diameter (nm)
 E :

Energy (J/kg)
 gd _{f} :

Gravitational acceleration (ms^{1} )
 Gr :

Grashof number
 h :

convection heat transfer coefficient
 k :

Thermal conductivity (W/m K)
 P :

Pressure (Pa)
 Pr :

Prandtl number
 q":

Average heat flux at the solidfluid interface (W/m^{2})
 r :

Radial direction (m)
 r_{i} :

Tube radial inner (m)
 r_{o} :

Tube radial outer (m)
 Re :

Reynolds number
 Ri:

Richardson number
 T :

Temperature (K)
 T*:

Temperature dimensionless
 t :

Thicknesses (m)
 V :

Velocity (m/s)
 Z :

Axial direction.
 Greek letters :

α: Tube inclination
 β:

Volumetric expansion coefficient (K^{1})
 θ:

Angular coordinate
 ϕ:

Volume fraction
 μ:

Dynamic viscosity
 ν:

Kinematic viscosity
 ρ:

Density
 τ:

Shear stress.
 Subscripts :

b: Bulk
 dr:

Drift
 eff:

Effective
 o:

Outer condition
 p:

Particle
 f:

Base fluid
 i:

Inner condition
 k:

Summation index
 m:

Mixture
 nf:

Nanofluid
 s:

Solid
 w:

Interface
 0:

Inlet condition.
Declarations
Authors’ Affiliations
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