HYDRODYNAMIC AND THERMAL CHARACTERISTICS OF
NANOFLUIDS IN UNIFORMLY HEATED TWO-DIMENSIONAL
AND AXI-SYMMETRIC PASSAGES
Mehrdad Raisee*, Zahra Niroobakhsh* and Azadeh Jafari** Department of Mechanical Engineering, Faculty of Engineering, Tehran University, Tehran, Iran.
e-mail: mraisee@ut.ac.ir
Keywords: Nanofluid, Nanoscale Particles, Heat Transfer Enhancement,
Abstract. In the present paper, the effects of adding nanoscale metallic particles γAl2O3on the hydrodynamic and thermal characteristics of laminar liquid flow (water and ethylene glycol) through two-dimensional and axi-symmetric passages are numerically investigated. The present numerical results are obtained using a 2D finite-volume code which solves the governing equations in polar and Cartesian coordinate systems. The pressure field is obtained with the well-known SIMPLE algorithm. Advective volume-face fluxes are approximated using the upstream quadratic interpolation scheme, QUICK. It is assumed that the passages examined in this investigation are under constant wall heat flux boundary condition. For each passage, numerical results are obtained at two Reynolds numbers of 100 and 250. The nanoparticle volume concentration is varied between 1 to 10%. It is found that the addition of nanoparticles increases both the wall heat transfer coefficient and wall shear stress. In general, the inclusion of nanoparticles to the base fluid has more pronounce effect on the wall shear stress than on the wall heat transfer coefficient. Results show that the ethylene glycol γAl2O3nanofluid gives higher heat transfer enhancement and shear stress rise than the waterγAl2O3 nanofluid.
1 INTRODUCTION
liquids for heat dispersion. The oil-and-carbon nanofluid is expensive but shows promise for industrial use because it is easier to produce than other nanofluids
As mentioned above nanofluids are made by suspending nanoscale particles of materials such as carbon, copper, aluminium, copper oxide and aluminium oxide in liquids such as oil, water, ethylene glycol and radiator fluid (a mixture mostly of water and ethylene glycol). Heat transfer enhancement of nanofluids depends on various parameters such as size and shape of the particles, their concentration, and the thermal properties of the particles as well as the base-fluid. Although the nanofluids have great potential for enhancing heat transfer, research work in this concept is still in the primary stage. Several existing published articles (Xuan and Li [1], Lee et al. [2] and Xuan and Roetzel [3]) are focused on prediction and measurement techniques of thermal conductivity of the nanofluids. These studies examined some current fluids such as water, engine oil and ethylene glycol, and nanoscale materials such asγAl2O3,
Cu, and SiO2. For example; Xuan and Li [1] experimentally shown that the addition of
2.5-7.5% copper oxide nanoparticles to the water increases its heat conduction coefficient by about 24-78%. However, the heat-transfer capability of ethylene glycol grew by 40 percent when only a 0.3 volume-percent of 10-nanometer-diameter spheres of copper were suspended in it.
As for as the numerical predictions of nanofluids in practical confined flows is concerned, only very recently a few papers have published.
Maïga
et al. [4] have recently presented numerical results on laminar and turbulent convective heat transfer of nanofluids in uniformly heated tube. Their results showed that the inclusion of nanoparticles to a base fluid such as water increases considerably the wall-heat-transfer in both the laminar and turbulent regimes. They also found that the ethylene glycol γAl2O3 mixture gives a far better heat transfer enhancement than the water γAl2O3mixture. The same research group in Roy et al. [5] presented the numerical results for hydrodynamic and thermal fields of water γAl2O3 nanofluid in a radial laminar flow cooling system. They showed that considerable heat transfer enhancement is possible with a 10% nanoparticle volume fraction nanofluid. On the other hand, an increase in wall shear stress was also reported with an increase in particle volume concentration.In this paper, the effects of adding nanoscale metallic particles of γAl2O3 on hydrodynamic and thermal characteristics of developing laminar liquid-flow (water and ethylene glycol) through uniformly heated 2D and axi-symmetric passages is numerically investigated. The main objectives of this study are to understand the mechanism of heat transfer enhancement in nanofluids and to examine the effect of particle volume concentration on the heat transfer coefficient.
2 DESCRIPTION OF PROBLEM
the passage length (L) is long and, thus at the exit plane of the computational domain the nanofluid is hydrodynamically and thermally fully-developed. For each geometry, numerical results are obtained for water γAl2O3 and ethylene glycol γAl2O3at two different Reynolds number of 100 and 250. In this study, the Reynolds number is based on the bulk velocity (Uin)
and hydraulic diameter, i.e. D for pipe and 2H for channel. For each case, the nanoparticle volume concentration (φ) ranged between 1 to 10%. The following relations are used to compute the thermodynamic and transport properties of the nanofluids under investigation:
p bf nf φ ρ φρ ρ =(1− ) + (1) b nf nf p p p ( )c c c = 1−φ +φ (2) bf nf φ φ µ µ =(123 2+7.3 +1) water 3 2O Al γ (3) bf nf φ φ µ µ =(306 2−0.19 +1) ethylene glycol 3 2O Al γ (4) bf nf k k =(4.97φ2+2.72φ+1) water 3 2O Al γ (5) bf nf k k =(28.905φ2+2.8273φ+1) ethylene glycol 3 2O Al γ (6)
where indices “b”, “bf” and “nf” refer to particle, based-fluid, and nanofluid respectively. Equations (1) and (2) are general relations use to compute the density and specific heat for a classical two-phase mixture, while equations (3) to (6) have been proposed by Maїga et. al. [4] based on the available experimental data.
3 GOVERNING EQUATIONS
In two-dimensional Cartesian and polar-symmetric coordinates the continuity equation takes the form:
0 )] ( ) ( [ 1 = ∂ ∂ + ∂ ∂ V r x U r x r m c m c m c ρ ρ (7)
where x and y represent the coordinates in the streamwise and cross-stream (or radial) directions respectively, rc is the radius of curvature, and index m equals 1 for the
axi-symmetric flows and zero for the plan cases.
Similarly, in 2D Cartesian and polar coordinate systems the conservation laws of momentum and energy for the steady, Newtonian, incompressible flow, are written as:
y-Momentum: )] y V r ( y ) y U r ( x [ r y P )] y V r ( y ) x V r ( x [ r )] V r ( y ) UV r ( x [ r m c m c m c m c m c m c m c m c m c ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ − ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ + ∂ ∂ µ µ µ µ ρ ρ 1 1 1 2 (9) Energy: )] y Pr r ( y ) x Pr r ( x [ r )] V r ( y ) U r ( x [ r m c m c m c m c m c m c ∂ Θ ∂ ∂ ∂ + ∂ Θ ∂ ∂ ∂ = Θ ∂ ∂ + Θ ∂ ∂ ρ ρ 1 µ µ 1 (10) whereUandVdenote the streamwise and cross-stream velocity components respectively, Pis the pressure, Θstands for temperature and Pr=µcp/kis the fluid Prandtl number.
4 NUMERICAL METHOD AND BOUNDARY CONDITIONS
The governing equations of flow and temperature fields in plan (or polar-cylindrical) are given in previous section. The general forms of these equations for a dependent variable Φ may be written as:
Φ Φ Φ + ∂ Φ ∂ Γ ∂ ∂ + ∂ Φ ∂ Γ ∂ ∂ = Φ ∂ ∂ + Φ ∂ ∂ S r y r y x r x V r y U r x m c m c m c m c m c ) ( )] [ ( ) ( )] ( [ ρ ρ (11)
where ΓΦ is the diffusion coefficient and SΦ denotes the total source term.
In the present study the above transport equations are solved using finite-volume methodology in a semi-staggered grid system. In such a grid distribution, both velocity components are computed and stored in the same nodal position and the velocity nodes are located at the corners of the scalar control volume. The pressure field is linked to that of velocity through the well-known SIMPLE pressure correction algorithm. To avoid stability problems associated with pressure-velocity decoupling the Rhie and Chow [6] interpolation scheme is also employed. The third order QUICK differencing scheme of Leonard [7] is employed for approximation of the convective terms in all transport equations. As shown in Figure 2, in this scheme, the cell face value of Φ is evaluated from a quadratic function passing through two bracketing nodes (one each side of the face) and a node on the upstream side.
The typical QUICK expression for the east face value of Φ can be written as:
< + Φ > + Φ = Φ − + 0 F Q 0 F Q e e E e e P e (12)
where Φpand ΦEare the values of Upwind approximation and QUICK scheme corrections
[
EE P E]
e Q− = −Φ +3Φ −2Φ 8 1 (14) As shown in Figure 3, the grid used for computations consist of 150×62 grid nodes in the stream wise (x) and cross-stream (y) directions respectively. As can be noted, the grid points are highly clustered in the vicinity of walls and also in the entrance region in order to resolve high gradients in these regions.Flow and heat transfer through channels examined in this investigation are governed by elliptic partial differential equations, and these require the prescription of boundary conditions along the entire perimeter of the solution domain. Along the solid walls, the velocity components are set to zero. As mentioned earlier, the walls of passages are under uniform heat flux boundary condition. To implement this condition, an extra source term is included in the discretised temperature equation for the near-wall temperature control volumes of the form: (15) cell p w A c q SΘ = ′′
whereAcellis the wall area of the control volume and cp is the specific heat capacity of the
fluid.
The wall temperature is subsequently determined from the values of the near-wall temperature and heat flux according to:
(16) p w I w c y q µ ∆ ′′ + Θ = Θ Pr
whereΘI is the temperature at the node nearest to the wall, and ∆y stands for the distance
from the wall.
At the inlet of computational domain, uniform flow and temperature fields were imposed. A zero stream-wise gradient condition is applied across the outlet boundary for all variations except temperature and pressure. For the thermal field outlet condition, the second derivation of temperature is set equal to zero. The outlet pressure is set equal to the sum of the pressure of upstream nodes and a uniform bulk pressure correction.
5 RESULTES AND DISCUSSION
In this section, numerical predictions for convective heat transfer of two nanofluids
considered in this paper are presented and discussed.
5.1 VALIDATION OF COMPUTER CODE
top and bottom walls of the channel.
Figure 4 shows the velocity profile obtained in the fully-developed region of the channel is in complete agreement with the well-known quadratic profile, i.e.u/U 6[y/H (y/H)2]
in = − ,
obtained from the analytical solution of Navier-Stokes equations. Moreover, Figure 5 indicates that the computed local Nusselt number for developing laminar flow at Re=100 is an excellent agreement with the analytical value of this quantity (Nu=hDh/k=8.235) in the
fully-developed part of the channel. Thus, these results indicate the validity of the computer code used in the present investigation.
The computer code was then used for the numerical simulation of the problem under investigation using two nanofluids namely: water γAl2O3 and ethylene glycol γAl2O3. In the following the effects of adding nanoparticles on the wall shear stress and also wall heat transfer enhancement are presented and discussed.
5.2 EFFECTS OF WATER
γ
Al
2O
3 NANOFLUIDFigures 6 and 7 respectively show the axial distribution of nanofluid-to-base-fluid wall shear-stress ratio i.e.,τr=τnf /τbf for the channel and pipe at two Reynolds numbers of 100 and
250 respectively. In all cases, it is observed that the shear-stress-ratio take the value of unity at the passage inlet. Subsequently, due to the axial development of the velocity field , for each value ofϕ ,τrsharply increases and reaches a constant value. Note that at the lower Reynolds
number (Re=100), the shear-stress-ratio reaches its asymptotic value more quickly compared to the higher Reynolds number (Re=250). This is because the entrance length for Re=100 is shorter than that for Re=250.
The effects of particle volume concentration on the averaged wall shear-stress-ratio (defined asτr =τnf /τbf) at Re=100 and 250 are shown in Figures 8 and 9 respectively. It is
clearly seen that for all cases investigated, the presence of nanoparticles substantially increases the wall stress-ratio. In general, for all cases examined, the least wall shear-wall shear-stress-ratio rise is 1.1 occurring for the lower value of φ=1% while the greatest increase is 2.9 corresponding toφ=10%. These results indicate the nanofluid pressure drop is significantly higher than the base fluid. It is worth mentioning that, similar results for wall shear-stress-ratio were reported by Maïga et al. [4].
Attention is now directed toward the heat transfer predictions. Figures 10 and 11 show the effects of particle volume concentration φ on the axial distribution of wall heat-transfer coefficient ratio, defined ashr =hnf /hbf. The results clearly show that the addition of
nanoparticles into a base fluid has a favorable effect on the wall heat-transfer coefficient. Similar to shear-stress profiles, the wall heat transfer coefficient ratio takes the value of 1 at the inlet of the passage. Subsequently, hrgradually increases and approaches to its
boundary layer is thinner than the hydrodynamic boundary layer; resulting a longer thermal entrance length. Comparison of results obtained for Re=100, with those obtained for Re=250 also shows that the hrprofiles for the lower Reynolds number reach to their constant values
more quickly than those returned for the higher. This is also due to the fact that entrance length for Re=250 is longer than that for Re=100.
The influences of particle volume concentration φ on the averaged wall heat-transfer coefficient agumentation are shown in Figures 12 and 13 for Re=100 and Re=250 respectively. The influence of adding nanoparticles to the base fluid on heat transfer enhancement is clearly visible in these figures. The lower heat transfer enhancement is about 1.02 occurring at Re=250 for φ=1%, whilst the higher rise is around 1.3 occurring at Re=100 for φ=10%. This indicates that for the laminar flow the addition of nanoparticles is more beneficial for the lower Reynolds numbers than for the higher.
5.3 EFFECTS OF ETHYLENE GLYCOL
γ
Al
2O
3 NANOFLUIDHaving discussed the effects of adding nanoparticlesγAl2O3 to the water, attention is now directed on the effects of addition of γAl2O3to ethylene glycol. In general, results obtained for ethylene glycol γAl2O3nanofluid are very similar to those presented in the previous subsection for water γAl2O3 nanofluid. Therefore, only predictions for the channel flow at Re=250 are presented.
Figure 14 shows the effects of particle volume concentration on the local and averaged wall shear-stress-ratios. Similar to trend observed for the water γAl2O3 nanofluid, the inclusion of the nanoparticles γAl2O3 to the ethylene glycol increases the wall shear-stress-ratio.
Figure 15 demonstrates the influence of nanoparticles γAl2O3 on the wall heat-transfer enhancement. As seen before, the heat transfer levels are increased with the inclusion of the nanoparticles to the ethylene glycol. To further investigate the reason of higher heat transfer rates of nanofluid, the radial distribution of nanofluid temperature (normalized with local bulk temperatureTb) in the fully developed region and axial distribution of wall temperature are
presented in Figure 16. As can be noted as a result of higher diffusion rates, the nanofluid temperature has decreased with the augmentation of φ , in particular in the near-wall region of the channel. Thus, indicating clearly a higher heat transfer rate with increasing particle concentration.
In Figure 17 the global effects of the particle concentration on the wall shear-stress-ratio and heat transfer enhancement of two nanofluids are compared. It is noted that the wall shear-stress rise and average heat transfer enhancement are more pronounced for the ethylene glycol
3 2O
Al
γ than for the water γAl2O3nanofluid. One can also observe that these effects are more important for a higher particle loading.
6 CONCLUSIONS
hydrodynamic and thermal characteristics of ethylene glycol γAl2O3 and water γAl2O3mixtures through 2D and axi-symmetric passages. The numerical code employed in this investigation was first validated by comparing the numerical results for fully-developed channel flow with the well-established analytical solution. The numerical code then employed to study how the addition of nanoparticles influences the wall shear stress and heat transfer coefficient. For each passage, numerical results are obtained at two Reynolds numbers of 100 and 250 and the nanoparticle volume concentration is varied between 1 to 10%. It is found that the addition of nanoparticles increases both the wall heat transfer coefficient and wall shear stress and inclusion of nanoparticles has a more pronounce effect on the wall shear stress than on the wall heat transfer coefficient. Results show that the ethylene glycol γAl2O3mixture gives better heat transfer enhancement with higher shear stress rise than the water γAl2O3 nanofluid.
AKNOWLEGHMENT
The authors would like to thank the University of Tehran for providing financial support of this study.
REFERENCES
[1] Y. Xuan and Q. Li, Heat transfer enhancement of nanofluids, Int. J. Heat Fluid Flow, 21, 58-64 (2000).
[2] S. Lee, S.U.-S. Choi, S. Li and J.A. Estman, Measuring thermal conductivity of fluids containing oxide nanoparticles, J. Heat Transfer, 121, 280-289 (1999).
[3] Y. Xuan and W. Roetzel, Conceptions for heat transfer correlation of nanofluids, Int. J. Heat Mass Transfer, 43, 3701-3707 (2000).
[4] S.El B.Maїyga, C.T. Nguyen, N.Galanis, and G. Roy, Heat transfer behaviors of nanofluids in a uniformly heated tube, J.Superlatttics and microstructures, 35, 543-557 (2004)
[5] G. Roy, C.T. Nguyen, and P. Lajoie, Numerical investigation of laminar flow and heat transfer in a radial flow cooling system with the use of nanofluids, J. Superlattices and Microstructures, 35, 497-511 (2004).
[6] C.M. Rhie and W.L. Chow, Numerical study of the turbulent flow past an airfoil with trailing edge separation, AIAA J., 21, 525-532 (1983)
(a)
(b)
a) Fe >0 and Fw>0
b) Fe <0 and Fw<0 Figure 2: The QUICK scheme.
0.0 0.5 1.0 1.5
U/U
in 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0y/
H
Numerical TheoreticalFigure 4: Comparison of predicted velocity profile for fully-developed laminar flow between two parallel plates with analytical solution at Re=100.
10.0 20.0 30.0 40.0 50.0
x/H
0.0 10.0 20.0 30.0 40.0 50.0Nu
Numerical Theoreticalx/L 0 0.25 0.5 0.75 1 1 1.5 2 2.5 3 1 % 2.5 % 5 % 7.5 % 10 % x/L 0 0.25 0.5 0.75 1 1.0 1.5 2.0 2.5 3.0 1 % 2.5 % 5 % 7.5 % 10 %
Figure 6: Effects of particle volume concentration on axial wall shear-stress-ratio distribution of water γAl2O3 at Re=100. x/L 0 0.25 0.5 0.75 1 1.0 1.5 2.0 2.5 3.0 1 % 2.5 % 5 % 7.5 % 10 % x/L 0 0.25 0.5 0.75 1 1.0 1.5 2.0 2.5 3.0 1 % 2.5 % 5 % 7.5 % 10 %
0 2.5 5 7.5 10 1.0 1.5 2.0 2.5 3.0 0 2.5 5 7.5 10 1.0 1.5 2.0 2.5 3.0
Figure 8: Effects of particle volume concentration on averaged wall shear-stress-ratio of water γAl2O3 at Re=100.
0 2.5 5 7.5 10 1.0 1.5 2.0 2.5 3.0 0 2.5 5 7.5 10 1.0 1.5 2.0 2.5 3.0
x/L hr 0 0.25 0.5 0.75 1 1 1.1 1.2 1.3 1.4 1.5 1 % 2.5 % 5 % 7.5 % 10 % x/L hr 0 0.25 0.5 0.75 1 1.0 1.1 1.2 1.3 1.4 1.5 1 % 2.5 % 5 % 7.5 % 10 %
Figure 10: Effects of particle volume concentration on axial wall heat transfer coefficient ratio distribution of waterγAl2O3 at Re=100. x/L hr 0 0.25 0.5 0.75 1 1.0 1.1 1.2 1.3 1.4 1.5 1 % 2.5 % 5 % 7.5 % 10 % x/L hr 0 0.25 0.5 0.75 1 1.0 1.1 1.2 1.3 1.4 1.5 1 % 2.5 % 5 % 7.5 % 10 %
0 2.5 5 7.5 10 1.0 1.1 1.2 1.3 1.4 0 2.5 5 7.5 10 1.0 1.1 1.2 1.3 1.4
Figure 12: Effects of particle volume concentration on averaged wall heat transfer coefficient ratio of water γAl2O3 at Re=100. 0 2.5 5 7.5 10 1.0 1.1 1.2 1.3 1.4 0 2.5 5 7.5 10 1.0 1.1 1.2 1.3 1.4
x/L 0 0.25 0.5 0.75 1 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1 % 2.5 % 5 % 7.5 % 10 % 0 2.5 5 7.5 10 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Figure 14: Effects of particle volume concentration on wall shear-stress-ratio ratio of ethylene glycolγAl2O3 at Re=250. x/L r 0 0.25 0.5 0.75 1 1.0 1.2 1.4 1.6 1.8 1 % 2.5 % 5 % 7.5 % 10 % 0 2.5 5 7.5 10 1.0 1.1 1.2 1.3 1.4 1.5
x/L Tw -Tin /Tb -Tin 0 0.25 0.5 0.75 1 0.0 0.3 0.5 0.8 1.0 1.3 1.5 0 % 1 % 2.5 % 5 % 7.5 % 10 % r/R T/T b 0 0.25 0.5 0.75 1 0.5 0.8 1.0 1.3 1.5 0 % 1 % 2.5 % 5 % 7.5 % 10 %
Figure 16: Effects of particle volume concentration on wall and radial temperature distributions of ethylene glycolγAl2O3at Re=250.
0 2.5 5 7.5 10 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Ethylene Glycol- Water-0 2.5 5 7.5 10 1.0 1.1 1.2 1.3 1.4 1.5 Ethylene Glycol-
