Heat Mass Transfer DOI 10.1007/s00231-013-1196-x

ORIGINAL

Numerical study of turbulent flow and heat transfer of Al2O3–water mixture in a square duct with uniform heat flux Okyar Kaya

Received: 13 August 2012 / Accepted: 16 June 2013 Ó Springer-Verlag Berlin Heidelberg 2013

Abstract Turbulent flow and convective heat transfer of a nanofluid made of Al2O3 (1–4 vol.%) and water through a square duct is numerically studied. Single-phase model, volumetric concentration, temperature-dependent physical properties, uniform wall heat flux boundary condition and Renormalization Group Theory k-e turbulent model are used in the computational analysis. A comparison of the results with the previous experimental and numerical data revealed 8.3 and 10.2 % mean deviations, respectively. Numerical results illustrated that Nu number is directly proportional with Re number and volumetric concentration. For a given Re number, increasing the volumetric concentration of nanoparticles does not have significant effect on the dimensionless velocity contours. At a constant dimensionless temperature, increasing the particle volume concentration increases the size of the temperature profile. Maximum value of dimensionless temperature increases with increasing x/Dh value for a given Re number and volumetric concentration. List of symbols aP Coefficient of P cell C1e Turbulent model constant = 1.42 C2e Turbulent model constant = 1.68 Cp Specific heat of nanofluid under constant pressure (J/kg K) Cpnp Specific heat of nanoparticle under constant pressure (J/kg K) Cl Turbulent model constant = 0.0845

O. Kaya (&) Department of Mechanical Engineering, Pamukkale University, 20070 Kinikli, Denizli, Turkey e-mail: [email protected]; [email protected]

C dh g I k l le ll Nu P Pr q00 R0 Re Rey S T Tb T? u u0 u y

Constant part of the source term Hydraulic diameter (m) Gravitational acceleration (m/s2) Turbulent intensity = u0 =u Turbulent kinetic energy (m2/s) Turbulent mixing length (m) Length scale of turbulent kinetic energy dissipation (m) Length scale of viscosity (m) Nusselt number Pressure (Pa) Prandtl number of nanofluid Wall heat flux (W/m2) Effect of strain in e equation (kg/ms4) Re number Re number for a cell having distance y from the nearest wall Modulus of mean rate of strain tensor (1/s) Temperature (K) Cross-sectional weighted average of the local fluid  1 R  temperature (K) ¼ UA uTdA Mixed mean temperature (K) Time averaged mean velocity (m/s) Instantaneous velocity component (m/s) Velocity (m/s) Non-dimensional viscous sublayer thickness

Greek symbols  1 a Inverse effective Pr number ¼ Pr ae Inverse effective Pr number for dissipation rate of   turbulent kinetic energy ¼ Pr1 e ak

Inverse effective Pr number for turbulent kinetic   energy ¼ Pr1 k

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Heat Mass Transfer

a t

e j g k l m / Uv q

Inverse effective Pr number for turbulent flow   ¼ Pr1 t Turbulent kinetic energy dissipation rate (m2/s3) Von Karman constant = 0.42 Rate of strain in turbulent flow = Sk/e Thermal conductivity of nanofluid (W/mK) Molecular viscosity of nanofluid (kg/ms) Kinematic viscosity of nanofluid (m2/s) Conservation of mass, momentum and energy equations Volume concentration of nanoparticles Density of nanofluid (kg/m3)

Subscripts bf Base fluid eff Effective i Inlet m Mean nb Neighbor cell np Nanoparticle t Turbulent 1 Introduction Cooling and heating systems generally use water, oil or ethylene glycol for heat transfer in their loop. The use of nanofluids, a mixture of nanometer-sized solid particles (\100 nm) and a base fluid, has been suggested to improve the efficiency as well as reduce the size, weight and cost of these systems. Among other researchers who studied nanofluids, Namburu et al. [1] numerically analyzed the turbulent flow and heat transfer with three different nanofluids, CuO, Al2O3 and SiO2 mixed with water or ethylene glycol, through a uniformly heated circular tube. They achieved new correlations for viscosity up to 10 % volumetric concentration. Yadav et al. [2] studied the effect of internal heat source, boundary conditions and nanofluid parameters on the onset of Darcy-Brinkman convection in a porous layer. Xuan and Li [3] and Pak and Cho [4] experimentally studied turbulent flow and convective heat transfer with nanofluids through a tube. They proposed a formula for convective heat transfer in nanofluids using experimental data. Luciu et al. [5] formulated temperature and volumetric concentration dependent physical property using experimental data. Maiga et al. [6] numerically studied the effects of Reynolds number and volumetric concentration (water–Al2O3) on turbulent heat transfer and wall shear stress. They evidenced that the inclusion of nanoparticles increased heat transfer coefficient as well as wall shear stress. A new correlation for heat transfer coefficient for nanofluids was also proposed.

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Ghaffari et al. [7] numerically studied turbulent flow of a nanofluid (water–Al2O3) in a horizontal curved tube with constant wall temperature. They presented the effects of buoyancy, centrifugal forces and nanoparticle concentration on secondary flow and skin friction coefficient. Lotfi et al. [8] used three different models—single phase, two-phase Eulerian and two-phase mixture—to numerically study a nanofluid (water–Al2O3) flowing through a horizontal tube. They concluded that the mixture model is the most reliable among others. Zeinali Heris et al. [9] investigated laminar flow and heat transfer of Al2O3– water nanofluid with constant wall temperature inside a circular tube. According to their results, the increase in heat transfer coefficient due to presence of nanoparticles was much higher than the prediction of single phase heat transfer correlation used with nanofluid properties. Mirmasoumi and Behzadmehr [10] used a two-phase model to investigate hydrodynamic and thermal behaviors of a nanofluid (water–Al2O3) in a horizontal tube over a wide range of Grasshof and Reynolds numbers. In the fully developed region, the nanoparticle concentration did not have any significant effect on the skin friction coefficient. In another numerical analysis, Mirmasoumi and Behzadmehr [11] evaluated the effects of mean diameter of the nanoparticles on hydrodynamic and thermal parameters. Using particles with smaller diameter (dp = 10 nm and 40 nm) increases the uniformity of the particles distribution at the tube cross-section and so that the single phase approach could be adopted. Rashmi et al. [12] numerically investigated natural convection heat transfer for Al2O3–water in a horizontal cylinder of L/Dh = 1.0, considering the nanofluid as single phase. They found that with the increase in particle volume fraction heat transfer decreases for natural convection. In order to reduce energy consumption, the use of passive solar energy applications is very important. The heat flux in the solar water heating system and also in domestic heating and in heat exchangers is generally uniform and nanofluids can be alternatively used as heat transfer fluids in these systems. In the literature, however, no study has been reported about turbulent flow and heat transfer of nanofluids inside a horizontal square duct. The scope of the present study is to simulate turbulent flow and heat transfer of the Al2O3– water mixture nanofluid in a square duct and to validate the results with the previous experimental and numerical data. Fluent commercial computational fluid dynamic software is employed to simulate turbulent flow and heat transfer in a uniformly heated square duct and the numerical results are demonstrated in terms of Nu and Re numbers.

Heat Mass Transfer

2 Mathematical modeling and numerical method 2.1 Assumptions A steady, forced turbulent flow and convective heat transfer of a nanofluid in a straight, square cross-sectioned duct was to be analyzed. The size of the nanoparticles was less than 40 nm so that the relative velocity between nanoparticles and the base fluid would be small [1]. For thermal equilibrium, similar temperatures of nanoparticles and base fluid were assumed. Because of no motion slip between the phases, the mixture got easily fluidized. Nanoparticles dispersed well within the base fluid with zero relative velocity, and the effective mixture behaved like a single-phase fluid. According to Pak and Cho [4] and Xuan and Li [3], the assumption of single phase is valid for nanofluids with smaller diameter (i.e. dnp = 13 nm) nanoparticles. Consequently, classical theories of single-phase fluids, including conservation equations of mass, momentum and energy could then be applied to a nanofluid. The physical and thermal properties of the nanofluid were evaluated as functions of temperature and volumetric concentration of nanoparticles in the base fluid. Computations have been carried out in a geometry of x/Dh = 100 as used in the study of Maiga et al. [6]. 2.2 Governing equations The configuration of a straight, square cross-sectioned duct is shown in Fig. 1. It was assumed that nanofluid velocity and temperature at the inlet of the duct are uniform and represented by Vi and Ti, respectively. Turbulent flow in the duct was maintained so as the Re number not to fall below 10,000. Duct walls were subjected to a constant heat flux, q00 , of 500,000 W/m2. A RNG k-e turbulence model was used to simulate turbulent flow and heat transfer in the duct, as well as it provides an analytical derivation for effective viscosity taking into account the effects of low Re number and has an additional term in e equation to improve the accuracy of strained flows. Turbulent intensity, I, and hydraulic diameter, dh, were also specified. Numerical Nu numbers for the volumetric concentrations of 2 and 4 % at different x/Dh values are given in Table 1. Dimensionless velocity = V/Vm contours at different x/Dh values is also given in Fig. 2. From Table 1 and Fig. 2, it can easily be seen that the flow reaches fully developed regime before the exit of the square duct. Therefore fully developed flow and temperature fields and zero normal gradients for all flow variables, except for pressure were assumed at the outlet (x/Dh = 100). Consequently, outflow boundary condition at the outlet section was chosen.

The governing equations of turbulent flow and heat transfer in a square duct were written in master cartesian coordinate system as follows: Continuity equation is given by oðquÞi ¼ 0: oxi

ð1Þ

Momentum equation is given by    oðqui uj Þ oP o oui ouj 2 ouk ¼ þ l þ  leff oxj oxi oxj eff oxj oxi 3 oxk ð2Þ

þ qgi : Energy equation is defined as   o o oT ðqui CP TÞ ¼ keff oxi oxi  oxi  oui oui ouj 2 ouk þ l þ  leff 3 oxj eff oxj oxi oxk

ð3Þ

where effective thermal conductivity is obtained from keff ¼ at Cp leff :

ð4Þ

Effective viscosity is leff ¼ l þ lt : Turbulent kinetic energy equation is   oðqui kÞ o ok ¼ ak leff þ l S2 oxi oxi oxi  t 1 oT oq  gi at l t qe q oxi oT P

ð5Þ

ð6Þ

where S is the modulus of mean rate of strain tensor given by pffiffiffiffiffiffiffiffiffiffiffiffi ð7Þ S ¼ 2Sij Sij where Sij is defined as   1 oui ouj Sij ¼ þ : 2 oxj oxi

ð8Þ

Equation of dissipation rate of turbulent kinetic energy is given by   De o oe e e2 ¼ ae leff q þ C1e lt S2  C2e q  R0 ð9Þ Dt oxi oxi k k where R0 is the effect of strain in e equation and is defined as R0 ¼

Cl qg3 ð1  g=g0 Þ e2 1 þ bg3 k

ð10Þ

where g ¼ Sk=e

ð11Þ

Turbulent viscosity is obtained from

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Heat Mass Transfer Fig. 1 Geometrical configuration of the duct in the present study

Table 1 Numerically computed Nu numbers for different x/Dh values 4 % Al2O3 V = 3 m/s

2 % Al2O3 V = 1 m/s Nu

x/Dh

Nu

70

127

70

209

80

127

80

209

90

127

90

209

k2 : e

k3=2 le

ð17Þ

where

x/Dh

lt ¼ qCl

e¼

ð12Þ

The subscripts i, j, k and the model constants g0, b, Cl, C1e, C2e in the equations are equal to 1, 2, 3 and 4.38, 0.012, 0.085, 1.42, 1.68, respectively. at, ak, ae are determined using Eq. (13),      a  1:3929 0:6321  a þ 2:3929 0:3679 l     ¼ ð13Þ a  1:3929 a þ 2:3929 leff 0 0



  Rey le ¼ Cl y 1  exp  Ae

ð18Þ

in that Cl ¼ jC3=4 l

ð19Þ

Here, Al in Eq. (16) and Ae in Eq. (18) are equal to 70 and 2Cl, respectively. If Rey [ 200, the RNG k-e turbulent model is to be employed. The non-dimensional viscous sub-layer thickness, y , in Eq. (16) is defined as 1=2

y ¼

qCl1=4 kP yP : l

ð20Þ

where a0 for at, ak, ae is equal to 1/Pr, 1, 1, respectively. The method of two-layer zonal non-equilibrium wall function [4] was used in numerical computations for nearwall regions. According to this model, Rey number is to be computed via pffiffiffi q ky : ð14Þ Rey ¼ l

During the simulations, FLUENT requires specification of the transported scalar quantities. Therefore it is appropriate to specify a uniform value of the turbulence intensity at the boundary where inflow occurs. The inlet turbulent kinetic energy is calculated by

If Rey \ 200, turbulent viscosity, lt, and dissipation rate of turbulent kinetic energy, e, could be obtained from Eqs. (15) and (17), respectively. To determine turbulent viscosity, we thus use pffiffiffi lt ¼ qCl kll ð15Þ

where turbulent intensity is estimated by

where 



Rey ll ¼ C1 y 1  exp  Al

 :

I¼

u0 %100 ffi 0:16ðRedh Þ1=8 : u

ð16Þ

ð21Þ

ð22Þ

Dissipation rate of inlet turbulent kinetic energy is ei ¼ c3=4 l

Dissipation rate of turbulent kinetic energy is given by

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3 ki ¼ ðui IÞ2 2

k3=2 l

ð23Þ

where turbulent length scale, l, is 0.007 dh. The details of the above equations (Eqs. 1–23) and the coefficients can be found in Fluent User’s guide [13].

Heat Mass Transfer Fig. 2 Dimensionless velocity = V/Vm contours at different x/Dh

Re=50000 % 4 Al2O3 x/Dh=50

x/Dh =70

x/Dh =80

x/Dh =100

2.3 Thermal and physical properties of nanofluids The Al2O3 nanoparticles used in the study had the following properties: density, qnp = 3,880 kg/m3; specific heat, CPnp = 0.773 kJ/kg K. It was assumed that the nanoparticles dispersed well within the base fluid. The thermophysical properties of the nanofluid were derived by the following relations: q ¼ ð1  Uv Þqbf þ Uv qnp

ð24Þ

CP ¼ ð1  Uv ÞCPbf þ Uv CPnp

ð25Þ

Density and specific heat of the nanofluid were determined by using Eqs. (24) and (25) as a function of volumetric concentration [4].

Temperature-dependent dynamic viscosities of the nanofluid for different volume concentrations were determined by using the following Nguyen et al. [14] formulas: lð1 %Þ ¼ 3:65785  1011 T4  4:88267  108 T3 þ 2:45398  105 T2  5:510714  103 T þ 0:467545089 ð26Þ lð2 %Þ ¼ 3:97752  1011 T4  5:30937  108 T3 þ 2:66844  105 T2  5:992306  103 T þ 0:508404721 ð27Þ

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Heat Mass Transfer

Rn/

lð3 %Þ ¼ 4:5148  1011 T4  6:02656  108 T3 5 2

3

þ 3:02889  10 T  6:801744  10 T þ 0:577079809 ð28Þ lð4 %Þ ¼ 4:38576  109 T3 þ 4:44807  106 T2  1:513857  103 T þ 0:173517495

ð29Þ

Temperature-dependent conductivities of the nanofluid for different volumetric concentrations were computed by the following equations [5]: kð1 %Þ ¼ 7:29423  106 T2 þ 5:851204  103 T  0:468564118

ð30Þ 6 2

R5/

 104

ð35Þ

where Rn/ is the residual value of / for the nth iteration, and R5/ is the maximum residual value of / for the first five iterations. Different turbulence models were tested for a given Re number (Re = 46,000), and the results are provided in Table 2. Precise results were obtained with RNG k-e turbulence model, and so RNG k-e model was the preferred turbulence solver method in the present study. 2.4.1 Grid independence test

3

kð2 %Þ ¼ 7:49502  10 T þ 6:011645  10 T  0:480642606 kð3 %Þ ¼ 7:6996  106 T2 þ 6:175092  103 T  0:4929312

ð31Þ

ð32Þ

kð4 %Þ ¼ 7:90806  106 T2 þ 6:341632  103 T  0:505435331 ð33Þ 2.4 Numerical method Fluent [13] software was used to solve all equations. For the governing equations, finite volume method was used to first convert these into algebraic equations to be then solved numerically. Subsequently, these algebraic equations were solved iteratively via Gauss Siedel relaxation procedure in Fluent [13] software. A non-uniform grid system was used to discretize square duct volume. The following pre-processing parameters were chosen in the software: Segregated Implicit Solver to solve the governing equations, Semi Implicit Method for Pressure Linked Equations (SIMPLE) algorithm to enforce mass conservation and to obtain pressure field, Second Order Upwind Scheme for interpolating momentum and energy parameters, and Pressure Staggered Option (PRESTO) for interpolating pressure. In order to accelerate convergence of the solver, Under Relaxation Factors (URF) were set for pressure, velocity, temperature, turbulent kinetic energy and dissipation rate to 0.3, 0.7, 0.8, 0.8 and 0.8, respectively. The algebraic equation for variable Ø at each node was written as X aP /P ¼ anb /nb þ c ð34Þ nb

where aP and anb are linearized coefficients for center and neighbor cells, and c is constant part of the source term. Except for energy (10-8), convergence for all equations was given by Eq. (35).

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The discretization grid was non-uniform in x, y and z directions. It was finer near the duct entrance and walls, where the velocity and temperature gradients are considerable. Nu numbers for different nodes in x, y and z directions are given in Table 3. As shown in table, Nu numbers for 60 9 90 9 90 grids, 70 9 90 9 90 grids, 90 9 90 9 90 grids, 70 9 100 9 100 grids and 70 9 110 9 110 grids are equal to 228, and consequently, 70 9 90 9 90 non-uniform grids were selected as the optimal grid number along the x, y, z directions. 2.4.2 Validation test Numerical simulations for turbulent flow of water were carried out. Hydraulic diameter and length of the square duct were 0.01 m and 1 m, respectively. Water enters the duct with different uniform axial velocities of Vi = 1, 2, 3, 4, 5, 6, 7 and 8 m/s and with a constant temperature of Ti, which is equal to 293 K. In order to validate the computational model, Nu numbers computed numerically for fully developed turbulent flow was compared with the Nu numbers derived using formulas Eqs. (36) and (37), given by Dittus and Boelter [15] and Gnielinski [16], respectively. Nu ¼ 0:023Re0:8 Pr n

ð36Þ

for 0.7 \ Pr \ 160, n = 0.3 for cooling, n = 0.4 for heating, Re [ 10,000, L/D C 10. Nu ¼ 0:012ðRe0:87  280ÞPr 0:4

ð37Þ

for 1.5 \ Pr \ 500, 3 9 103 \ Re \ 106. A comparison of computed Nu numbers with empirical formulas is shown in Fig. 3. The average deviations of numerically computed Nu numbers from Dittus and Boelter [15] and Gnielinski [16] formulas were 8.52 and 0.05 %, respectively. This result agrees with Bejan’s [17] interpretation that the Gnielinski [16] correlation (Eq. 37) gives

Heat Mass Transfer Table 2 Comparison of numerical Nu numbers computed by using different turbulence models with the empirical Nu numbers for water

Table 3 Numerically computed Nu numbers for different grid numbers

Turbulence model

Nu computed

Nu Dittus and Boelter [15]

Nu Gnielinski [16]

Standard k-e

200

213

227

6.1

11.9

RNG k-e

228

213

227

-7.0

-0.4

Realizable k-e

184

213

227

13.6

18.9

Number of grids

Nu

70 9 70 9 70

259

70 9 80 9 80

227

10 9 90 9 90

224

30 9 90 9 90

230

50 9 90 9 90

230

60 9 90 9 90

228

70 9 90 9 90

228

90 9 90 9 90

228

70 9 100 9 100

228

70 9 110 9 110

228

Comparison of Nu Number from Dittus-Boelter and Gnielinski formulas with the computed values for water

600

% Comp.-Dittus and Boelter

3 Results and discussions Subsequent to the validation tests, similar computational models were applied on the nanofluid for different concentrations and Re numbers. For the purpose of comparing the results with the previous studies, similar heat flux of 5 9 105 W/m2 was applied on the duct walls. 3.1 Comparison of numerically computed Nu numbers with the previous correlations The present numerical simulation was compared with two correlations, the first being Pak and Cho’s [4] correlation, which was a proposed equation to calculate the fully developed heat transfer coefficient for a nanofluid and given by: Nu ¼ 0:021Re0:8 Pr 0:5

ð38Þ 4

500

Nu

400 Present work Dittus-Boelter Gnielinski

300 200 100 0 0

20000

40000

60000

80000

100000 120000

Re

Fig. 3 Comparison of numerically computed Nu numbers with empirical formulations

more accurate results (±10 %) than does the Dittus and Boelter [15] correlation (Eq. 36). Our results indicate that the choice of methods and the models used in numerical computations were appropriate. Turbulent flow of the candidate nanofluid in a square duct has constant heat flux (q00 = 5 9 105 W/m2); its boundary condition could be simulated using the same models and methods for different volumetric concentrations and Re numbers.

% Comp.Gnielinski

5

for 6.54 \ Pr \ 12.33 and 10 \ Re \ 10 . Pak and Cho [4] derived the correlation by curvefitting their experimental data, and in the present study, the maximum deviation of the empirical correlation (Eq. 38) from the experimental results was about 4.8 %. A comparison of the numerically computed Nu numbers with Eqs. (38) and (39) for 1, 2, 3 and 4 % volumetric concentrations of Al2O3–water mixture is shown in Fig. 4. The average deviation of numerically computed Nu numbers from Pak and Cho’s [4] correlation (Eq. 38) was about 8.3 % over the range of Re numbers studied. The second correlation for comparison with the present numerical results was provided by Maiga et al. [6] and given by Eq. (39). Nu ¼ 0:085Re0:71 Pr 0:35

ð39Þ

for 6.6 \ Pr \ 13.9, 104 \ Re \ 5 9 105 and 0 \ u \10. Maiga et al. [6] expected their correlation (Eq. 39) to show a mean deviation of relative error of 12 % from numerical results. A comparison of the present numerical study with Eq. (39) for different volumetric concentrations is shown in Fig. 4, and the mean deviation from Eq. (39) is about 10.2 %.

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Heat Mass Transfer

Fig. 4 Comparison between the computed values of Nu numbers and the equation given by Pak and Cho [3] and Maiga et al [5] for Al2O3–water mixture

600

% 3 Al2O3

600

500

500 400

Nu

Nu

400 300

300 Present Study

200

200

% 1 Al2O3 % 2 Al2O3

Maiga et al Pak and Cho

100

% 3 Al2O3

100

% 4 Al2O3

Gnielinski

0

0 0

20000

40000

60000

80000

100000

Re

Fig. 5 Comparison of Nu number for Al2O3 (3 %) with different correlations

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0

20000

40000

60000

80000

100000

Re

Fig. 6 The influence of Al2O3 nanoparticle volume concentration on the Nusselt number for different Re numbers

Heat Mass Transfer Re=20000

600

Re=40000 Re=60000

500

Re=70000

Nu

400 300 200 100 0 0

1

2

3

4

5

Fig. 7 Effect of parameters / and Re number on Nu number

Fig. 8 Effect of x/Dh on dimensionless velocity = V/Vm contours

From Fig. 4, it may be read that the deviation of Maiga et al. [6] correlation (Eq. 39) from the present study is bigger than the deviation of Pak and Cho [4] correlation (Eq. 38). This was due to that Maiga et al. [6] used constant thermophysical properties in their numerical computations, whereas in the present study, temperaturedependent dynamic viscosities and heat conduction coefficients were used. Pak and Cho [4] and Maiga et al. [6] obtained correlations for Nu number (Eqs. 38 and 39) by curve fitting with their experimental and numerical data respectively. Therefore, one can observe from the Fig. 4 that the empirical data of [4] and [6] are linear but the numerical data of the present study is not linear. At comparison of the results of different correlations with the results of the present study is shown in Fig. 5.

Re=50000 x/Dh=80 %1 Al2O3

%2 Al2O3

%3 Al2O3

%4 Al2O3

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Heat Mass Transfer Fig. 9 Effect of Reynolds number on dimensionless velocity = V/Vm contours

x/Dh =80 % 4 Al2O3 Re=10000

Re=50000

Re=80000

Re=100000

It may be concluded that the results of the present study and the Gnielinski [16] correlation are the closest to the experimental data of Pak and Cho [4]. 3.2 Effect of Re number and nanoparticle volumetric concentration on Nu number Nu numbers for different Re numbers and volumetric concentrations are demonstrated in Fig. 6. From the figure, it can be observed that higher the volumetric concentration of Al2O3, higher is the Nu number, and hence higher is the heat transfer. Also, higher the Re number, higher is the Nu number for all volumetric concentrations.

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In low Re number, the increments of Nu number for different volumetric concentrations of Al2O3 nanofluid are not too much, but at a Re number bigger than 60,000, the increment in Nu number especially for 4 % concentration of Al2O3 nanofluid is higher than for 1, 2 and 3 % concentrations. The increase in Nu number is due to the increase in Pr number at higher concentrations [1]. Figure 7 shows the variation of Nu number as function of parameters Re and /. One can see from the figure that Nu number increases with both of these parameters. For a given Re number say Re = 20,000, Nu number increases from 116 (/ = 1 %) to 140 (/ = 4 %) and for a fixed volumetric concentration, say / = 3 % Nu increases from

Heat Mass Transfer Fig. 10 Effect of volumetric concentration on dimensionless temperature contours

Re=50000 x/Dh=80 %1 Al2O3

%2 Al2O3

%3 Al2O3

%4 Al2O3

128 to 410 for Re augmenting from 20,000 to 70,000. It is concluded that if higher rates of heat transfer is required, especially higher volumetric concentrated nanofluids with higher Re number could be preferred. 3.3 Effect of volumetric concentration on dimensionless velocity contours Figure 8 shows the effect Al2O3 on dimensionless observe from the figure increasing the volumetric

of volumetric concentration of velocity contours. One can that for a fixed Re number, concentration of nanoparticles

does not have significant effect on the dimensionless velocity contours. 3.4 Effect of Re number on dimensionless velocity contours The effect of Re number on dimensionless velocity contours for 4 % volumetric concentration and for x/Dh equal to 80 is shown in Fig. 9. It can be understood that for a given value of dimensionless velocity, as the Re number increases from 10,000 to 50,000, the dimensionless velocity contours shift towards the center but the position

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Heat Mass Transfer Fig. 11 Effect of Reynolds number on dimensionless temperature contours for x/Dh = 0.8

(a) % 1 Al2O3 Re=10000

Re=50000

Re=80000

Re=100000

did not change between the Re numbers of 50,000 and 100,000. 3.5 Effect of volumetric concentration on temperature contours Figure 10 shows the effect of volumetric concentration on dimensionless temperature contours for Re of 50,000 and x/Dh of 80. The dimensionless temperature contours are calculated by the following formula: (T - Tw)/ (Tb - Tw). It can be understood from the figure that for a given value of dimensionless temperature, the contours shift towards the walls of the duct as the volumetric concentration increases. In other words, if the volumetric concentration of Al2O3 increases, the rate of heat transfer increases.

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3.6 Effect of Re number on temperature contours Figure 11a, b shows the effect of Re number on dimensionless temperature contours for 1 and 4 % volumetric concentrations of Al2O3. As the Re number increases, bigger values of temperature contours occur in the center of the duct and the radius of these dimensionless contours decreases. As the volumetric concentration increases for the similar Re number, the maximum value of dimensionless temperature contour decreases. 3.7 Effect of x/Dh on temperature contours Figure 12 shows the effect of x/Dh on dimensionless temperature contours. Higher the x/Dh value, higher is the value of the maximum dimensionless temperature and also

Heat Mass Transfer Fig. 11 continued

(b) % 4 Al2O3 Re=10000

Re=50000

Re=80000

Re=100000

.

the position of the maximum dimensionless temperature contour gets closer to the duct wall in the flow cross-sectional area.

4 Conclusion Turbulent convection heat transfer in a horizontal square duct with constant wall heat flux was numerically studied. A nanofluid consisting of Al2O3 and water was used as the flowing fluid in simulations and whose physical properties were temperature-dependent. In order to demonstrate the validity of numerical computations, a comparison with the previous studies was done.

Average deviations of the numerically computed Nu numbers in the present results are about 8.3 and 10.2 % from Pak and Cho [4] and from Maiga et al. [6], respectively. Numerical data clearly demonstrated the relation between Re number and volumetric concentrations of the nanoparticles and Nu number; higher the Re number and volumetric concentration, higher is the Nu number. According to numerical results, the position dimensionless temperature contours gets closer to the wall with increasing volumetric concentrations but the position of dimensionless velocity contours does not change. Higher the x/Dh value, higher is the value of the maximum dimensionless temperature and also the position of the maximum dimensionless temperature

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Heat Mass Transfer Fig. 12 Effect of x/Dh on dimensionless temperature contours for Re = 50,000

% 4 Al2O3 x/Dh=50

x/Dh=70

x/Dh=80

x/Dh=100

=

contour gets closer to the duct wall in the flow crosssectional area.

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