Heat transfer and friction factor characteristics of [Al.sub.2][O.sub.3], CuO and Ti[O.sub.2] nanofluid in plain tube with longitudinal strip inserts by using computational fluid dynamics.

Reddy, M. Chandra Sekhara ; Rao, V. Vasudeva ; Sundar, L. Syam 等

Introduction

A liquid is coolant is widely used in heat exchangers and transportation vehicles to prevent the overheating or heat transfer rate of equipments like electronic devices and heat exchangers. However a convectional heat transfer fluid such as water or ethylene glycol generally has poor thermal properties. So many authors are put efforts for dispersing small particles with high thermal conductivity in the liquid coolant have been conducted to enhance the thermal properties of the convectional heat transfer fluids. Choi [1995] and his team developed nano-sized particles and obtained higher thermal conductivity by engineering the particle dispersion in liquids, subsequently the researchers Masuda et al. [1993], Lee et al. [1999], Wang et al. [1999], Eastman et al. [1999, 2001], Das et al. [2003] mostly concentrating on the determination of effective thermal conductivity of nanofluid.

Previous investigations on the convective heat transfer enhancement of nanofluids have been reported as follows: Xuan and Li [2003] have first time presented the empirical correlation for the estimation of Nusselt number in laminar and turbulent flow condition using nanofluids Cu particles. Wen and Ding [2004] observed that [Al.sub.2][O.sub.3] nanoparticles when dispersed in water can significantly enhance the convective heat transfer in the laminar flow regime and the enhancement increases with Reynolds number, as well as particle concentration compared to base fluid. Experiments with [Al.sub.2][O.sub.3]/water nanofluid in the laminar flow range of Reynolds number in the range of 700 and 2050 has been conducted by Heris et al. [2007] under isothermal wall boundary condition and observed enhancements of heat transfer to take place with increase in Peclet number and volume concentration. Pak and Cho [1998] developed the regression for the estimation of Nusselt number in plain tube with [Al.sub.2][O.sub.3], Ti[O.sub.2] nanofluid under turbulent flow condition.

Some researchers are numerically investigated on the convective heat transfer enhancement with nanofluids have been reported as follows: Maiga et al. [2005] numerically investigated the heat transfer of water/[Al.sub.2][O.sub.3] and ethylene glycol/[Al.sub.2][O.sub.3] nanofluid under laminar flow condition. Namburu et al. [2009] have been numerically obtained the heat transfer enhancement with CuO, [Al.sub.2][O.sub.3], Si[O.sub.2] compared to base liquid under turbulent flow condition. Numerical analysis of laminar flow heat transfer of [Al.sub.2][O.sub.3]/ethylene glycol and [Al.sub.2][O.sub.3]/water nanofluids in tube has been reported by Palm et al. [2004] and Roy et al. [2004] and observed wall shear stress to increase with volume concentration and Reynolds number. Putra et al. [2003] reported natural convection heat transfer with [Al.sub.2][O.sub.3]/water and CuO/water.

Experimental results are showing that further heat transfer enhancement of nanofluid in plain tube is possible with inserts (like twisted tape and longitudinal strip). Sharma et al. [2009], Sundar and Sharma [2010a, 2010b] first time presented the empirical correlation for the estimation of Nusselt number and friction factor in transition and turbulent flow condition using water and different volume concentration of [Al.sub.2][O.sub.3] nanofluid.

Nanofluids in circular tube with and without longitudinal strip insert data and empirical correlations are not available in the literature under turbulent flow condition. The present study focuses with the estimation of nanofluids heat transfer and pressure drop with longitudinal strip inserts in a plain tube by using commercially available FLUENT. 6.0. software. The data obtained from the numerical analysis is compared with the data available in the literature.

Thermo-physical properties of nanofluid and geometry

The following equations of Pak and Cho [1998] are used for calculating the thermophysical properties of [Al.sub.2][O.sub.3] and Ti[O.sub.2] nanofluid. The following data of Xuan and Li [2003] is used for the thermo-physical properties of CuO nanofluid.

CuO nanofluid

S.No. Volume Density, Thermal
 fraction, P, conductivity, K,
 kg/[m.sup.3] W/m K
 (%)
1 0.3 1014.22 0.6204
2 1.0 1051.61 0.6466
3 1.5 1078.32 0.6798
4 2.0 1105.03 0.712

S.No. Absolute Specific
 viscosity, [mu] * heat, Cp,
 [10.sup.-3], J/kg K
 N-s/[m.sup.2]

1 1.1152 4170.92
2 1.219 4143.99
3 1.360 4124.98
4 1.5021 4105.98

(Source: Xuan and Li [2003])

[Al.sub.2][O.sub.3] nanofluid

[[rho].sub.nanofluid] = (1 - [phi]) [[rho].sub.base fluid] + [phi] [[rho].sub.particle] (1)

[([C.sub.p]).sub.nanofluid] = (1-[phi]) [([C.sub.p]).sub.base fluid] + [phi][([C.sub.p]).sub.particle] (2)

[[mu].sub.nanofluid] = [[mu].sub.base fluid] (1 + 39.11 [phi] + 533.9 [[phi].sup.2]) (3)

[k.sub.nanofluid] = [k.sub.base fluid] (1 + 7.47 [phi]) (4)

Ti[O.sub.2] nanofluid

[[mu].sub.nanofluid] = [[mu].sub.base fluid] (1 + 5.45 [phi] + 108.2 [[phi].sup.2]) (5)

[k.sub.nanofluid] = [k.sub.base fluid] (1 + 2.920 [phi] - 11.99 [[phi].sup.2]) (6)

Mathematical Modeling

Assumptions

The nanopaticles in the base fluid may be easily fluidized and consequently the effective mixture behaves like a single-phase fluid Xuan and Li [2003]. It is also assumed that the fluid phase and nanoparticles are in equilibrium with zero relative velocity. This may be realistic as nanoparticles are much smaller than micro particles and the relative velocity decreases as the particle size decreases. The resultant mixture may be considered as a convectional single-phase fluid. The thermal and physical properties are temperature dependent under the operating conditions. The effective thermo-physical properties are dependent upon the temperature and volume concentration. Furthermore the assumption for single phase for a nanofluid is validating with the experimental results of Pak and Cho [1998]. Under these assumptions, the classical theory of single-phase fluid can be applied to nanofluid.

Governing equations

The problem under investigation is a three-dimensional steady, forced turbulent convection flow of nanofluid flowing inside a straight circular tube having diameter of 0.00853m inner diameter, 0.00913m outer diameter and length of 1.5m. The fluid enters the circular tube with uniform axial velocity and temperature. Plane the governing equation for the fluid flow are Shih [1984]:

div([rho][bar.V]=0 (7)

div([rho][bar.V][bar.V]) = -grad([bar.P]) + [mu] [[nabla].sup.2] [bar.V] - div([rho] u' u') (8)

div([rho] V [C.sub.p] T) = div (k grad [bar.T] - [rho][C.sub.p] u' t') (9)

In the above equations, the symbols [bar.V], [bar.P] and [bar.T] represent the time averaged flow variables, while the symbols u' and t' represent the fluctuations in velocity and temperature. The terms in the governing equations [rho] u' u' and [rho] [C.sub.p] u' t' represent the turbulent shear stress and turbulent heat flux. The terms are unknown and must be approximately expressed in terms of mean velocity and temperature.

Turbulent modeling

For closure of the governing equations of fluid flow, empirical data or approximate models are required to express the turbulent stresses and heat flux quantities of the related physical phenomenon. In the present numerical analysis, k = [epsilon] turbulent model proposed by Launder and Spalding [1972] was adopted. k = [epsilon] Turbulent model introduces two additional equations namely turbulent kinetic energy (k) and rate of dissipation ([epsilon]). The equations for turbulent kinetic energy (k) and rate of dissipation ([epsilon]) are given by:

div([rho] [bar.V] k)= div {[([mu] + [[mu].sub.t])]/[[sigma].sub.k] grad k} + [G.sub.k] - [rho][epsilon] (10)

div([rho] [bar.V] [epsilon])= div{[([mu] + [[mu].sub.t])]/[[sigma].sub.3] grad [epsilon]} + [C.sub.1[epsilon]] ([epsilon]/k]) [G.sub.k] + [C.sub.2[epsilon]] [rho] ([[epsilon].sup.2]/k) (11)

In the above equations, ([G.sub.k]) represents the generation of turbulent kinetic energy due to mean velocity gradients, ([[sigma].sub.k]) and ([[sigma].sub.[epsilon]]) are effective Prandtl numbers for turbulent kinetic energy and rate of dissipation, respectively; ([C.sub.1[epsilon]]) and ([C.sub.2[epsilon]]) are constants and ([[mu].sub.t]) is the eddy viscosity and is modeled as

[[mu].sub.t]= ([rho][C.sub.[mu]][k.sup.2]/[epsilon]) (12)

Further information is available in Launder and Spalding [1972] and Fluent [2005] for turbulence modeling.

Boundary conditions

The governing equations of the fluid flow are non-linear and coupled partial differential equations, subjected to the following boundary conditions. At the tube inlet section, uniform axial velocity [V.sub.in] temperature [T.sub.in] turbulent intensity and hydraulic diameter have been specified. At the outlet section, the flow and temperature fields are assumed fully developed the flow and temperature fields are assumed fully developed (L/D = 175.84). Outflow boundary condition has been implemented for the outlet section. This boundary condition implies zero normal gradients for all flow variables except pressure. On the upper wall of the tube, the no slip boundary condition was imposed. The wall is subjected to a uniform heat flux.

Results and Discussions

Nusselt number for water, [Al.sub.2][O.sub.3], Ti[O.sub.2] and CuO nanofluid in plain tube

The tube has a diameter of 0.00853m inner diameter, 0.00913m outer diameter and a length of 1.5m. The fluid enters the tube with a constant inlet temperature [T.sub.in] of 300K and uniform axial velocity [V.sub.in]. Constant Wall Heat Flux (CWHF) of 5658W/[m.sup.2] is applied on the outer periphery of the tube and the GAMBIT model with boundary conditions is shown in Fig. 1. The Reynolds number was varied from 2000-25000. In order to validate the computational model, the numerical results were compared with the theoretical data available for the conventional fluids.

[FIGURE 1 OMITTED]

Nusselt number correlations for single phase fluid

General correlations available in the literature for the estimation of Nusselt number of single phase fluid are given below:

Dittus-Boelter [1930] correlation

Nu = 0.023 [Re.sup.0.8] [Pr.sup.0.4] (13)

Gnielinski [1976] correlation

Nu = 0.021 [Re.sup.0.8] [Pr.sup.0.5] (14)

The numerical Nusslet number of water in plain tube at different Reynolds number is shown in Fig. 2 along with the data obtained from the Dittus-Boelter [1930] and Gnielinski [1976] and it observed that numerical Nusselt number is in very good agreement. The numerical analysis is conducted in laminar and turbulent flow condition in both the regions the numerical Nusselt number is close agreement with the literature values.

[FIGURE 2 OMITTED]

The numerical Nusselt number of 1.0% volume concentration of [Al.sub.2][O.sub.3], CuO and Ti[O.sub.2] nanofluid is shown in Fig. 3 along with the data of water. From the fig it is observed that [Al.sub.2][O.sub.3] nanofluid is having high Nusselt number compared to water and other nanofluids under the same Reynolds number and same volume concentration. The thermophysical properties of [Al.sub.2][O.sub.3] nanofluid is high compared to CuO and Ti[O.sub.2] nanofluid under same volume concentration and the properties play vital role for heat transfer augmentation. The numerical Nusselt number of 2.0% volume concentration of [Al.sub.2][O.sub.3], CuO and Ti[O.sub.2] nanofluid is shown in Fig. 4. The Nusselt number of all nanofluids increases with increase of Reynolds number and increase of percentage of volume concentration.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Pressure drop for water, [Al.sub.2][O.sub.3], Ti[O.sub.2] and CuO nanofluid in plain tube

The pressure drop across the tube is measured with the equation of

[DELTA] p = f(L/D)([rho][V.sup.2]/2) (15)

Friction factor correlation for single phase fluid

General correlation available in the literature for the estimation of friction factor of single phase fluid is given below:

Blasius [1908] correlation

f = 0.316 [Re.sup.-0.25] (16)

The friction factor for water obtained from the numerical analysis is shown in Fig. 5 along with the data of Pak and Cho [1998] and Blasius [1908] and it is found that the obtained friction factor is very good agreement.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

The friction factor of [Al.sub.2][O.sub.3], CuO and Ti[O.sub.2] nanofluid at 1.0% volume concentration is shown in Fig. 6 along with the data of base fluid and it is observed that [Al.sub.2][O.sub.3] nanofluid is having high friction factor compared to other nanofluids. The thermo-physical properties of [Al.sub.2][O.sub.3] nanofluid is high compared to CuO and Ti[O.sub.2] nanofluid. The absolute viscosity causes the enhancement in friction factor for all the nanofluids, comparatively [Al.sub.2][O.sub.3] nanofluid is having more. The friction factor of [Al.sub.2][O.sub.3], CuO and Ti[O.sub.2] at 2.0% volume concentration is shown in Fig. 7 along with the friction factor of 1.0% volume concentration. From the figure it is observed that the friction factor increases with increase of Reynolds number and increase of volume concentration.

Nusselt number of water, [Al.sub.2][O.sub.3], Ti[O.sub.2] and CuO nanofluid in plain tube with longitudinal strip inserts

Schematic diagram of the longitudinal strip inserts in a plain tube, prototype of plain tube with longitudinal inserts of aspect ratio, AR = 2 modeled in GAMBIT software is shown in Figs. 8a-8b and dimensions of the inserts is shown in the Table 1. The boundary conditions of plain tube are incorporated for the estimation of Nusselt number and friction factor. The velocity boundary condition is used as inlet boundary condition and the velocity of different aspect ratios of longitudinal strip inserts are calculated based on the hydraulic diameter. The pressure inlet, heat flux boundary condition is used as outlet boundary condition and wall boundary condition.

[FIGURE 8a OMITTED]

[FIGURE 8b OMITTED]

[FIGURE 9 OMITTED]

The obtained Nusselt number from the numerical analysis of [Al.sub.2][O.sub.3], CuO and Ti[O.sub.2] nanofluid in plain tube with different aspect ratios of (AR = 1, 2, and 4) longitudinal strip inserts at 2.0% volume concentration is shown in Fig. 9 along with the plain tube data. The results shown that Nusselt number of [Al.sub.2][O.sub.3] nanofluid in plain tube with longitudinal strip inserts of aspect ratio, AR = 1 is having high Nusselt number compared to the other nanofluids under the same Reynolds number and same percentage of volume concentration.

Friction factor for water, [Al.sub.2][O.sub.3], Ti[O.sub.2] and CuO nanofluid in plain tube with longitudinal strip inserts

Eq. (15) is used to estimate the friction factor of water, [Al.sub.2][O.sub.3], CuO and Ti[O.sub.2] nanofluid at 2.0% volume concentration in plain tube with different aspect ratios (AR = 1, 2 and 4) I shown in the Fig. 10 and it is observed that the friction factor increases with increase of Reynolds number, increase of volume concentration and it is also increases with decrease of aspect ratio of the longitudinal strip inserts. From the figure it is observed that [Al.sub.2][O.sub.3] nanofluid in plain tube with longitudinal strip inserts of AR = 1 is having high friction factor compared to other nanofluids at same Reynolds number and same volume concentration.

[FIGURE 10 OMITTED]

Conclusions

The following conclusions are drawn from the numerical analysis

1. Heat transfer coefficient and friction factor of [Al.sub.2][O.sub.3], Ti[O.sub.2] and CuO nanofluids in circular tube is studied numerically.

2. The Nusselt number of [Al.sub.2][O.sub.3] nanofluid increases 1.69 times, TiO2 nanofluid increases 1.04 times and CuO nanofluid increases 1.147 times to the base fluid respectively at 2.0% volume concentration and at 25,000 Reynolds number.

3. The increase in heat transfer coefficient of [Al.sub.2][O.sub.3] nanofluid increases 1.91 times, Ti[O.sub.2] nanofluid increases 1.072 times and CuO nanofluid increases 1.33 times to the base fluid respectively at 2.0% volume concentration and at 25,000 Reynolds number.

4. Compared to three different nanofluids [Al.sub.2][O.sub.3] nanofluid is having high heat transfer rates.

5. The increase in friction factor of [Al.sub.2][O.sub.3] nanofluid increases 1.091 times, Ti[O.sub.2] nanofluid increases 1.031 times and CuO nanofluid increases 1.54 times to the base fluid respectively at 2.0% volume concentration and at 25,000 Reynolds number.

6. [Al.sub.2][O.sub.3], CuO and Ti[O.sub.2] nanofluids heat transfer coefficient increases with increase of the volume concentration and Reynolds number.

7. Prandtl number of nanofluid increases with decrease in operating temperature, because the viscosity plays a predominant role. Pressure loss increases with increase in the volume concentration of the nanofluid.

References

[1] Choi, S.U.S. (1995) 'Enhancing thermal conductivity of fluid with nanoparticles. In: Siginer, D.A., Wang, H.P. (Eds.), Developments and Applications of Non-Newtonian Flows', FED-V.231/ MD-V.66. ASME, New York, pp. 99-105.

[2] Das, S.K, Putra, N., Thiesen, P., Roetzel, W. (2003) 'Temperature dependence of thermal conductivity enhancement for nanofluids', Journal of Heat Transfer, Vol. 125, pp. 567-574.

[3] Dittus F.W., and Boelter. L.M.K. (1930) 'Heat transfer for automobile radiators of the tubular type', University of California Publications in Engineering, Vol.2, p. 443.

[4] Eastman, J.A., Choi, S.U.S., Li, S., Soyez, G., Thompson, L.J., DiMelfi, R.J. (1999) 'Novel thermal properties of nanostructured materials, Journal of Metastable' Nanocrystal Materials, 2(6), pp. 629-634. [5] Eastman, J.A., Choi, S.U.S., Li, S., Yu, W., Thompson, L.J. (2001) 'Anomalously increase effective thermal conductivities of ethylene glycol based nanofluids containing copper nanoparticles', Applied Physics Letter, 78(6), pp. 718-720.

[6] Gnielinski, V. (1976) 'New equations for heat and mass transfer in turbulent pipe and channel flow', International Chemical Engineering, Vol. 16, pp. 359368.

[7] Heris, S.Z., Esfahany, M.N.,.Etemad, S.Gh. (2007) 'Experimental investigation of convective heat transfer of [Al.sub.2][O.sub.3]/water nanofluid in circular tube', International Journal of Heat and Fluid Flow, Vol. 28, pp. 203-210.

[8] Fluent. 5.0. User Manual.

[9] Launder, B.E., and Spalding, D.B. (1972) 'Mathematical models of turbulence', Academic Press, New York.

[10] Maiga, S.E.B., Palm, S.J., Nguyen, C.T., Roy, G., Galamis, N. (2005) 'Heat transfer enhancement by using nanofluids in forced convention flows', International Journal of Heat and Fluid Flow, Vol. 26, pp. 530-546.

[11] Namburu, P.K., Das, D.K., Tanguturi, K.M., Vajjha, R.K. (2009) 'Numerical study of turbulent flow and heat transfer characteristics of nanofluids considering variable properties', International Journal of Thermal Sciences, Vol. 48, (2), pp. 290-302.

[12] Palm S.J., Roy, G., Nguyen C.T. (2004) 'Heat transfer enhancement in radial flow cooling system-using nanofluid' In: Proceeding of the ICHMT Inter. Symp. Advance Comp. Heat Transfer, Norway, CHT-04-121.

[13] Pak, B.C., Cho, Y.I. (1998) 'Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles', Experimental Heat transfer, Vol. 11, pp. 151-170.

[14] Putra, N., Roetzel, W., Das, S.K. (2003) 'Natural convection of nanofluids', Heat and Mass Transfer, Vol. 39 (8), pp. 775-784.

[15] Roy, G., Nguyen, C.T., Lajoie, P.R. (2004) 'Numerical investigation of laminar flow and heat transfer in a radial flow cooling system with the use of nanofluids', Superlattices and Microstructures, Vol. 35 (3), pp. 497-511.

[16] Sharma K.V., Sundar, L.S., Sarma, P.K. (2009) 'Estimation of heat transfer coefficient and friction factor in the transition flow with low volume concentration of [Al.sub.2][O.sub.3] nanofluid flowing in a circular tube and with twisted tape insert', International Communications in Heat and Mass Transfer, Vol. 36, pp. 503-507.

[17] Shih, T. M. (1984) 'Numerical heat transfer', Hemisphere Publishingcorporation, New York.

[18] Sundar, L.S., Sharma, K.V. (2010a) 'Turbulent heat transfer and friction factor of [Al.sub.2][O.sub.3] nanofluid in circular tube with twisted tape inserts', International Journal of Heat and Mass Transfer, Vol. 53, pp. 1409-1416.

[19] Sundar, L.S., Sharma K.V., (2010b) 'Heat transfer enhancements of low volume concentration [Al.sub.2][O.sub.3] nanofluid and with longitudinal strip inserts in a circular tube', International Journal of Heat and Mass Transfer, Vol. 53 (1920), pp. 4280-4286.

[20] Wang, X., Xu, X., Choi, S.U.S. (1999) 'Thermal conductivity of nanoparticlesfluid mixture', Journal of Thermophysics heat transfer, Vol. 13(4), pp. 474-480.

[21] Wen, D., Ding, Y. (2004) 'Experimental investigation into convective heat transfer of nanofluid at the entrance region under laminar flow conditions', International Journal of Heat and Mass Transfer, Vol. 47 (24), pp. 5181-5188.

[22] Xuan, Y., Li, Q. (2003) Investigation on convective heat transfer and flow features of nanofluids, Journal of Heat Transfer, Vol. 125, pp. 151-155.

* M. Chandra Sekhara Reddy, V. Vasudeva Rao and L. Syam Sundar

Department of Mechanical Engineering, Sreenidhi Institute of Science and Technology, Yamnampet, Ghatkesar, Hyderabad-501 301, A.P., India

* Corresponding Author E-mail: mekalacs@gmail.com

Table 1: Dimensions of longitudinal strip inserts.

S. No. Parameter Aspect Ratio, AR = W/H, m

 1 2 4
1 W (width) 0.01 0.010 0.010
2 H (height) 0.01 0.005 0.0025