Numerical validation of experimental heat transfer

Applied Thermal Engineering 73 (2014) 294e304

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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng

Numerical validation of experimental heat transfer coefficient with SiO2 nanofluid flowing in a tube with twisted tape inserts W.H. Azmi d, 1, K.V. Sharma a, *, P.K. Sarma b, Rizalman Mamat d, 1, Shahrani Anuar d, 1, L. Syam Sundar c a

Department of Mechanical Engineering, Universiti Teknologi PETRONAS, Seri Iskandar, 31750 Tronoh, Perak, Malaysia GITAM University, Rishikonda, Visakhapatnam, India Center for Mechanical Technology and Automation, Department of Mechanical Engineering, University of Aveiro, 3810-193 Aveiro, Portugal d Faculty of Mechanical Engineering, Universiti Malaysia Pahang, 26600 Pekan, Pahang, Malaysia b c

h i g h l i g h t s Experiments are undertaken with SiO2 nanofluid up to 4% volume concentration. Numerical model is developed for nanofluid flow with twisted tape insert for a wide range of Reynolds number. SiO2 nanofluid gives maximum heat transfer coefficient of 94.1% at 3.0% concentration at a twist ratio 5. Coefficient of eddy diffusivity and Prandtl index is obtained as a function of Re, concentration and twist ratio. Numerical results are in good agreement with exp data with SiO2, Al2O3 and Fe3O4 nanofluids.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 27 November 2013 Accepted 23 July 2014 Available online 1 August 2014

A numerical model has been developed for turbulent flow of nanofluids in a tube with twisted tape inserts. The model is based on the assumption that van Driest eddy diffusivity equation can be applied by considering the coefficient and the Prandtl index in momentum and heat respectively as a variable. The results from the numerical analysis are compared with experiments undertaken with SiO2/water nanofluid for a wide range of Reynolds number, Re. Generalized equation for the estimation of nanofluid friction factor and Nusselt number is proposed with the experimental data for twisted tapes. The coefficient and the Prandtl index in the eddy diffusivity equation of momentum and heat is obtained from the numerical values as a function of Reynolds number, concentration and twist ratio. An enhancement of 94.1% in heat transfer coefficient and 160% higher friction factor at Re ¼ 19,046 is observed at a twist ratio of five with 3.0% volumetric concentration when compared to flow of water in a tube. A good agreement with the limited experimental data of other investigators is observed with Al2O3 and Fe3O4 nanofluids indicating the validity of the numerical model for use with twisted tape inserts. © 2014 Elsevier Ltd. All rights reserved.

Keywords: SiO2 nanofluid Twisted tape Numerical model van Driest eddy diffusivity equation Heat transfer coefficient Friction factor

1. Introduction Tape inserts are commonly used for heat transfer enhancement in several applications involving heat recovery, solar heating, air conditioning and refrigeration systems, chemical reactors, etc.

* Corresponding author. Tel./fax: þ60 053687163. E-mail addresses: [email protected] (W.H. Azmi), sharma.korada@ petronas.com.my, [email protected] (K.V. Sharma), [email protected] (P.K. Sarma), [email protected] (R. Mamat), [email protected] (S. Anuar), [email protected] (L. Syam Sundar). 1 Tel.: þ60 9 4246338/fax: þ60 9 4242202. http://dx.doi.org/10.1016/j.applthermaleng.2014.07.060 1359-4311/© 2014 Elsevier Ltd. All rights reserved.

Passive heat transfer augmentation using twisted tapes, longitudinal inserts, wire coil insert, etc for a wide range of Reynolds and Prandtl numbers have been reported by Bergles [1]. The twisted tape causes the flow to swirl, providing longer path and residence time thereby enhancing heat transfer. They are commonly employed in heat exchangers due to ease in their manufacture and low cost. However, the pressure drop with the insert is higher due to reduced flow area and additional resistance offered by the tape surface area when compared to flow in a plain tube. Experimental investigations were undertaken by various investigators with pure fluids [2e9]. Smithberg and Landis [2] undertook experiments with air as the working medium. They

W.H. Azmi et al. / Applied Thermal Engineering 73 (2014) 294e304

observed increased pressure drop due to vortex flow caused by the twisted tape which continuously mixes the fluid flowing at the core and developed a semi-empirical model for the estimation of friction factor. Thorsen and Landis [3] extended the analysis to develop a correlation equation for the estimation of Nusselt number using the experimental data of water. Lopina and Bergles [4] developed a superposition model for the estimation of Nusselt number accounting for increased speed due to tape insert and centrifugal buoyancy due to flow in the tube. They observed an increase of 20% in the Nusselt number with tight fit in comparison to reduced width tapes. Manglik and Bergles [5] conducted experimental investigations with twisted tape inserts using water and ethylene glycol in the turbulent range of Reynolds number. They developed a correlation equation for the estimation of Nusselt number based on an asymptotic model for isothermal condition of the wall. Ayub and Al-Fahed [6] observed the effect of gap between the tube and tape insert on the pressure drop under turbulent flow of water. The pressure drop increased with tape width. Naphon [7] has undertaken experimental studies in the turbulent range for flow in tubes and with twisted tape inserts. A correlation equation was developed which is valid for the Reynolds number and twist ratio in the range of 7000 < Re < 23,000 and 3.1 H/D 5.5. Akhavan-Behabadi et al. [8] conducted experiments with condensing refrigerant R-134a at different mass flow rates and developed correlations for the determination of heat transfer coefficients. Chang et al. [9] developed a set of empirical equations for the estimation of turbulent heat transfer and pressure drop with smooth and serrated twisted tapes. They observed heat transfer enhancements with smooth walled tapes to be greater than with serrated tapes at large twist ratios. The thermal performance at smaller twist ratios with smooth and serrated tapes was similar. The theoretical modeling considered the effect of turbulence due to spiral flow and heat conduction in the tape. Date [10] undertook numerical analysis to predict friction factor and heat transfer for fully developed laminar and turbulent flow. The numerical procedure of Gosman et al. [11] was modified by Date [10] for the convergence of results. Chiu and Jang [12] undertook three dimensional numerical and experimental analysis with tapes of three twist angles and horizontal longitudinal strips with and without holes. They performed the experiments with air as the working medium and observed enhancement in heat transfer coefficient and pressure drop with longitudinal inserts and twisted tapes. The heat transfer coefficients with twisted tapes was higher compared to longitudinal inserts for both types of inserts having holes and without. Herwig and Kock [13] developed a tool from thermodynamic point of view for evaluating heat transfer performance under turbulent flow in a pipe with twisted tape inserts. Turbulence modeling of the flow phenomenon indicated a reduction in the overall entropy production in a certain range of twist ratios when compared to flow in a plain tube. Recent interest in the use of nanofluids for possible heat transfer augmentation has drawn the attention of many investigators [14e25]. Experiments to determine turbulent heat transfer coefficients and pressure drop for flow of nanofluids in a tube has been initiated by Pak and Cho [14]. Computational analysis has been undertaken by Roy et al. [15], Palm et al. [16], Khaled and Vafai [17], Maïga et al. [18], Bianco et al. [19], Fard et al. [20], and Namburu et al. [21] for different nanofluids, particle sizes, operating temperatures in the laminar and turbulent range of Reynolds number. A theoretical model for the evaluation of nanofluid heat transfer coefficient has been presented by Sarma et al. [22] for turbulent flow in a tube. They developed an eddy diffusivity equation of momentum and heat and showed good agreement of

295

numerical results with the experimental data of Al2O3/water nanofluid. Experiments with nanofluids flow over twisted tape in the turbulent range of Reynolds number are due Sundar and Sharma [23]. They conducted experiments with Al2O3/water nanofluid in the range of 10,000 < Re < 22,000 and 0 H/D 83. The results indicated enhancements in heat transfer coefficient and friction factor of Al2O3/water nanofluid to be respectively 33.5% and 109.6% with 0.5% concentration and twist ratio five compared to flow of water in a tube. In another paper, Sundar et al. [24] conducted experiments with Fe3O4/water nanofluid and obtained enhancements of 51.9% and 123.1% in heat transfer coefficient and friction factor respectively compared to flow of water in a tube under similar operating conditions. Chandrasekar et al. [25] conducted experiments with Al2O3/water nanofluid with wire coiled inserts in the laminar Reynolds number range of 600 < Re < 2275. Sekhar et al. [26] conducted experiments with Al2O3 nanofluid and twisted tapes in mixed laminar flow range valid for solar thermal applications. Salman et al. [27] undertook CFD analysis to predict heat transfer and friction factor with Cu/water nanofluid using conventional and parabolic-cut twisted tapes of twist ratio 2.93, 3.91 and 4.89. Numerical analysis for the determination of nanofluid heat transfer coefficient and friction factor with twisted tape insert in the turbulent range of Reynolds number has not been undertaken till now. The turbulence models of discrete zone and continuous type available in the literature are applicable to pure fluids. However, the flow due to the presence of solid particles in a nanofluid can be different from that of pure fluid, as the particles can absorb a portion of the turbulent kinetic energy. Further, the models are applicable to fluid flow in a plain tube. Hence, the turbulence models are to be modified for nanofluid flow over a twisted tape insert, to accommodate for the deviations in the flow characteristics. The model of van Driest with the turbulent characteristics reflected in the eddy diffusivity equation is considered in the analysis. The equation is applicable as the nanofluids considered in the range of concentration are assumed homogenous. However, the coefficient K and the index of Prandtl number x in the equation of momentum and heat respectively is treated as variables, as the nanofluid characteristics deviate from that of a pure liquid. The experimental data of heat transfer coefficients and pressure drop available in the literature for flow over twisted tape insert is limited for nanofluids. Hence, experiments are undertaken with SiO2/water nanofluid for a wide range of Reynolds number and concentration. The results from the numerical model for different operating conditions are compared with the experimental data and for predicting the flow characteristics. 2. Properties of nanofluids The properties of SiO2 nanofluid at different concentrations are required in the analysis. The thermal conductivity and viscosity of nanofluid is determined using KD2 Pro thermal property analyzer and Brookfield LVDV-III Ultra Rheometer respectively. The experimental values of viscosity and thermal conductivity available in the literature are used in the development of Eqs. (1) and (2) by Sharma et al. [28]. The values of viscosity and thermal conductivity of SiO2 nanofluids are observed to be in good agreement with the equations given by:

mr ¼

0:038 dp 0:061 mnf f 11:3 T 1 þ nf 1þ ¼ 1þ 100 mw 70 170

(1)

296


kr ¼

0:2777 knf f 1:37 T ¼0:8938 1 þ 1 þ nf 100 kw 70 dp 0:0336 ap 0:01737 1þ 150 aw

(2)

The equations for density and specific heat of nanofluid, based on mixture relation used in the analysis, is given as.

rnf ¼ 4rp þ ð1 4Þrw Cnf ¼

(3)

ð1 4ÞðrCÞw þ 4ðrCÞp

(4)

ð1 4Þrw þ 4rp

Equations (1)e(4) are used in the numerical analysis to determine the properties along with the property relations of water given in Table 1.

outlet, and the surface temperatures are recorded under steady state condition. The experimental values of friction factor and heat transfer coefficient are evaluated using Darcy pressure drop and Newton's law of cooling equations, respectively. The maximum errors in the experimental data for friction factor and heat transfer coefficient are 4.4% and 0.92%, respectively. The uncertainty analysis is detailed by Azmi et al. [29]. The reliability and repeatability of the values with water is ensured before undertaking experiments with SiO2 nanofluid in the concentration range of 0.5e4.0% at various Reynolds number. About 100 ml of nanofluid is drawn before and after the conduct of experiment at every concentration to evaluate the state of dispersion by measuring its electrical conductivity. The experimental setup and the procedure undertaken for the determination of friction factor and heat transfer coefficient for flow of water and nanofluid in a tube is detailed by Azmi et al. [30]. The experimental procedure is repeated for water and nanofluid with tapes of twist ratios 5, 10 and 15 inserted in the tube.

3. Experimental procedure 4. Formulation of the mathematical model The experimental setup is integrated with a circulating pump, flow meter, heater, control panel, thermocouples, pressure transducer, chiller, collecting tank, and the test section. The heaters enclose a copper tube of 1.5 m with inner diameter and outer diameter respectively, 16 mm and 19 mm which constitutes the test section. The total length of fluid flow in the tube is approximately 4.0 m which ensures fully developed turbulent flow conditions. The schematic diagram of the experimental setup Fig. 1a presented by Azmi et al. [29] is shown along with the twisted tape configuration Fig. 1b. A 0.5 horse power pump connected to a collecting tank of 0.03 m3 capacity is used to circulate the working fluid through the test section. The outer diameter of the test section is wrapped with two nichrome heaters each of 1500 W rating. The tube is enclosed with ceramic fiber insulation to minimize heat loss to the surroundings. Seven K-type thermocouples are fixed at different locations, five on the surface of the tube wall at 0.25, 0.5, 0.75, 1.0 and 1.25 m from the inlet and the other two are located at the inlet and outlet to measure the temperatures of the working fluid. A flow meter which works in the range of 5e16 LPM is connected between the pump and the inlet to test section. A chiller of 1.4 kW rating is located between the test section and the collecting tank. A constant power of 600 W is supplied to the heater, while the chiller is adjusted to obtain a fluid bulk temperature of 30 C with a deviation of ±1 C at all flow rates and nanofluid concentration. A pressure transducer connected across the test section records the pressure drop. A data logger records the surface and fluid temperatures every five seconds to determine the steady state nature of the experiment. The uncertainties in the measuring instruments are given by Azmi et al. [29]. With the experimental setup, tests are undertaken with distilled water to determine the pressure drop and heat transfer coefficients at various flow rates. The flow rate, the pressure drop across the length of the tube, the temperatures of the fluid at the inlet and

The modeling of turbulence due to twisted tape insert using eddy diffusivity equation has been initiated by Sarma et al. [31]. They included the effect of centrifugal forces and other secondary flows due to tape insert by considering the constant K in the eddy diffusivity Eq. (5) of van Driest [32] as a function of twist ratio and Reynolds number. The damping constant Aþ in Eq. (5) remained unaltered at 26 for flow over twisted tapes. The analysis employed the friction factor relation of Smithberg and Landis [2] in the estimation of heat transfer coefficient. They obtained a good agreement of the numerical results with their experimental values and that of others. The turbulence characteristics of fluid flow with twisted tape insert in a tube has been presented with regression Eq. (6) undertaken with the numerical data.

oi2 vuþ εm h þ n ¼ Ky 1 exp yþ =Aþ n vyþ K ¼ 3:3891Re0:0914

0:3737 H D

and

(5)

Aþ ¼ 26

(6)

Equation (6) is valid in the range of 10,000 < Re < 100,000 and 4 < H/D < 15 and applicable to pure fluids. The present analysis is undertaken for nanofluid flow over tapes. The numerical procedure of Sarma et al. [22] is adapted in the evaluation of the turbulent characteristics. The salient assumptions made in the development of the mathematical model are listed. a) The SiO2/water nanofluid behaves as a homogenous fluid having uniform properties. b) The properties of the nanofluids can be estimated with Eqs. (1)e(4) and that of water given in Table 1. c) The total momentum exchange of the nanofluid is due to molecular and eddy viscous forces given by

Table 1 Regression equations for the estimation of water properties. Properties Density [36] Viscosity Thermal conductivity Specific heat

Regression equation " rw ¼ 1000 1:0

#

ðTw 4:0Þ2 119000þ1365Tw 4ðTw Þ2

mw ¼ 0.00169 4.25263e 5 Tw þ 4.9255e 7 (Tw)2 2.0993504e 9 (Tw)3 kw ¼ 0.56112 þ 0.00193 Tw 2.60152749e 6 (Tw)2 6.08803e 8 (Tw)3 Cw ¼ 4217.629 3.20888 Tw þ 0.09503 (Tw)2 0.00132 (Tw)3 þ 9.415e 6 (Tw)4 2.5479e 8 (Tw)5


297

Fig. 1. (a) Schematic diagram of the experimental setup of Azmi et al. [29]. (b) Configuration of twisted tape of Azmi et al. [29].

" t ¼ mnf 1 þ

εm nnf

#

vu vy

(7)

d) The viscosity of the nanofluid is observed to be independent of the shear rate and hence Newtonian theory is applicable for concentration less than 4.0%. The shear distribution across the tube is assumed to be linear given in non dimensional form as

" # t yþ ¼ 1 þ tw R

(8)

! where K is a constant

(10)

g) The eddy diffusivity of heat is related to momentum through the relationship

" # x εH εm Prnf ¼ anf nnf

(11)

where, x ¼ f(f, Rþ and H/D) h) The influence of nanoparticle in the base liquid can be assessed, since it is established for pure fluids

e) The thermal exchange is due to a combination of molecular and eddy diffusivities given by

vT q ¼ rnf Cnf anf þ εH vy

εm vuþ ¼ f Kyþ uþ ; K 2 yþ2 þ nnf vy

or

q ¼ knf 1 þ

εH anf

!

vT vy

" # εH εm Prf for Prf > 1 ¼ af nf

(12)

(9) 4.1. Nanofluid friction factor

f) By dimensional reasoning, the eddy viscosity can be assumed as a function of

Experimental values of friction factor for flow of SiO2 nanofluid in a tube with tape inserts are determined. A regression equation is

298


developed with the present data of water, SiO2 and the data of Azmi et al. [29] with TiO2 with an average deviation of 5.0%, standard deviation of 5.7% and a maximum deviation of 16.4% given by.

fnf f 0:05 ¼ 1:4 0:001 þ 100 fSL 1:2 H Rep n where fSL ¼ 4 0:046 þ 2:1 0:5 D 2þp " 0:5 # H where n ¼ 0:2 1 þ 1:7 D

(13)

(13a)

valid in the range of 6800 < Re < 30,000, 5.00 Pr 7.24, f 4.0% and 5 H/D 15. The experimental values of nanofluid friction factor are in good agreement with the values estimated with Eq. (13) in the turbulent range as shown in Fig. 2. 4.2. Nanofluid Nusselt number

Fig. 2. Validation of experimental data with Eq. (13).

The Nusselt number estimated with the experimental values of SiO2 nanofluid at different concentrations is subjected to regression using 323 data points. Equation (14) is obtained with an average deviation of 4.1%, standard deviation of 5.1% and maximum deviation of 15.3% given by.

Nunf ¼

hnf D D 1:3 0:4 1þ ¼ 0:073Re0:702 Prnf knf H

(14)

4.3. Evaluation of momentum eddy diffusivity Equations (10) and (11) are solved based on the assumption that nanofluid obeys Newtonian relation between shear stress and the rate of shear deformation to determine the value of K in eddy diffusivity Eq. (5). Equation (8) with the aid of Eq. (7) can be written in non dimensional form as.

1

ZRþ Re ¼ 4

. uþ 1 yþ Rþ dyþ

3. The friction factor is estimated from the relation

"

fth

Rþ ¼8 Re

#2 (18)

4. The value of K from the numerical analysis is accepted if the deviation from the relation [(fnf fth) 100/fnf] is less than 0.001 for the given input value of Rþ. If the condition is not satisfied, linear extrapolation is employed for evaluating the value of K till such time the criterion is satisfied. Steps 1 to 4 are repeated and the computations carried out for the Reynolds number range of 3000 < Re < 100,000. The variation

! ! yþ εm = 1 þ for the condition uþ ¼ 0 at yþ ¼ 0; Rþ ynf (15)

Combining Eqs. (5) and (15), the velocity profile can be obtained from.

duþ ¼ dyþ

(17)

0

valid in the range of 6800 < Re < 30,000, 5.00 Pr 7.24, f 4.0% and 5 H/D 15. The Nusselt number estimated with Eq. (14) is in good agreement with the experimental data of water and nanofluid as shown in Fig. 3.

duþ ¼ dyþ

2. The dimensionless Reynolds number for flow of nanofluid in a tube is given by

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u u

2 yþ t 2 þ2 þ þ 1 þ 1 þ 4 K y 1 exp y A 1 Rþ

2 2 K 2 yþ2 1 exp yþ Aþ (16) þ

The value of A ¼ 26 in Eq. (16) given by van Driest [32] is retained for nanofluid also. The assumption is made due to homogeneous nature of the nanofluid. The numerical procedure adopted by Sarma et al. [22] is reiterated in brief for clarity. Steps in the evaluation of K: 1. The velocity profile can be obtained from Eq. (16) for an assumed initial value of K as 0.03 till yþ / Rþ.

Fig. 3. Validation of experimental data with Eq. (14).


299

of friction factor with Reynolds is compared with other investigators [4,5,29] as shown in Fig. 4.

4.4. Evaluation of eddy diffusivity of heat Equation (11) relates eddy diffusivity of momentum with heat. The value of Prandtl index x in Eq. (11) is dependent on temperature profiles across the tube. The temperature profile depends on nanofluid Prandtl number in addition to Reynolds number and twist ratio H/D. With tape insert, the flow is both hydro dynamically and thermally developed. Hence, neglecting the convective component, the energy equation considering eddy conduction in non dimensional form can be written as.

v vyþ

"

εm x Pr 1þ nnf

!

# vT þ ¼ 0 where T þ ¼ ðT TC Þ=ðTW TC Þ vyþ (19)

with the boundary conditions that at yþ ¼ 0, Tþ ¼ 1, and yþ ¼ Rþ, Tþ ¼ 0. The Nusselt number is estimated from the relation.

Nuth ¼ 2Rþ

vT þ ðTW TC Þ þ vy yþ ¼0 ðTW TB Þ

(20)

The temperature correction term (T W TC)/(T W TB) is evaluated along with the velocity and temperature profiles with the relation.

Z ðTW TC Þ ¼Z ðTW TB Þ 0

Rþ

Rþ 0

. uþ 1 yþ Rþ dyþ

. uþ 1 T þ 1 yþ Rþ dyþ

Fig. 5. Comparison of Nusselt number with other investigators for water.

6. Equation (21) is used for obtaining the temperature correction factor. 7. For a given value of Rþ, H/D and f, the absolute value of Nusselt number estimated with Eq. (20) is referred as Nuth. 8. The experimental value of Nusselt number is referred as Nuexp, determined from the correlation Eq. (14). 9. The Prandtl index ‘x’ is accepted, if the deviation between the values obtained from theory and experiment calculated with the relation [(Nuexp Nuth) 100/Nuexp] is less than 0.001. Otherwise, linear extrapolation is employed for evaluation of' ‘x’ till such time the criterion is satisfied.

(21)

Steps in the evaluation of Prandtl index x: The procedure for obtaining the velocity profile is outlined in section 4.3. The subsequent analysis is a conjoint with the previous program for the determination of K.

Steps 6 to 9 are repeated and the computations in the Reynolds number range of 3000 < Re < 100,000 carried out to determine Prandtl index x for various values of concentration f and twist ratio H/D. The variation of Nusselt number with Reynolds is compared with other investigators [4,5,29] in Fig. 5. 5. Results and discussion

5. Equation (19) is solved with the initial value of x ¼ 1 (applicable for water) and the temperature gradient vT þ =vyþ yþ ¼0 estimated for a given twist ratio.

Experiments are undertaken in the turbulent range of Reynolds number with water and SiO2 nanofluid for flow in a tube with

Fig. 4. Comparison of friction factor with other investigators for water.

Fig. 6. Experimental friction factor for flow in tube and with twisted tape.

300


Fig. 7. Comparison of experimental friction factor with numerical results.

Fig. 9. Temperature profiles of water and nanofluid for different H/D ratios.

twisted tape inserts and compared with the numerical results. The values of friction factor and Nusselt number estimated with Eqs. (13) and (14) are compared with other investigators as shown in Figs. 4 and 5 for water at a twist ratio 9 and Pr ¼ 5. A satisfactory agreement between the investigators can be observed. The deviation amongst the values obtained by the investigators for friction factor and Nusselt number may be due to different values of pipe and tape roughness used by them. The experimental values of friction factor for flow of water and nanofluid in a tube and with tape insert of twist ratio 5 is shown in Fig. 6. Evidently the values of friction factor obtained with tape inserts are greater than the values obtained with a plain tube. The experimental values of friction factor are compared for water and nanofluid with the numerical results in Fig. 7 for two values of twist ratio, H/D ¼ 5 and 10. A good agreement of the experimental data with the numerical results can be observed. The variation of dimensionless velocity and temperature with non-dimensional radial distance is shown in Figs. 8 and 9 respectively at a Reynolds number of 25,000. The velocity profiles for water and nanofluid volume concentration of 4.0% is shown for

three twist ratios in Fig. 8. It can be observed that lower velocities are obtained with nanofluid (shown as lines 2a, 2b and 2c) compared to flow of water at all twist ratios. This may be due to particle drag on the bulk movement of the fluid inducing resistance to flow. As the twist ratio increases, the flow characteristics approach that of a plain tube with greater velocities. The local temperature variation of water and 4.0% nanofluid concentration is shown for twist ratios H/D ¼ 5 and 25 in Fig. 9. The thermal conductivity and diffusivity of nanofluid is greater than water which might be reason for obtaining lower temperatures. It can be observed that the temperature of water and nanofluid does not vary significantly with twist ratio. The variation can be observed in the temperature gradients of water and nanofluid for a twist ratio. A decrease in twist ratio increases turbulence and hence lower temperatures can be expected with nanofluid and water. The characteristics of turbulence reflected through the eddy diffusivity values is shown for water and nanofluid for 4.0% concentration in Fig. 10 for twist ratios of H/D ¼ 5, 10 and 25. The eddy diffusivity increases with concentration due to increase in the turbulence caused by greater number of nanoparticles. An increase

Fig. 8. Velocity profiles of water and nanofluid for different H/D ratios.

Fig. 10. Variation of eddy diffusivity with radial distance.


Fig. 11. Variation of K in the eddy diffusivity expression of van Driest with Reynolds number for 5 H/D 25 and 0 f 4.0.

in eddy diffusivity can be observed with a decrease in twist ratio as shown in Fig. 10 for Re ¼ 25000. The values are higher for nanofluid compared to that of water at all twist ratios. The trend supports the enhancement in shear resistance and wall heat flux of nanofluid in comparison to water. The coefficient of eddy diffusivity K in van Driest Eq. (5) varies with Reynolds number, nanofluid concentration and twist ratio as shown in Fig. 11. The values of K for water are lower than nanofluid at a twist ratio. The value decreases with increase in twist ratio for water and nanofluid. An increase in the value of twist ratio H/D to 25, the characteristics of flow approach that of a plain tube. This is evident from K approaching a constant value of 0.4 which is applicable to single phase fluids such as water. The variation of the index of Prandtl number x in the eddy diffusivity of heat with Reynolds number for the three twist ratios is shown in Fig. 12. It can be observed that x increases with twist ratio and does not vary significantly with the Reynolds number. The values increase with a decrease in nanofluid concentration at a twist ratio. The values of x approach 1.0 for water at Re ¼ 10,000,

Fig. 12. Variation of Prandtl exponent x with concentration and twist ratio, H/D.

301

Fig. 13. Validation of coefficient K and Prandtl exponent x.

when the twist ratio is enhanced from 5 to 15, which is in agreement with the value of 0.91 proposed by Cebeci [33]. Based on the validation of experimental values, the numerical data of water and nanofluid is subjected to regression in two forms; as a function of (Re, f and H/D) and (Rþ, f and H/D) in the development of equations for coefficient K in Eq. (5) and the index of Prandtl x in Eq. (11). The equations obtained are given by.

f 0:04767 H 0:3741 K ¼ 4:957Re0:0936 0:001 þ 100 D

(22)

0:1097 f 0:05039 H 0:3956 0:001 þ K ¼ 4:318 Rþ 100 D

(23)

f 0:08542 H 0:6527 0:1 þ x ¼ 0:1156Re0:04084 0:001 þ 100 D (24)

Fig. 14. Comparison of SiO2 Nusselt numbers with numerical results for H/D ¼ 5.

302


Fig. 15. Comparison of SiO2 Nusselt numbers with numerical results for different twist ratios at f ¼1.5%.

0:0509 f 0:0842 H 0:6427 0:001 þ 0:1 þ x ¼ 0:1115 Rþ 100 D (25) Equations (22) and (23) are obtained with an average deviation of 2.2% and standard deviation of 3.0%. Equations (24) and (25) are obtained with an average deviation of 5.5% and standard deviation of 7.1%. The Eqs. (22)e(25) are valid in the range of 0 f 4.0%, 200 Rþ 4000, 3000 Re 100,000, 5 H/D 15. The values estimated with Eqs. (22)e(25) are validated with the results from theory as shown in Fig. 13. The experimental data of Nusselt number obtained at various concentrations for H/D ¼ 5 is enclosed by the lines drawn with f ¼ 0 and f ¼ 4.0% in Fig. 14. It can be observed that the experimental Nusselt number increase with concentration up to 3.0% and decrease thereafter. A decrease in Nusselt number is also observed at other H/D ratios of 10 and 15, when the concentration is increased to more than 3.0%. Similar observation of decrease in

Fig. 17. Comparison of Fe3O4 Nusselt numbers with numerical results for H/D ¼ 5.

Nusselt number with TiO2/water nanofluid by Duangthongsuk and Wongwises [34] at 1.0% concentration and Al2O3/water by Pak and Cho [14] at 2.78% concentration at a bulk temperature of 25 C has been observed. The numerical values of Nusselt number estimated for SiO2 nanofluid are shown in comparison with the experimental data for three twist ratios in Fig. 15. A good agreement between the theory and experiment is obtained for nanofluid volume concentration of f ¼ 1.5% for the twist ratios undertaken. The numerical model is further validated with the experimental data of Sundar and Sharma [23] and Sharma et al. [35] undertaken with Al2O3/water nanofluid at 0.5% concentration for different H/D ratios in Fig. 16. The experimental data with Fe3O4/water nanofluid undertaken by Sundar et al. [24] at 0.6% concentration is shown in comparison with the numerical results in Fig. 17 for H/D ¼ 5. The numerical results are in good agreement with the experimental values of Nusselt numbers shown through Figs. 14e17, thus validating the model and the reliability of the numerical results. 6. Conclusions The experimental data undertaken for flow over twisted tape inserts with SiO2 nanofluid are in good agreement with the numerical results. The property Eqs. (1)e(4), along with Eqs. (13) and (14) respectively for friction factor and Nusselt number employed in the analysis, predicts Al2O3 and Fe3O4 experimental data of Sundar and Sharma [23], Sundar et al. [24] and Sharma et al. [35] thus validating the model and supporting the generality of the equations presented. The coefficient K and the index of Prandtl x in the eddy diffusivity equation of momentum and heat respectively can be estimated with Eqs. (22)e(25). The experimental results indicate 29.6% enhancement in heat transfer coefficient and 23.7% greater friction factor at 3.0% concentration when compared to water at a twist ratio of 5.0. The values are 94.1% and 160% greater compared to flow of water in a tube. The heat transfer coefficients decrease when undertaken for concentrations greater than 3.0% at the nanofluid bulk temperature of 30 C. Acknowledgements

Fig. 16. Comparison of Al2O3 Nusselt numbers with numerical results for different twist ratios.

The financial support by Universiti Malaysia Pahang under GRS100354 and RDU130391 are gratefully acknowledged. The corresponding author thanks the Jawaharlal Nehru Technological University Hyderabad for the academic support rendered in this regard.


303

Nomenclature

References

Aþ C CFD dp D f fSL h H ID k K kr L LPM Nu OD Pr q Q R Rþ Re T Tþ u uþ u* V V y yþ

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constant in van Driest expression specific heat, (J/kg K) computational fluid dynamics diameter of nanoparticle, (nm) tube inner diameter, (m) 2 Darcy friction factor, ððD=LÞð2DP=rV ÞÞ friction factor Eq. (13a) of Smithberg and Landis heat transfer coefficient, (W/m2 K) helical pitch of the twisted tape for 180 rotation, m inner diameter, mm thermal conductivity, (W/m K) coefficient in eddy diffusivity equation of van Driest thermal conductivity of nanofluid to water ratio, (knf/kw) tube length, (m) liter per minute Nusselt number, (hD/k) outer diameter, mm Prandtl number, (mC/k) heat flux, Q/(pDL), (W/m2) heat input, (W) radius of the tube, (m) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dimensionless radius, ððR=nÞ ðtW =rÞÞ Reynolds number, ðrVD=mÞ temperature, ( C) non-dimensional temperature, (T W T/T W TC) velocity, (m/s) non-dimensional ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (u/u*) pvelocity, shear velocity, ð ðtW =rÞÞ; (m/s) input voltage, (V) average velocity, m/s distance measured normal to the wall, (m) dimensionless distance measured normal to the wall, (yu*/n)

Greek symbols thermal diffusivity, (k/rCp) (m2/s) thickness of strip, (m) pressure drop, (Pa) εH thermal eddy diffusivity, (m2/s) εm momentum eddy diffusivity, (m2/s) F volume concentration, (%) 4 volume fraction, 4 ¼ (F/100) m absolute viscosity, (kg/m s) mr ratio of nanofluid to water viscosity, (mnf/mw) r density, (kg/m3) t shear stress, (N/m2) n kinematic viscosity, (m2/s) x Prandtl exponent

a d DP

Subscripts B mean bulk C central axis of the tube exp experiment f base fluid nf nanofluid p particle r ratio reg regression th theory w water W wall

304


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