3d numerical investigation of thermal characteristics of nanofluid flow through helical tubes using...

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3D Numerical Investigation of Thermal Characteristicsof Nanofluid Flow through Helical Tubes Using Two-Phase Mixture ModelSajjad Ahangar Zonouzia, Habib Aminfarb & Mousa Mohammadpourfardc

a Department of Mechanical Engineering, Razi University, Kermanshah, Iranb Faculty of Mechanical Engineering, University of Tabriz, Tabriz, Iranc Department of Mechanical Engineering, Azarbaijan Shahid Madani University, Tabriz, IranAccepted author version posted online: 03 Sep 2014.Published online: 03 Nov 2014.

To cite this article: Sajjad Ahangar Zonouzi, Habib Aminfar & Mousa Mohammadpourfard (2014) 3D Numerical Investigationof Thermal Characteristics of Nanofluid Flow through Helical Tubes Using Two-Phase Mixture Model, International Journal forComputational Methods in Engineering Science and Mechanics, 15:6, 512-521, DOI: 10.1080/15502287.2014.952847

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International Journal for Computational Methods in Engineering Science and Mechanics, 15:512–521, 2014Copyright c© Taylor & Francis Group, LLCISSN: 1550-2287 print / 1550-2295 onlineDOI: 10.1080/15502287.2014.952847

3D Numerical Investigation of Thermal Characteristicsof Nanofluid Flow through Helical Tubes Using Two-PhaseMixture Model

Sajjad Ahangar Zonouzi,1 Habib Aminfar,2 and Mousa Mohammadpourfard3

1Department of Mechanical Engineering, Razi University, Kermanshah, Iran2Faculty of Mechanical Engineering, University of Tabriz, Tabriz, Iran3Department of Mechanical Engineering, Azarbaijan Shahid Madani University, Tabriz, Iran

This article presents numerical investigation of the heat trans-fer behavior of Al2O3/water nanofluid flow through helical tubes.Temperature-dependent thermal conductivity and the Brownianmotion of nanoparticles in the nanofluid have also been taken intoaccount. The results obtained indicate that the addition of nanopar-ticles into the base fluid will enhance the overall heat transfer co-efficient in helical tubes. It was also concluded that the amountof increase in overall heat transfer coefficient with the addition ofnanoparticles to the base fluid in a helical tube with smaller pitchcircle diameter is more than a helical tube with higher pitch circlediameter.

Keywords Helical tube, Mixture model, Nanofluid, Mixed convec-tion, Constant temperature

1. INTRODUCTIONThe increment of heat transfer coefficient is one of the most

important technical aims for industries and research. It has beenwidely reported in the literature that heat transfer in helicalcoils is much higher as compared to a straight tube due toa complex flow pattern inside these tubes. Thus, due to theircompact structure and higher heat transfer coefficient, helicallycoiled tubes are widely used in industrial applications such as forheat exchangers, refrigeration, power generation, and chemicalreactors.

The characteristics of flow (i.e., pressure drop and heat trans-fer of helical tubes) have been investigated by many researchers,

Address correspondence to Mousa Mohammadpourfard, De-partment of Mechanical Engineering, P.O. Box 3751–71379,Azarbaijan Shahid Madani University, Tabriz, Iran. E-mail: [email protected]

Color versions of one or more of the figures in the article can befound online at www.tandfonline.com/ucme.

both numerically [1–4] and experimentally [5–8]. The effect oftorsion on the flow in a helical tube of circular cross-section wasexperimentally investigated by Yamamoto et al. [9] for a rangeof Reynolds numbers from about 500 to 20,000. In addition, theheat transfer enhancement in helical coil systems was reportedby Prabhanjon et al. [10] and Berger et al. [11]. Jayakumar et al.[12] investigated thermal hydraulic characteristics of air-watertwo-phase flows in helical pipes and evaluated the effects of thegeometrical parameters on thermal and hydraulic behavior ofturbulent flow.

Another method for improvement of heat transfer efficiencyis the use of nanofluids instead of commonly used fluids suchas water, ethylene glycol, and engine oil, all of which haverelatively low thermal conductivities. Nanofluids are promisingheat transfer fluids containing a small quantity of solid nano-sized particles (usually less than 100 nm) that are uniformlyand stably suspended in a liquid. The dispersion of a smallamount of solid nanoparticles in conventional fluids changestheir thermal conductivity remarkably. Many investigations havealso been carried out on the thermal conductivity improvementof nanofluids in recent years. Heris et al. [13] investigated thelaminar flow of Al2O3(20 nm)/water nanofluid and, instead ofthe commonly analyzed constant wall heat flux boundary con-dition, they considered the constant wall temperature boundarycondition. Sasmito et al. [14] showed that the addition of smallamounts of nanoparticles, up to 1%, significantly improves theheat transfer performance of coiled tubes.

Some researchers also investigated the effect of temperatureon thermal conductivity [15–19], and they showed the depen-dency of thermal conductivity to temperature so that takingvariable thermal conductivity into account in nanofluid analysissignificantly improved the accuracy. Variations of temperatureaffect the Brownian motion of nanoparticles, which is the re-sult of the random motion of nanoparticles and leads to dramaticchanges of thermal conductivity of nanofluids with temperature.

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NANOFLUID FLOW THROUGH HELICAL TUBES 513

Many models were developed for the determination of thermalconductivity of nanofluids based on the Brownian motion ofnanoparticles.

Jang and Choi [20] modeled the thermal conductivity ofnanofluids as a function of not only thermal conductivities ofthe base fluid and nanoparticles, but also as a function of the tem-perature and size of the nanoparticles. Xu et al. [21] and Koo andKleinstreuer [22] proposed another thermal conductivity modelfor nanofluids based on the Brownian motion of nanoparticlesand composed of a static and a dynamic part. Chon et al. [23]investigated the thermal conductivity of Al2O3/water nanofluidexperimentally for nanoparticle sizes ranging between 11 nmand 150 nm and temperature validity ranges of 21 to 71◦C.They showed that there is a nonlinear increment of thermal con-ductivity with temperature and higher thermal conductivity withsmaller particle sizes.

The present numerical study is carried out for six varioushelical geometries (see Figure 1), four different nanoparticlevolume fractions, and three different flow rates. This work isinvestigated under constant wall temperature boundary condi-tion, which can have applications in heat exchangers with phasechange, such as condensers. Gravity effects were also taken intoaccount in this study. Since gravitational force is exerted in thevertical direction and the inlet of the tube considered to be atthe top of the tube, gravity will help the enhancement of heattransfer in this case. Simultaneous effects of centrifugal force,gravitational force, and nanoparticle volume fraction in the basefluid on heat transfer augmentation are studied.

The main aim of this study is to investigate the thermalbehavior of nanofluids’ flow in helical tubes in different particlevolume fractions and mass flow rates to evaluate the effectsof geometrical parameters of helical tubes on the heat transferperformance.

2. THEORETICAL FORMULATION

2.1 Governing EquationsThe dimensional conservation equations for steady-state con-

dition are as follows:Continuity equation:

∇ · (ρm�vm) = 0. (1)

Momentum equation:

∇ · (ρm�vm�vm) = −∇p + ∇ · (μm∇�vm) + ∇ · (αPρP�vdr,p�vdr,p)

−ρm,0(T − T0)βm�g (2)

Energy equation:

∇ · [(αpρP cp,p�vp + (1 − αp)ρf cp,f �vf )T ] = ∇ · (km∇T ) (3)

Volume fraction:

∇ · (αpρP �vm) = −∇ · (αpρP �vdr,p) (4)

where

�vm = αpρP �vp + (1 − αp)ρf �vfρm

(5)

�vdr,p = �vp − �vm (6)

are the mass-averaged velocity and drift velocity, respectively,and αpis the volume fraction of nanoparticles.

A two-phase mixture model can be used for two-phase flowswhere the phases are strongly coupled and move at differentvelocities. The model solves the continuity, the energy, and themomentum equation for the mixture. The momentum equationfor the mixture is the sum of the momentum equations for theindividual phases. The volume fraction equation is the continuityequation for the particles phase. Clearly, the volume fractionequation and the other equations of the flow are coupled. Theslip velocity is defined as the velocity of a secondary phase (p)with respect to the velocity of the primary phase (f):

�vpf = �vp − �vf . (7)

The drift velocity is related to the slip velocity

�vdr,p = �vpf − αpρP

ρm

(�vf − �vp). (8)

Considering Stokes drag coefficient and forces act on a singleparticle, the slip velocity is defined similar to Jafari et al. [24].∑ �Fp = 0 → (ρp − ρf )Vp�g − 3πμf dp�vpf = 0 (9)

�vpf = ρpd2p

18μf

ρp − ρf

ρp

�g. (10)

Thermophysical properties of nanofluids:The mixture physical properties in the above equations are

used as follows:Mixture density:

ρm = αpρp + (1 − αp)ρf (11)

Mixture dynamic viscosity [25]:

μm =(

1 + 5

2αp

)μf (12)

Mixture thermal expansion coefficient [26]:

βm =⎡⎣ 1

1 + (1−αp)ρf

αpρP

βp

βf

+ 1

1 + αpρP

(1−αp)ρf

⎤⎦ βf (13)

Mixture specific heat:

cp,m = αpρpcp,p + (1 − αp)ρf cp,f

αpρp + (1 − αp)ρf(14)

Mixture thermal conductivity:In constant wall temperature cases, the importance of taking

the variations of thermal conductivity and thermal dispersioninto account in nanofluid heat transfer analysis is emphasized.

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514 S. A. ZONOUZI ET AL.

FIG. 1. (a-f) Schematic geometrics of physical models; (g) boundary conditions; (h) grid.

The thermal conductivity equation used here for nanofluid hasbeen developed by the Koo and Kleinstreuer model [22] and isconsidered to be composed of two parts, in which the first partincludes Maxwell’s theory [27] and the second part takes theeffect of Brownian motion into effect.

knf = kp + 2kw − 2(kw − kp)αp

kp + 2kw + (kw − kp)αpkw

+ 5 × 104βαpcp,wρw

√κT

ρpdp

f (T , αp) (15)

In the above formula, β and f(T , αp

)are dependent on the

temperature and volume fraction of nanoparticles and are givenby:

β = 0.0017(100αp)−0.0841 (16)

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1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

288 298 308 318 328 338 348

k nf/k

bf

Temperature(K)

Exp (1%)Exp (4%)1%2%3%4%

FIG. 2. Thermal conductivity ratio results of the Koo and Kleinstreuer modelas a function of temperature at various values of particle volume fractions andtheir comparison with experimental results of Chon.

f (T , αp) = (−0.5017αp + 0.058)T + (140.991αp − 17.0377)

(17)

Figure 2 depicts the variations of the nanofluid conductiv-ity with temperature at different nanoparticle volume fractionsbased on the above model, which is in good agreement with theexperimental results of Chon et al. [23].

Thermophysical properties of base fluids:Because of the considerable variations in water thermophys-

ical properties values with temperature, they have been consid-ered as functions of temperature as follows:

Density:

ρw = −3.1 × 10−3T 2 + 1.5917T + 801.92 (18)

Viscosity:

μw = 0.1917e−0.0181T (19)

Thermal conductivity:

kw = −0.5981 + 0.00653T − 8.354 × 10−6T 2 (20)

TABLE 1Grid independency test (PCD = 100 mm; Pitch = 20 mm;

m = 0.01; at the center point of tube’s outlet)

Node number (r × θ × z) vz/V0 T/T0

r direction 24 × 32 × 1400 1.1541067 1.158041932 × 32 × 1400 1.1534125 1.156990740 × 32 × 1400 1.1519739 1.1565819

θ direction 32 × 24 × 1400 1.1635956 1.156701032 × 32 × 1400 1.1534125 1.156990732 × 40 × 1400 1.1469362 1.1576287

z direction 32 × 32 × 1300 1.146408 1.156992132 × 32 × 1400 1.1534125 1.156990732 × 32 × 1500 1.1569775 1.1569733

The specific heat and thermal expansion coefficient of waterare equal with:

cp,w = 4200, βw = 0.000379

2.2 Heat Transfer PerformanceThe total heat transfer rate is given as:

Q = mcp,m(Tm,out − Tm,in) (21)

where m is mass flow rate and Tm,in and Tm,out are mixed meantemperature at inlet and outlet, respectively.

The overall convective heat transfer coefficient (h) is calcu-lated as follows:

h = Q

A�TLM

, �TLM = �Ti − �To

Ln �Ti

�To

(22)

where �Ti = (Ts − Ti) and �To = (Ts − To) .

2.3 Boundary ConditionsThe above-mentioned nonlinear and coupled partial differ-

ential governing equations subjected to the following boundaryconditions (as seen in Figure 1g):

• At the tube inlet (i.e., z = 0):

m = min, T = Tin (23)

• Walls: At walls, constant wall temperature is appliedand we have no slip condition for velocities:

T = Twall, u = 0 (24)

• At the tube outlet (i.e., x = l): at the exit plain, a pres-sure boundary condition is applied.

2.4 Numerical MethodThe set of 3D coupled nonlinear differential equations was

discretized with the control volume technique. The control-

0

5

10

15

20

25

30

0 50 100 150 200 250 300 350 400 450

Nu

z/D

Experimental data [28]

Numerical result

Re = 1620

FIG. 3. Comparison of the Nusselt number with the experimental numbers.

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FIG. 4. Distribution of (a) axial velocity profile; (b) temperature distribution inside a helical tube with PCD = 200 mm, Pitch = 20 mm, m = 0.07 at x = l/2for pure water flow.

volume-based technique consists of the division of the domaininto discrete control volumes using a computational grid andintegration of the governing equations on the individual con-trol volumes to construct algebraic equations for the discretedependent variables (“unknowns”), such as velocities, pressure,temperature, and conserved scalars. Next, the discretized equa-tions are liberalized and the resultant linear equation system issolved to yield updated values of the dependent variables. For theconvective and diffusive terms, a second-order upwind methodwas used, while the SIMPLE procedure was introduced forthe velocity–pressure coupling. In the pressure-based approachwhich we have used in our current numerical work, the pressurefield is extracted by solving a pressure or pressure-correctionequation which is obtained by manipulating continuity and mo-mentum equations. The pressure equation is derived from thecontinuity and the momentum equations in such a way that thevelocity field, corrected by the pressure, satisfies the continu-

ity. Since the governing equations are nonlinear and coupled toone another, the solution process involves iterations wherein theentire set of governing equations is solved repeatedly until thesolution converges.

A structured non-uniform grid has been used to discretize thecomputational domain (see Figure 1h). Several different griddistributions have been examined to ensure that the calculatedresults are grid-independent (see Table 1). The used grid forthe present calculations consisted of 32, 32, and 1400 nodes,respectively in the radial, circumferential, and axial directions.As shown in Table 1, increasing the grid numbers does notsignificantly change the dimensionless velocity and temperatureat the mentioned point in the table.

To demonstrate the validity and precision of the model andthe numerical procedure, a comparison with the previously pub-lished, experimental work using water-Al2O3 nanofluid has beendone. Figure 3 indicates the comparison of the numerical results

FIG. 5. Distribution of (a) axial velocity profile; (b) temperature distribution inside a straight tube atx = l/2 for pure water flow.

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TABLE 2Properties of studied nanoparticles

Property ρp cp,p kp βp

Value 3900 kg/m3 785 J/kgK 36 w/m. K 0.000008 1/K

1300135014001450150015501600165017001750180018501900195020002050210021502200225023002350240024502500

70 100 150 200

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t tr

ansf

er c

oef

fici

ent(

W/m

2 K)

Pitch Circle Diameter(mm)

mass flow rate = 0.01 kgs-1

Pitch = 20 mm0%1%2%3%

FIG. 6. Comparison of the heat transfer coefficients of helical tubes withdifferent PCDs.

for Nusselt number with the experimental results of Kim et al.[28] in a horizontal tube. As seen, there is very good agreementbetween them.

3. RESULTS AND DISCUSSIONThe results are presented for water-based nanofluid-

consisting Al2O3 particles with 47 nm mean diameter (sphericalshape). The physical properties of the Al2O3 nanoparticles arepresented in Table 2. Al2O3/water nanofluids are widely utilizedfor the experimental investigation of heat transfer of nanofluidsdue to their relatively easy production and low cost. Physicalmodels are helical circular tubes with the lengths of 1260 mmand diameters of 10 mm (see Figure 1). This study investigatesthe thermal behavior of helical tubes with different geometriesin constant length of tube.

295

300

305

310

315

320

325

330

335

340

345

0 200 400 600 800 1000 1200 1400

T mea

n(K

)

Tube length(mm)


Pitch = 20 mm

PCD = 70 mmPCD = 100 mmPCD = 150 mmPCD = 200 mmstraight

FIG. 7. Mixed mean temperature along the tube length for pure water.

295

300

305

310

315

320

325

330

335

340

345

0 200 400 600 800 1000 1200 1400

T mean(K)

Tube length (mm)


PCD = 100 mm Pitch = 20 mm

helical tube-pure waterhelical tube-2% Al2O3straight tube-pure waterstraight tube-2% Al2O3

FIG. 8. Comparison of mixed mean temperatures along tube length betweena helical tube and a straight tube of the same length.

3.1 The Pitch Circle Diameter EffectThe effect of PCD of the coil is to govern centrifugal force

on the moving fluid and this will, in turn, affect the secondaryflows along the pipe cross-section. The centrifugal force causesthe flow rate in the core of the tube to begin to move to the outerbend. Therefore, as shown in Figure 4a, the maximum axialvelocity zone appears at the outer part and causes a decreasein the temperature near the outer bend (see Figure 4b). Thedistortion of symmetry about the horizontal plane is due tothe interaction between vertical flow due to gravity force andhorizontal flow resulting from centrifugal force.

The presence of secondary flow with high velocities is ex-pected to have a direct impact on the heat transfer rate. As thepitch circle diameter is increased, the centrifugal force decreasesand the helical tube approaches a straight tube. Figures 5a and5b depict axial velocity profile and temperature distribution offlow through a straight tube, respectively.

Figure 6 shows that the overall heat transfer coefficient de-creases as pitch circle diameter increases so we have highertemperatures at the same distance from the inlet with a helicaltube which has lower pitch circle diameter compared to a heli-cal tube with higher pitch circle diameter (PCD) (see Figure 7).Thus, with the addition of 1% Al2O3 nanoparticles to the basefluid, as shown in Figure 6, the amount of increase in the over-all heat transfer coefficient of a helical tube with lower PCD

0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

PCD = 0.1 straight

Ove

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W/m

2 K)


Pitch = 20 mm

0%

1%

2%

3%

FIG. 9. Comparison of heat transfer coefficient between a helical tube and astraight tube of the same length.

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FIG. 10. (a-d) Temperature distributions at x = l/8, x = 3l/8, x = 5l/8 and x = 7l/8; (e-h) streamlines at x = l/8, x = 3l/8, x = 5l/8 and x = 7l/8 of nanofluidflow (2% volume fraction of nano-particles) along tube length.

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0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0 200 400 600 800 1000 1200 1400

k eff

Tube length(mm)

PCD = 100 mmpitch = 20 mm

0%1%2%3%

FIG. 11. Variations of the effective thermal conductivity of nanofluid alongthe tube length at different particle volume fractions.

is more than a helical tube with higher PCD. This is becauseof the temperature-dependent behavior of nanofluids and thevariations of thermal conductivity with temperature due to theBrownian motion of nanoparticles, which leads to an increase inthermal conductivity at higher temperature and will enhance theoverall heat transfer coefficient. Furthermore, Figure 6 showsthat the decrease of overall heat transfer coefficient by increas-ing the PCD is also a general trend for different nanoparticlevolume fractions of nanofluid flow in helical tubes.

Figure 7 depicts mixed mean temperatures along tube lengthfor a helical tube and a straight tube, which shows higher tem-peratures in helical tubes in comparison with straight tubes atany same length of the tube. Figure 8 compares the temperaturesbetween a helical tube and a straight tube along the tube length.As shown in Figure 9, the overall heat transfer coefficients in thehelical tube are much higher than in the straight tube, and onecan see that adding 1% Al2O3 nanoparticles to the base fluid willincrease the overall heat transfer coefficient in the helical tubesmore than in the straight tube. Figure 10 depicts temperaturedistributions and streamlines of nanofluid flow (1% nanoparti-cle volume fraction) along the tube length for the helical pipe indifferent cross-sections.

1200

1400

1600

1800

2000

2200

2400

2600

0 0.5 1 1.5 2 2.5 3 3.5

Ove

rall

hea

t tr

ansf

er c

oef

fici

ent(

W/m

2 K)

Volume fraction(%)

PCD = 70 mmPitch = 20 mm

mass flow rate = 0.01



kgs -1

kgs -1

kgs -1

FIG. 12. Variation of heat transfer coefficient of nanofluid flow through ahelical pipe with PCD = 70 mm.

1200

1400

1600

1800

2000

2200

0 0.5 1 1.5 2 2.5 3 3.5

Ove

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t tr

ansf

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oef

fici

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W/m

2 K)

Volume fraction(%)





kgs -1

kgs -1

kgs -1


3.2 The Effect of Nanoparticle ConcentrationsThe amount of nanoparticles suspended in the base fluid

plays a significant role in the determination of heat transfer per-formance of helical tubes. In this study, four different nanopar-ticle volume fractions of Al2O3 (0, 1, 2, and 3%) have beeninvestigated. It is clearly seen that the addition of small amountsof nanoparticles significantly improves the overall heat transferperformance.

Figure 6, which shows the overall heat transfer coefficientfor helical tubes with different PCDs, indicates that the rate ofincrement in overall heat transfer coefficient decreases with anincrease of nanoparticle volume fraction so that, at a volumefraction of 1%, we have greater increase in overall heat transfercoefficient. Also, the addition of nanoparticles of more than 2%of volume fraction does not greatly increase the overall heattransfer coefficient, while 2% volume fraction of nanoparticlescan be economically efficient.

The reason for the reduction in the rate of increment of over-all heat transfer coefficient in larger amounts of nanoparticlevolume fractions in a helical tube is that by adding nanoparti-cles to the base-fluid, the specific heat of nanofluid reduces and,in order to keep the flow rate constant, the inlet velocity shouldbe decreased. Therefore, with the addition of nanoparticles tothe base fluid, the influence of decreasing velocity and specific

1200

1300

1400

1500

1600

1700

1800

1900

2000

2100

0 0.5 1 1.5 2 2.5 3 3.5

Ove

rall

hea

t tr

ansf

er c

oef

fici

ent(

W/m

2 K)

Volume fraction(%)

PCD = 150 mmPitch= 20 mm




kgs -1

kgs -1

kgs -1


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1100

1200

1300

1400

1500

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1700

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2000

0 0.5 1 1.5 2 2.5 3 3.5

Ove

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t tr

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oef

fici

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W/m

2 K)

Volume Fraction(%)


mass flow rate = 0.01mass flow rate = 0.007mass flow rate = 0.004

kgs -1

kgs -1

kgs -1


heat of nanofluid on the heat transfer coefficient can be compet-itive with increasing thermal conductivity of the nanofluid (seeFigure 11). Moreover, mass flow rate and length of the helicaltube are the other effective parameters in determining the ef-ficient volume fraction of nanoparticles which is needed to beadded to the base fluid in order to have the highest heat transfercoefficient.

In Figures 12–15, the overall heat transfer coefficients areseparately shown for PCD = 0.07 m, 0.1 m, 0.15 m, and 0.2 min different nanoparticle volume fractions and flow rates. Asseen at a given nanoparticle volume fraction, increasing flowrate augments the heat transfer coefficient. Moreover the effectof increasing the flow rate on enhancement of the heat transfercoefficient in the helical tubes is more prominent since an in-crease of the flow rate will increase the centrifugal forces’ andsecondary flows’ intensity.

3.3 Effect of PitchWhile the curvature of the coil governs the centrifugal force,

the pitch causes Corioli’s forces leading to torsion on the fluidflow through the pipe, so that the region of higher velocity shiftsdownward as the pitch is increased [12] and, when the pitchis zero, the velocity field is symmetrical about the horizontal

1800

1900

2000

2100

2200

2300

2400

0 0.5 1 1.5 2 2.5 3 3.5

Ove

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oef

fici

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W/m

2 K)

Volume fraction(%)


Pitch = 20 mm

pitch=0.02

pitch-0.04

pitch=0.06

FIG. 16. Variations of overall heat transfer heat transfer coefficient for differ-ent pitch sizes.

central plane. It is necessary to mention that the effect of pitchon heat transfer performance of the helical tubes is much weakerthan the pitch circle diameter effect. Figure 16 shows that, at agiven volume fraction of nanoparticles, the overall heat transfercoefficient generally enhances with increase of pitch; you cansee the reduction in rate of increment of overall heat transfercoefficient with the addition of larger amounts of nanoparticles.

4. CONCLUSIONIn this article, the results of a 3-D numerical investigation of a

laminar, developing, mixed convection of a nanofluid consistingof water and Al2O3 spherical-shape nanoparticles through he-lical pipes of different PCDs and pitches have been presented.Simultaneous effects of centrifugal force, gravitational force,and nanoparticle volume fraction in the base fluid on heat trans-fer augmentation are studied at different mass flow rates un-der constant wall temperature boundary condition and physicalproperties supposed to be temperature-dependent, such as ther-mal conductivity of nanofluid. The following conclusions havebeen obtained from the results:

• Decrease of overall heat transfer coefficient with in-crease of PCD is a general trend for nanofluid flowwith different volume fractions of nanoparticles in he-lical tubes.

• The rate of enhancement of overall heat transfer coef-ficient decreases with increase of volumetric fractionof nanoparticles.

• The amount of increase in overall heat transfer coef-ficient by addition of 1% volume fraction of Al2O3

nano-particles to the base fluid in a helical tube withlower PCD is more than a helical tube with higherPCD.

• As the pitch circle diameter is increased, the centrifu-gal force decreases and the helical tube approaches astraight tube so temperatures and heat transfer coeffi-cients in the helical tube are much higher than in thestraight tube. The addition of 1% Al2O3 nanoparticlesin the base fluid will increase the overall heat transfercoefficient in helical tubes more than in the straighttube.

NOMENCLATUREρ density (kgm−3)cp specific heat (Jkg−1 K−1)μ dynamic viscosity (kgm−1 s−1)k conductivity (wm−1 K−1)β thermal expansion coefficient (K−1)�v velocity (ms−1)T temperature (K)P pressure (Pa)�vpf slip velocity vector (ms−1)�vdr drift velocity vector (ms−1)

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αp particle volume fractiondp nanoparticle diameter (m)l tube length (m)D tube diameter (m)r radial directionθ circumferential directionPCD pitch circle diameterm mass flow rate (kgs−1)Tw eall temperature (K)x axial direction(distance from inlet of the tube)g gravitational acceleration (ms−2)

subscriptf pertaining to base fluidp pertaining to nanoparticlem pertaining to mixture0 pertaining to inlet conditionsw pertaining to water

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3d numerical investigation of thermal characteristics of nanofluid flow through helical tubes using...

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