Scientific Research and Essays Vol. 5(22), pp. 3418-3429, 18 November, 2010 Available online at http://www.academicjournals.org/SRE ISSN 1992-2248 ©2010 Academic Journals Full Length Research Paper

Effects of geometrical parameters and internal wall velocity on turbulent flow and heat transfer in helical

square duct with moving wall

Okyar Kaya1* and Ismail Teke2

1Department of Mechanical Engineering, Pamukkale University, 20070, Kinikli, Denizli/Turkey.

2Department of Mechanical Engineering, Yildiz Technical University, 34349, Besiktas, Istanbul/Turkey

Accepted 19 August, 2010

In this study the effect of geometrical dimensions and the internal wall velocity on turbulent flow and heat transfer in a helical square duct with movable wall are numerically analyzed. RNG k-� turbulent model and finite volume method are used in the numerical solutions. The validity of the numerical solutions are tested by previous experimental studies with different boundary conditions and it has been determined that the helical duct curvature ratio and the internal wall velocity have significant effects but the dimensionless helical pitch has not on the internal wall Nusselt number. Key words: Helical square duct, RNG k-� turbulent model, finite volume method, curvature ratio, internal wall velocity, dimensionless pitch.

INTRODUCTION The geometry of heat exchanger is very important and several criteria must be satisfied in the selection of it including the ease of manufacturing, cost and desired heat transfer surface area. In order to increase the rate of heat transfer, helical channels with a finite pitch have been extensively used in various industrial applications.

Compared to the straight tube, the flow field and heat transfer characteristics in a helical coiled tube is very complicated because of the centrifugal force which occurred in the channel. Geometric structure of a helical duct is formed of three parameters; hydrodynamic dia-meter, pitch and curvature. It is possible to find a large number of studies about the effects of these three geometric parameters on helical flow and heat transfer in the literature. Laminar flow and heat transfer in helical coils was the most studied topic. Germano (1982), Tuttle (1990), Liu and Masliyah (1993), Zabielski and Mestel (1998) studied helical ducts and they were mainly focused on stationary ducts with a circular cross section. Ko (2005) analyzed optimal curvature ratio for laminar forced convection in helical coiled tube for the minimal entropy generation, Yu et al. (2003) performed three *Corresponding author. E-mail: okaya@pau.edu.tr. Tel: +90-258-2963160. Fax: +90-258-2963262.

dimensional laser Doppler anemometry measurements and numerical calculations on developed laminar flow in helical coiled pipes. They claimed comprehensive rela-tionships between axial flow and secondary flow. Ko and Ting (2006) numerically studied entropy generation induced by laminar forced convection in a curved rectangular duct optimal De number and aspect ratios for different heat fluxes were proposed.

Chen et al. (2006) examined the characteristics of fluid flow and convective heat transfer in a helical square duct rotating with a constant angular velocity about the center of curvature. The variations of flow structure and temperature distribution with the force ratio and the torsion were examined. They also studied the effects of rotation and torsion on the friction factor and Nusselt number.

Joye et al. (1993) experimentally studied the heat transfer coefficients for the aspect ratios smaller and bigger than one in rectangular helical channels. Much higher heat transfer rates were obtained with the aspect ratios which are smaller than one.

Bolinder (1996) studied the effects of torsion and curvature on the flow in helical rectangular ducts. The researcher claimed that torsion has a stronger impact on the flow for aspect ratios greater than one and for increasing Re number, the secondary flow is dominated by effects due to curvature.

By using the numerical solutions, Bolinder and Sunden (1996) proposed correlations for laminar flow mean Nusselt number in helical rectangular ducts. Sakalis et al. (2005) studied the effects of torsion, curvature and the axial pressure gradient on the velocity components and the friction factor in helical square ducts. Their results show that the torsion deforms the symmetry of the two centrifugal vortices of the secondary flow and the friction factor decreases for torsion in the range 0 to 0.1 and increases as the torsion increases further.

Eason et al. (1994) investigated developing tempera-ture field, the variation of peripherally averaged Nusselt number and the effect of the Prandtl number on the temperature field for laminar flow and heat transfer in helical square ducts numerically.

Although there are a lot of studies about laminar flow in helical rectangular and square ducts, there is limited number of study about turbulent flow in the literature. Kadambi (1983) studied turbulent flow and heat transfer in helical rectangular ducts experimentally. Kadambi (1983) proposed emprical correlations for turbulent flow Nusselt number and pressure drop in helical rectangular duct by using the experimental data. Kaya and Teke (2005) experimentally and numerically studied turbulent forced flow and heat transfer in helical square ducts and proposed Nusselt number correlations. In their study the inner wall is stationary and isothermal, the other walls are adiabatic.

Chen and Jan (1993) solved continuity and Navier-Stokes equations by the Galerkin finite element method to study the torsion effect on the fully developed laminar flow in helical square ducts.

Helical ducts are widely used in heat exchanger industries. The walls of the duct are generally stationary. But for example in plastic pipe industry, cooling of plastic pipes can be done with helical ducts which has non-stationary internal wall. The internal wall of the helical rectangular duct is the outer surface of plastic pipe which is translational moving with a constant velocity, the other walls of the duct are stationary. This cooling system has resemblance to helical rectangular duct flow which has translational moving isothermal internal wall.

According to the literature survey results, turbulent flow and heat transfer for moving internal wall with uniform temperature and motionless and adiabatic top, bottom, external helical duct walls have not been studied before. The purpose of this study is to determine the effects of geometrical parameters and internal wall velocity on turbulent flow and heat transfer in a helical square duct numerically. The validity of numerical results was tested by comparing them with the previous experimental results. HELICAL CHANNEL GEOMETRY AND BOUNDARY CONDITIONS The pitch b, the radius a, and the curvature of the helical

Kaya and Teke 3419 square duct R are shown in Figure 1. The internal wall of the helical square duct was the external surface of a plastic pipe moving with a constant velocity and having constant temperature in z direction, on the other hand, top, bottom and external walls of helical square duct were motionless and assumed to be adiabatic.

The cooling water is assumed to enter the helical channel with uniform velocity Vg and constant temperature Tg. In order to create fully turbulent flow, the water flow rates were selected high enough to provide higher Reynolds numbers than the critical Reynolds number which was given by Srinivasan (1968) in Equation (1).

)121(2100Re 2/1δ+=kr (1) In this study, the effects of the curvature ratio �, the dimensionless pitch � and the helical internal wall velocity Vz on turbulent flow and heat transfer in a helical square duct with the given boundary conditions shown in Figure 1 were numerically studied. The accuracy of the numerical solutions was tested by using experimental results of Kaya and Teke (2005) and Kaya (2002) with different boundary conditions. Mathematical model The numerical solutions of turbulent flow and heat transfer in helical square duct were obtained by using the equations that were taken from the user’s guide of Fluent (1998). Governing equations have been arranged according to the master Cartesian coordinate system and are given below. The continuity equation can be written as follows:

0)(

=∂

∂

i

i

xuρ

(2)

Momentum equation is given by

ik

keff

i

j

j

ieff

jij

ji gxu

x

u

xu

xxP

x

uuρµµ

ρ+

∂∂

−∂∂

+∂∂

∂∂+

∂∂−=

∂∂

32

)()(

(3) Energy equation is defined as

���

�

���

�

∂∂−

∂∂

+∂∂

∂∂+

∂∂

∂∂=

∂∂

k

keff

i

j

j

ieff

j

i

ieff

iPi

i xu

x

u

xu

xu

xT

xTCu

xµµλρ

32

)()()(

(4) where effective thermal conductivity is obtained from

effλ = effpT C µα (5)

3420 Sci. Res. Essays

Figure 1. The geometry of the helical channel.

Effective viscosity is

tleff µµµ += . (6)

Turbulent kinetic energy equation is defined as

ρερρ

µαµµαρ −∂∂

∂∂−+

∂∂

∂∂=

∂∂

Pi

tTiti

effkii

i

TxT

gSxk

xxku

)(1

)()( 2 (7)

where S is the modulus of mean rate of strain tensor given by

S= ijij SS2 (8)

where Sij is defined as

)(21

i

j

j

iij x

u

xu

S∂∂

+∂∂

= . (9)

Equation of dissipation rate of turbulent kinetic energy is given by

'2

22

1)( Rk

CSk

CxxDt

Dt

ieff

i

−−+∂∂

∂∂= ερµεεµαερ εεε (10)

where R’ is the effect of strain in � equation and is defined as

k

CR

2

30

3'

1

)/1( εβη

ηηρηµ

+−

= (11)

where η is

εη /Sk= . (12) Turbulent viscosity is obtained from

ερµ µ

2kCt = . (13)

The subscripts i, j, k and the model constants η0, β, Cµ,

C1ε, C2ε in the equations above are equal to x, y, z in master Cartesian coordinate system and the constants are 4.38, 0.012, 0.085, 1.42, 1.68 respectively.

Inverse Prandtl numbers for αt, αk, αε which are written in equations (7) and (10) are calculated by the Equation (14) given below,

eff

mol

µµ

αα

αα =

++

−−

3679.0

0

6321.0

0 3929.23929.2

3929.13929.1 (14)

where α0 for αt, αk, αε is equal to 1/Pr, 1, 1 respectively. The two-layer zonal non-equilibrium wall function method was used in numerical computations for near wall regions. According to this model Rey number have to be computed via

µρ yk

y =Re . (15)

If Rey < 200, then turbulent viscosity µt and dissipation rate of turbulent kinetic energy ε were obtained from equations (16) and (18) respectively. Turbulent viscosity is calculated by

µµρµ lkCt = (16) where

���

�

���

�−−= )

Reexp(1**

µµ A

yCl yl . (17)

Dissipation rate of turbulent kinetic energy is

ε

εl

k 2/3

= (18)

where

��

���

�−−= )

Reexp(1

εε A

yCl yl (19)

4/3−= µκCCl (20)

where Aµ in Equation (17) and Aε in Equation (19) are equal to 70 and 30 respectively. If Rey > 200, then the RNG k-ε turbulent model is employed. Wall heat flux is computed by using the equation given below

*

2/14/1)("

T

PPPw

y

kCCTTq µρ−

= (21)

where *

Ty non dimensional thermal sublayer thickness is obtained from

��

�

��

��

>

<

��

�

��

��

��

���

� +=)(

)()ln(

1Pr

Pr

**

**

*

*

*

T

T

tT

yy

yyPEy

yy

κ (22)

Kaya and Teke 3421 Where

4/12/1 )PrPr

)(1PrPr

()()4/sin(

4/ t

t

vAP −=

κππ

(23)

where E, κ, Prt, Av in Eq. (22) are equal to 9.793, 0.42, 0.85, 26 respectively. The non dimensional viscous sublayer thickness y* in Eq. (22) is defined as

µρ µ PP ykC

y2/14/1

* = . (24)

The convective heat transfer coefficient on the wall is calculated by

)/(" ∞−= TTqh w . (25) At the inlet, cooling water temperature, turbulent kinetic energy, dissipation rate of turbulent kinetic energy, axial, radial and tangential velocities are equal to Ti, ki, �i, Vi, 0, 0 respectively. The inlet turbulent kinetic energy is calculated by

2)(23

Iuk ig = (26)

where turbulent intensity is estimated by

8/1'

)(Re16.0100% −− ≅=

hd

u

uI . (27)

The dissipation rate of inlet turbulent kinetic energy is

lk

ci

2/34/3

µε = (28)

where the turbulent length scale which is given in Equation (28) is defined as l = 0.007 dh. (29) The diffusion flux for axial, radial, tangential velocities and for temperature, turbulent kinetic energy and its dissipation rate were set to be zero (Eq. (30)) at the outlet:

0),,,,,( =∂∂ εθ kTuuun rs . (30)

Numerical method The governing equations were solved numerically by

3422 Sci. Res. Essays

Figure 2. Cross-sectional grid distribution of helical square duct.

using control volume finite element method in the computational fluid dynamics package program named as Fluent. A non-uniform grid system 50*50 grids recommended by Kaya and Teke (2005) shown in Figure 2 was used in the numerical solutions. Kaya and Teke (2005) conducted grid independency tests by dividing cross-sectional area into different number of 20*20, 30*30, 40*40, 45*45, 50*50 grids and as a result they found that there is less than 1% difference between 45*45 and 50*50 grids.

By using the entrance region length data suggested by Austin and Seader (1974), computational domain was extended to at least 11/6 helical toured to form fully developed flow. The helical square duct was divided cross-sectionally and axially into 50*50 and 140 cells respectively. The algebraic conservation equation for any variable Ø at each node were written as follows:

� +=nb

nbnbPP caa φφ (31)

The aP and anb values in Equation (31) are the linearized coefficients for center and neighbor cells and c is the constant part of the source term.

All of the algebraic equations are solved with Gauss Siedel iteration method, at the end of each iteration the variation of φ variable is controlled by the URF (Under Relaxation Factor) correction factor. ‘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, ‘Pre-ssure Staggered Option (PRESTO)’ for interpolating pressure, ‘Renormalization Group Theory (RNG)’ k-� turbulence model for simulating turbulent flow were selected as pre-processing parameters in the Fluent® software.

In helical duct flow strain is formed suddenly and must be taken into account. The most important difference between the RNG k-� model and the other k-� model is the R’ term which calculates the effect of strain on turbulent viscosity. RNG k-� model also calculates turbulent Prandtl number as a function of fluid’s physical properties, but in the others turbulent Prandtl number can be constant. Moreover, Kaya (2002) showed that RNG k-� model was the best candidate among k-� models compared with the experimental results. Because of these advantages RNG k-� model was chosen as the turbulent flow model in the numerical computations.

Except for energy (10-8), the convergence criterion for all equations was given in Equation (32).

45 10−≤

φ

φ

R

R n

(32)

where nRφ is the residual value of φ for the nth iteration

and 5

φR is the maximum residual value of φ for the first

five iterations. All of the numerical calculations were based on the temperature dependent properties of water.

Kaya and Teke 3423

Table 1. Comparison of numerical Nu number of helical internal wall with the previous experimental Nu number in helically coiled square duct which has one motionless uniform temperature and three motionless adiabatic walls.

Re NuNumerical NuExperimental (Kaya and Teke, 2005)

15329 116.75 116.17

18204 128.95 125.47

19114 133.25 128.55

20644 147.44 144.76

21605 157.13 153.01

25103 173.79 175.42

26878 191.35 178.87

Table 2. Comparison of experimental and numerical heat transfer from internal wall of helical rectangular duct to cooling water in a helical square duct with uniform temperature and velocity internal wall and adiabatic, motionless other walls.

VD QNUMERICAL QEXPERIMENTAL (KAYA, 2002)

0.15 1666 1442

0.25 1588 1401

0.275 1233 1133 RESULTS AND DISCUSSION In this study three dimensional turbulent flow and heat transfer in a helical square duct which has an internal wall with uniform temperature, velocity and adiabatic and motionless external, bottom and upper walls.

The validity of models and methods used in numerical computations were tested by the previous experimental studies. The experimental data of Kaya and Teke (2005) named as “Turbulent Forced Convection in a Helically Coiled Square Duct with One Uniform Temperature and Three Adiabatic Walls” were compared with numerical solutions applying the similar boundary conditions on Table 1.

It can be understood from Table 1 that in a helical square duct with motionless walls there is only 0 - 6.9% difference between the numerical and the experimental Nusset numbers of helical internal wall.

Also experimental data of turbulent flow and heat transfer in a helical rectangular duct with one uniform temperature and velocity, three motionless and adiabatic walls were compared with the similar boundary condi-tioned numerical solutions on Table 2. Table 2 depicts that there is nearly 8 - 13% difference between the numerical and experimental results. Due to the adiabatic assumptions made for external, top and bottom walls in numerical simulations, heat transfer is over predicted compared with the experiments. In the experi-ments the walls were not fully insulated so that the rate of heat transferred from the internal wall to cooling water is less

than in the numerical computations. Numerical and experimental results of motionless and

moving wall boundary conditions are very close to each other as seen in Tables 1 and 2. Thus, in this study similar methods and models were used in all of the numerical computations.

Velocity and temperature contours obtained from numerical solutions for different internal wall velocities at different cross-sections beginning from the inlet of helical square duct are given in Figures 3 and 4, respectively.

It can be seen from Figure 3 that velocity contours for Vz = 0 m/s at � = 510° and � = 570°, for Vz = 0.5 m/s and Vz = 1 m/s at � = 630° and � = 660° are very similar to each other except minor differences in the maximum velocity contours. Therefore it is understood that hydrodynamically fully developed flow in helical duct with motionless internal wall is formed earlier than in duct with movable internal wall. Due to the momentum transfer from the inner wall, hydrodynamically developed flow grows later in movable walled helical ducts than in motionless walled ones. Because of the centrifugal forces which occurred in helical square duct, velocity contours are much sparser and have low values on the internal wall side but on the external wall side they are more frequent and have high values as seen in Figure 3. Thus hydrodynamic boundary layer on the internal wall of helical square duct is thicker than on the external wall.

When the internal wall velocity of helical duct is equal to 0 m/s, the velocity contours of the lower and upper regions of flow cross-section are symmetric but when the

3424 Sci. Res. Essays

Re=26878

Vz=0 m/s Vz=0.5 m/s Vz=1 m/s

�=90°

�=270°

�=510°

�=570°

�=630°

�=660°

Figure 3. Cross-sectional flow velocity contours for different internal wall velocities at different angles beginning from the inlet of helical square duct.

internal wall is moving (Vz = 0.5 and Vz = 1 m/s) the symmetrical structure is disrupted and the velocity contours were directed towards the movement direction of internal wall.

Figure 4 demonstrates the velocity contours at �=660°,

whenever the internal wall is moving or motionless as the Reynolds number increases, the thickness of boundary layer decreases and the maximum velocity contours are seen to occupy more space. Maximum velocity contours of the lower and the upper cross-sectional regions of

Kaya and Teke 3425

Figure 4. Velocity contours for different Re numbers, internal wall velocities at �=660°.

helical duct with motionless internal wall are symmetric. On the other hand in ducts with moving internal walls, velocity contours on the upper cross-section are bigger than those of the lower cross-section.

Figure 5 displays the temperature contours of various cross-sections starting from the inlet of helical duct, for all of helical internal wall velocities the temperature contours begins not to change at �=600°. As in the velocity contours, temperature contours are being directed toward the upper right corner of flow cross-section and the upper and lower symmetrical structure is damaged. Tempera-ture contours are getting larger from the entrance of helical duct towards fully developed region and they are very close to each other near the internal wall but very sparse near the external wall as seen in Figure 5.

Figure 6 displays the temperature contours for different Reynolds numbers and internal wall velocities at � = 660°, as seen from the figure when Reynolds number and helical internal wall velocity Vz increase the temperature contours occupy larger spaces in the flow cross-section.

Figure 6 demonstrates the temperature contours of the

flow with motionless internal wall, as seen from the figure the contours are very close to each other near the internal wall, but they are getting sparse towards the external wall. But when the internal wall is movable, the temperature contours are very close to each other on the top right side of the flow cross-section that is the movement direction of internal wall and getting sparse towards the left bottom side of the flow cross-section.

In order to see the effects of geometrical parameters, the dimensionless ratio of hydraulic radius to coil radius � = a/R named as curvature ratio and helical pitch to helical coil perimeter ratio � = b/2�R named as dimensionless pitch are used in this study. These two dimensionless parameters � and �, keeping one of them fixed and the other one variable, various numerical computations were done for different helical inner wall velocities and Reynolds numbers, the results were compared graphically in Figure 7.

As seen from Figure 7 by increasing curvature ratio � and Vz the speed of helical internal wall, Nusselt number increases. Furthermore it is understood that for low Reynolds numbers when the speed of internal wall increases

3426 Sci. Res. Essays

Figure 5. Cross-sectional temperature contours at different angles from the inlet of helical channel for Re = 26878.

the increment in Nusselt number is too much but as the number of Reynolds increases the increment decreases. Figure 7 also demonstrates, increases in � and Vz values, causes an increase in helical internal wall Nusselt number.

Figure 8 displays the numerical helical internal wall Nusselt numbers for a fixed � but for different values of

�,Vz and Reynolds numbers. From these curves it was understood that when � increases, Nusselt number increases averagely less than % 1 because of this helical dimensionless pitch � has negligible effect on the helical internal wall Nusselt number. And also it can be seen from Figure 8 that for high number of Reynolds and flow velocities, helical internal wall Nusselt number has

Kaya and Teke 3427

Vz=0 m/s Vz=0.5 m/s Vz=1 m/s Re=15329

Re=18204

Re=21605

Re=26878

Figure 6. Cross-sectional temperature contours at �=660° for different Re numbers and helical internal wall velocities.

a) Re=15329 b) Re=18204

c) Re=21806 d) Re=26878

Re=15329

0

50

100

150

200

250

300

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

�

Nu

Vz=0 m/s

Vz=0.5 m/s

Vz=1 m/s

Re=18204

0

50

100

150

200

250

300

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

�

Nu

Vz=0 m/s

Vz=0.5 m/s

Vz=1 m/s

Re=21605

0

50

100

150

200

250

300

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

�

Nu

Vz=0 m/s

Vz=0.5 m/s

Vz= 1m/s

Re=26878

0

50

100

150

200

250

300

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

�

Nu

Vz=0 m/s

Vz=0.5 m/s

Vz=1 m/s

Figure 7. Effects of curvature ratio and internal wall velocity at � = 0.05232.

3428 Sci. Res. Essays

Re=15329

0

20

40

60

80

100

120

140

160

180

200

0 0.05 0.1 0.15 0.2

�

Nu

Vz=0 m/s

Vz=0.5 m/s

Vz=1 m/s

Re=18204

0

20

40

60

80

100

120

140

160

180

200

0 0.05 0.1 0.15 0.2

�

Nu

Vz=0 m/s

Vz=0.5 m/s

Vz=1 m/s

a) Re=15329 b) Re=18204

Re=21605

0

50

100

150

200

250

0 0.05 0.1 0.15 0.2

�

Nu

Vz=0 m/s

Vz=0.5 m/s

Vz=1 m/s

Re=26878

205

210

215

220

225

230

235

0 0.05 0.1 0.15 0.2

�

Nu

Vz=0 m/s

Vz=0.5 m/s

Vz=1 m/s

c) Re=21806 d) Re=26878

Figure 8. Effects of dimensionless pitch and internal wall velocity at � =0.05479.

higher values. Conclusion In this study the effect of geometrical dimensions, helical internal wall velocity Vz on turbulent flow and heat transfer in a helical square duct with one uniform temperature, velocity and three adiabatic, motionless walls are discussed. The following conclusions can be drawn from the current study: (1) The experimental and the numerical results of motionless or movable internal wall boundary conditions show a good agreement with each other. (2) Hydro dynamically fully developed flow in helical square duct with motionless internal wall is formed earlier than in duct with movable internal wall. (3) In helical square ducts with motionless internal wall, velocity contours are symmetric at the lower and upper cross-sectional regions but in the ducts with movable

inner wall they are not symmetric. (4) For all of the helical internal wall velocities, the temperature contours begins not to change at � = 600°. (5) As the curvature ratio � and the internal wall velocity Vz increases, the internal wall Nusselt number increases. (6) For high values of Reynolds number and internal wall velocities, helical internal wall Nusselt number increases. (7) Dimensionless pitch � has negligible effect on the inner wall Nusselt number for both motionless and movable inner wall boundary conditions. (8) For different helical pitch, curvature, hydraulic radius and internal wall velocity values, optimization can be done for the highest heat transfer rates. NOMENCLATURE a, Hydraulic radius [m]; b, helical coil pitch [m]; Cp, specific heat under constant pressure [J/kgK]; g, gravitational acceleration [m/s2]; h, convection heat transfer coefficient [W/m2K]; I, turbulent intensity

(=−uu /' ); k, turbulent kinetic energy [m2/s]; l, turbulent

mixing length [m]; Nu, nusselt number; Pr, prandtl number; Q, rate of heat transfer [W]; ''q , wall heat flux [W/m2]; R, radius of helical coil [m]; R’, represents the effect of strain in ε equation [kg/ms4]; Re, reynolds number; Rey, re number for a cell which has a distance of y from the nearest wall; S, modulus of mean rate of strain tensor [1/s]; T, temperature [K]; T�, mixed mean

temperature [K]; t, time [s]; −u , time averaged mean

velocity [m/s]; 'u Instantaneous velocity component [m/s]; V, Velocity [m/s]; y*, non-dimensional viscous

sublayer thickness; *

Ty , Non-dimensional thermal sublayer thickness. GREEK SYMBOLS �, Under relaxation factor; ��, Inverse Pr number for dissipation rate of turbulent kinetic energy, �k, Inverse Pr number for turbulent kinetic energy; �T, Inverse Pr number for turbulent flow; �, turbulent kinetic energy dissipation rate [m2/s3]; , the rate of strain in turbulent flow (= Sk / � ), κ , Von Karman constant (=0.42); , non-dimensionless pitch [=b/2�R]; �, curvature ratio [=a/R]; �, molecular viscosity [kg/ms]; ν , Kinematic viscosity [m2/s]; φ , the parameter used in conservation of mass, momentum and energy equations; ρρρρ, density of fluid [kg/m3]; �, the axial angle from the entrance of the helical channel [°]. SUBSCRIPTS Eff, Effective; G, inlet; Kr, critical; N, normal direction; Nb, neighbor cell; P, P center cell; R, radial direction; S, axial direction; T, turbulent; W, wall; z, component in z direction. REFERENCES Austin LR, Seader JD (1974). Entry region for steady viscous flow in

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