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Available online at www.sciencedirect.com

Chemical Engineering and Processing 47 (2008) 893905

Numerical study of turbulent forced convectionin coiled flow inverter

Monisha Mridha, K.D.P. Nigam

Department of Chemical Engineering, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi 110016, India

Received 21 July 2006; received in revised form 8 February 2007; accepted 9 February 2007

Available online 3 March 2007

Abstract

A numerical study is done to investigate turbulent forced convection in a new device of coiled flow inverter. The proposed device works onthe technique based on flow inversion by changing the direction of centrifugal force in helically coiled tubes thus enabling rotation of the plane

of vortex. The objective of the present study is to characterize the flow development and temperature fields in coiled flow inverter (CFI) under

turbulent flow for the range of 10,000


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894 M. Mridha, K.D.P. Nigam / Chemical Engineering and Processing 47 (2008) 893905

Table 1

Published studies for heat transfer for turbulent flow in curved tubes

Author Method Range of parameters Remarks

1/ NRe NPr

Jeschke [8] Experimental 6.1, 18.2 150,000 0.7 (NNu)(NPr)0.4 = 0.045(1 + (3.54/))(NRe)0.76. The work

was for a limited range of parameters

White [9] Experimental 15.15, 50, 2050 15,000100,000 7 fc = 0.08N1/4Re + 0.012/

. Heat transfer coefficient may be

predicted from fluid friction data

Kirpikov [10] Experimental 10, 13,18 10,00045,000 7 (NNu)(NPr)0.4 = 0.0456(NRe)0.85(1/)0.21. Heat transfer

coefficient was obtained using the wall to bulk temperature

difference

Ito [21] Experimental 16.4, 40,100, 250,

648

2,000400,000 7 Proposed an empirical equation for the critical Reynolds

number, NRe cr = 2 104()0.32

Seban and

McLaughlin [11]

Experimental 17, 104 6,00065,000 2.9657 (NNu)(NPr)0.4 = 0.023(NRe)0.85(1/)0.1. For turbulent flow,

the results for heat transfer coefficient were simplified and

average heat transfer coefficients for the periphery was

predicted more accurately using friction factors for curved tubes

Rogers and Mayhew

[12]

Experimental 10.8, 13.3, 20.12 3,00050,000 7 (NNu) = 0.021(NRe)0.85(NPr )0.4()0.1. Non-isothermal frictionfactors and heat transfer coefficients were estimated and were

recommended for design purposes

Mori and Nakayama

[13,14]

Theoretical

and

experimental

18.7, 40 10,000200,000 1 For NPr 1, NNu = (NPr/26.2(N2/3Pr 0.074))N

4/5Re ()

1/10 [1 + (0.098/{NRe()2}1/5)]; for NPr >1, NNuN

0.4Pr =

(1/41.0)N5/6Re ()

1/12 [1 + (0.061/{NRe()2.5}1/6)]. In thefirst order approximation, heat transfer in a curved pipe doesnt

differ for uniform wall temperature or uniform heat flux, in both

laminar and turbulent regions

Schmidt [15] Experimental The empirical formula presented is as following: NNufd /NNus

=1.0 + 3.6(1 )0.8, where NNus = 0.023N0.8Re N0.4Pr

Shchukin [16] Experimental 6.2104 NRec


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M. Mridha, K.D.P. Nigam / Chemical Engineering and Processing 47 (2008) 893905 895

Table 1 ( Continued)


1/ NRe NPr

Zheng et al. [17] Numerical 20 10,000100,000 0.7 An interaction phenomenon between turbulent forced

convection and thermal radiation of an absorbing-emitting gas

in a curved pipe at different temperature ratio, optical thickness,

and wall emissivity was studied. There was no influence ofthermal radiation, optical thickness, wall emissivity, and

temperature ratio on velocity fields but slightly affected the

temperature fields when only radiation-participating medium

was considered. The Nusselt number was affected by the wall

emissivity

Cioncolini and Santini

[18]

Experimental 6.9369 2,90016,000 7 Coil curvature was found effective in smoothing the emergence

of turbulence. Criterion for predicting turbulence emergence in

coiled pipes was proposed for different curvature ratios

Table 2

Published studies for enhanced heat transfer by chaotic advection


D/d Re Pr

Acharya et al. [24] Numerical and experimental 5 3,00010,000 7 hi = 512.04N0.138Re . There was an enhancement of68% of heat transfer coefficient in helical coil with

alternating axis as compared to conventional helical

coil with constant axis. The pressure drop for the

chaotic configuration was 1.52.5% more than the

regular helical coil

Mokrani et al. [25] Experimental 5.5 60200 130 The effect of chaotic advection in the chaotic heat

exchanger on temperature uniformity and overallefficiency was studied. Flatter temperature

distribution was found in the chaotic coil as

compared to the regular helical coil. The relative

enhancement of the chaotic heat exchanger was

1328%

Chagny et al. [26] Experimental 11 3030,000 6.5820 Comparison for heat transfer between chaotic type

heat exchanger and a helical coil type heat exchanger

was made. At low Reynolds numbers, heating was

more homogeneous and heat transfer was intensified

in the chaotic advection regime without any increase

in energy expenditure. There was no influence on

heat transfer for Prandtl numbers higher than 225

Acharya et al. [27] Numerical 5, 10 501,200 0.110 NNu = 0.7()0.18

N0.5Re N

0.375Pr for Pr 1, NNu =

0.7()0.18N0.5Re N0.3Pr for Pr 1. The modified coil

geometry had 720% more heat transfer with

respect to regular coil with little change in pressure

drop. There was little enhancement with Reynolds

number for Pe 60

Lemenand and Peerhossaini [28] Numerical 11 100300 30100 NNu = 1.045N0.303Re N0.287Pr N0.033bends . A simplifiedthermal model was implemented to simulate heat

transfer in a helically coiled and chaotic

configuration

Kumar and Nigam [29] Numerical 10 251,200 0.74150 The bent coil configuration displays a 2030% heat

transfer enhancement as compared to the straight

helical coil with 56% increase in pressure drop


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results reported by Saxena and Nigam [30] shows that the devise

was behaving like a plug flow reactor.

Recently, Kumar and Nigam [29] numerically studied the

hydrodynamics and heat transfer in helical coiled tubes and

coiled flow inverter with curvature ratio, = 0.1 for a working

range of 25NRe 1200. They reported that the coiled flowinverter displays a 2030% heat transfer enhancement as com-

pared to the straight helical coils under laminar flow conditions

with the small increase in pressure drop.

In the present study, an attempt has been made to predict

the hydrodynamics and heat transfer in the coiled flow inverter

for turbulent flow. The flow development and temperature fields

in coiled flow inverter with = 0.1 and pitch of 0 and 0.02m

have been investigated for the range of Reynolds number from

10,000 to 30,000. Simulations were done with fluids (air, water,

kerosene and ethylene glycol) of Prandtl number ranging from

0.7 to 150.

2. Mathematical formulation

2.1. Governing equations

The geometry and system of coordinates considered are

shown in Fig. 1. The circular pipe studied, which has a diameter

of dt, is coiled at radius of Rc (=dc/2). The distance between

the two turns (the pitch) is reported by H. The bends introduced

in between the helical coils are of 90 and each helical tubehas same length before and after the bend. At the inlet ( = 0),fluid enters at a temperature T0 with a velocity ofu0. The wall

of the pipe is heated under constant temperature, Tw. The flow

was considered to be steady, and constant thermal properties

were assumed, except for the density of air. The values of T0,Tw, and physical and thermal properties of different fluids have

been reported in Table 3. The ideal gas law assumption was used

to the flowing fluid air.

TheRNG k modelproposedby Yakhot and Orszag [31] was

used to model the turbulent flow and heat transfer in the helical

Fig. 1. (a) System of coordinates, (b) geometry of coiled flow inverter and (c)

unstructured grid on one cross-section of the coiled flow inverter.

Table 3

Properties used in the numerical simulation T0 =300K, Tw = 350KProperties Air Water Kerosene Ethylene glycol

(kg/m3) Ideal gas law 998.2 780 1111.4

Cp (J/kg K) 1006.43 4182 2090 2415

(W/m K) 2.4E2 6E1 1.49E1 2.52E1 (kg/m s) 1.7894E

5 1.003E

3 2.4E

3 1.57E

2

Molecular weight 2 8.96 18.0152 157.30 62.0482

coil andcoiled flow inverter because the RNGmodel included an

additional term in its equation that significantly improved the

accuracy forrapidly strained flows,such as those in curved pipes.

The effect of swirl on turbulence is included in the RNG model,

enhancing accuracy for swirling flows. Researchers [1,17,32]

have used k model to simulate the turbulent flow and heat

transfer in curved tube for the same range of flow rate as done in

the present work. The maximum difference between the present

study and the past numerical and experimental data [14,15] was

less than 5% for the same parameter range. The time-averaged,

fully elliptic three-dimensional differential governing equations

can be written in tensor form in the Cartesian system as follows:

state : p = RT (for air) (1)

mass :ui

xi= 0 (2)

momentum :(uiuj)

xj

= xj

eff

ui

xj+ uj

xi

2

3eff

uk

xk

p

xi(3)

energy :(uiCpT)

xi

= xi

T

eff

T

xi

+ duidxj

eff

ui

xj+ uj

xi

2

3eff

uk

xkij

(4)

turbulent kinetic energy :(uik)

xi

= xi

keff k

xi

+ tS2 + Gb (5)

dissipation rate of turbulent kinetic energy :(ui)

xi

= xi

eff

xi

+ C1

ktS

2 C22

k R (6)

The effective viscosity, eff can be defined as

eff= mol

1 +

C

mol

k

2(7)


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where mol is the molecular viscosity. The coefficients T, kand in Eqs. (4)(6) are the inverse effective Prandtl numbers

for T, k, and , respectively.

The inverse effective Prandtl numbers, T, kand arecom-

puted using the following formula derived analytically by the

RNG theory:

1.39290 1.39290.6321

+ 2.39290 + 2.39290.3679 = moleff (8)

where 0 is equal to 1/Pr, 1.0, and 1.0, for the computation of

T, k, , respectively.

When a non-zero gravity field and temperature gradient are

present simultaneously, the k model account for the genera-

tion of k (kinetic energy) due to buoyancy [Gb in Eq. (5)] and

the corresponding contribution to the production of (energy

dissipation) in Eq. (6). In FLUENT, the effects of buoyancy are

always included despite the fact that the effect of buoyancy is

not so significant at very high Reynolds number. The generation

of turbulence due to buoyancy is given by

Gb = git

Prt

T

xi(9)

where Prt is the turbulent Prandtl number for energy and gi is

the component of the gravitational vector in the ith direction. In

the caseof the RNG k model, Prt = 1/T, and , the coefficient

of thermal expansion, is defined as =1/(p/T)p.In the Eq. (6), R is given by

R = C3(1 /0)

1 + 32

k(10)

where = S. k/, 0 4.38, = 0.012. The model constants C,C1, and C2 are equal to 0.085, 1.42 and 1.68, respectively [17].The term S in Eqs. (5) and (6) is the modulus of the mean

rate-of-strain tensor, defined as S2SijSij , whereSij =

1

2

ui

xj+ uj

xi

(11)

The two-layerbased, non-equilibriumwallfunction was usedfor

the near-wall treatment of flow in the given geometry. The non-

equilibrium wall functions are recommended for use in complex

flows because of thecapability to partly account for the effects of

pressure gradients and departure from equilibrium. The numer-

ical results for turbulent flow tend to be more susceptible togrid dependency than those for laminar flow due to the strong

interaction of the mean flow and turbulence. The distance from

the wall at the wall-adjacent cells must be determined by con-

sidering the range over which the log-law is valid. The size of

wall adjacent cells can be estimated from yp( y+p /u), whereu (w/)0.5 = u(cf/2)0.5. In the present study, the y+p wastaken in the range of 3060.

2.2. Boundary conditions

No-slip boundary condition, ui = 0, andconstant temperature,

Tw, were imposed at the wall. At the inlet, uniform profiles for

all the dependent variables were employed:

u = u0, T = T0, k = k0, = 0 (12)The turbulent kinetic energy at the inlet, k0, and the dissipation

rate of turbulent kinetic energy at the inlet, 0, are estimated by

k0 =3

2 (u0I)2, 0 = C3/4

k03/2

L (13)

The turbulence intensity level, I, is defined as u/u 100%,where u is root-mean-square turbulent velocity fluctuation.

At the outlet, the diffusion fluxes for all variables in exit

direction are set to be zero:

n(ui,T ,k,) = 0 (14)

where n is used to represent the normal coordinate direction

perpendicular to the outlet plane.

2.3. Parameter definitions

The following non-dimensional parameters and variables

were used in order to characterize the heat transfer in chaotic

configuration:

NRe =u0dt

, NDe = NRe

, = dt

dc,

= Tb TwTin Tw

, Tb =

1

uA

A0

u TDA

,

f =w

(1/2)u20, fm =

1

2 2

0f d,

NNu, =qwdh

(Tw Tb), NNu,m =

1

A

A0

NNu, d (15)

where is the curvature ratio, Tb the bulk temperature, f and

NNu,, local friction factor and Nusselt number along the circum-

ference of the pipe,respectively,fm andNNu,m, the circumference

average friction factor and Nusselt number and u0 denotes the

velocity at the inlet of the tube.

3. Numerical computation

3.1. Numerical method

The governing equations for mass, momentum and heat

transfer in the helical pipe were solved in the master Carte-

sian coordinate system with a control-volume finite difference

method (CVFDM) similar to that introduced by Patankar [33].

Fluent 6.2 [36] program was used as a numerical solver for the

present three-dimensional simulation. An unstructured (block-

structured) non-uniform grid system was used to discretize the

governing equations. Fig. 1(c) illustrates the grid topology used

on one cross-section. The convection term in the governing

equations was modeled with the bounded second-order upwind

scheme and the diffusion term was computed using the mul-

tilinear interpolating polynomials nodes Ni(X, Y, Z). The final


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Table 4

Grid independent test (NDe = 6325, NPr= 0.7, and = 0.1)

Total grids (cross-

sectional axial)NNu,max NNu,m Mori and

Nakayama [14]

1110 440 121.91 82.701280 440 125.75 82.92 84.761710

440 127.15 83.01

1860 440 127.15 83.011710 510 127.15 83.01

discrete algebraic equation for variable at each node is a set

of nominally linear equations that can be written as

app =

nb

anbnb + C (16)

where the subscript nb denotes neighbor value. The coefficients

ap and anb, contain convection and diffusion coefficients. C is

the source of in the control volume surrounding point p. The

SIMPLEC algorithm introduced by Van Doormaal and Raithby

[34] was used to resolve the coupling between velocity and pres-

sure. To accelerate convergence, the under-relaxation technique

was applied to all dependent variables. In the present study, the

under-relaxation factor for the pressure,p, was 0.3; that for tem-

perature, T, was 0.9; that for the velocity component in the i

direction, ui, was 0.5; that for body force was 0.8; that for kand

, was 0.7.

3.2. Convergence criteria

The numerical computation is considered converged when

the residual summed over all the computational nodes at the nth

iteration, Rn, satisfies the following criterion:

Rn

Rm 106 (17)

where Rm refers to the maximum residual value of variable

summed over all the computation cells after the mth iteration

and Rn, the value at the nth iteration.

A grid refinement study was conducted to determine an ade-

quate distribution. Boundary layer has been considered while

meshing the geometry in GAMBIT. Table 4 presents a com-parison of the predicted results at different grid distributions

(cross-sectional axial) for a fully developed turbulent flowin coiled tube. The sectional number refers to the total num-

ber of elements on one cross-section ( = constant) of the pipe.

Table 4 indicates that for the computation domain (0 <


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Fig. 3. Computed velocity contours in coiled flow inverter (one bend) with

= 0.1,NDe

= 6325,NPr

= 0.7at differentcrosssections: (a) = 15

, (b) = 30

,

(c) = 60, (d) = 90, (e) =120, (f) =180, (g) =270, (h) =360 and(i) =720.

ent workers [1920,2223]. No such rotation of velocity fields

in the laminar fully developed flow was observed in the study

of Kumar and Nigam [29]. This suggests that effect of torsion is

less significant for laminar flow as compared to turbulent flow

conditions. A fully developed flow was obtained at =270 asit can be seen clearly that the velocity fields does not have much

change after =270.The velocity field from the outlet of coiled tube was intro-

duced at the inlet of coiled flow inverter with one bend. It can be

seen from Fig. 3 that the orientation of flow field gets changeddue to the 90 bend. The above result reveals that the velocityfields, which had maximum velocity in coiled tube, would have

minimum velocity after 90 bend. Similarly, the velocity vectorsthat were at minimum velocity would have maximum velocity

Fig. 4. Computed velocity contours in coiled flow inverter (two bends) with

= 0.1,NDe = 6325,NPr= 0.7at differentcrosssections: (a) = 15, (b) = 30,

(c) = 60, (d) = 90, (e) =120, (f) =180, (g) =270, (h) =360 and

(i) =720.

Fig. 5. Computed velocity contours for fluids of various Prandtl number in: (i)

torus (H= 0) and (ii) helical coil (H= 0.02 m) for NDe = 6325 and =360.

after inversion. The initial inversion in case of laminar flow [29]

was observed at 120 while in the present case it was 180. Thisobservation is expected due to intense turbulence. The flow then

again reassembled back to almost 90. It can be also seen thatthe velocity fields does not have much change after =270;indicating a fully developed flow. The oscillations of velocity

fields were observed due to torsion.

The velocity fields from the outlet of first bend were intro-

ducedattheinletofsecondbend. Fig.4 showscomputedvelocity

fields at different cross sections in coiled flow inverter havingtwo bends. The same phenomenon was observed after the sec-

ond bend as observed after the first bend. The orientation of flow

fields again changed almost to 90.Computations were also carried out in torus and helical coil

with definite pitch in order to see the effect of torsion. Fig. 5

shows the computed velocity fields of fluids with different

Prandtl number at the outlet of torus and helical coil. The fig-

ure shows that the velocity contours in both the geometry were

gradually shifted towards outer wall as was increased. The

contours in torus were symmetric to the centerline between the

outer most point to the innermost point. The velocity distribu-

tion at the outlet was found to be similar for fluids of higher

Prandtl number. It was also observed that the velocity contoursin helical coil were asymmetric in nature because of rotational

flow, which is generated due to torsion. There was no change

in behavior of velocity contours even at higher Prandtl number.

This suggests that buoyancy has a little role to play in developed

turbulent flow as compared to developed laminar flow [35].

The complete axial velocity profiles for helical pipe and

coiled flow inverter at different angular planes are shown in

Figs. 68. The axial velocity profiles on horizontal centerline

at different axial planes in coiled tube are shown in Fig. 6(i).

This figure shows that at low values of , the velocity profiles

werealmost symmetrical on the horizontal centerline. This result

agrees with the axial velocity field. With the increase of , the


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Fig.6. Developmentof axial velocityprofile on: (i) horizontal centerlineand (ii)

vertical centerline in coiled tube with = 0.1, NDe = 6325, NPr = 0.7 at different

axial planes: (a) = 15, (b) = 30, (c) = 60, (d) = 90, (e) =120, (f)

=180, (g) =270 and (h) =360.

axial velocity became asymmetrical. At the horizontal center-

line, the maximum velocity shifted to the outside of the pipe

because of the unbalanced centrifugal force on the main flow.

The velocity profiles in turbulent flow were shifted more toward

theouterwall as compared to thevelocityprofiles in laminar flow

presented by Kumar and Nigam [29]. It was also observed that

the velocity profiles on horizontal centerline were flatter than

in the case of laminar flow. The axial velocity profiles at the

vertical centerline are shown in Fig. 6(ii). The velocity became

slightly asymmetrical. The asymmetrical nature may be due to

the torsion effect acting on the fluid. The profiles on the ver-

tical centerline were also flatter in turbulent flow as compared

to laminar flow. The axial velocity profiles at different axial

planes on horizontal centerline in coiled tube with one bend are

shown in Fig. 7(i). It can be seen that the velocity profiles were

Fig.7. Developmentof axial velocityprofile on: (i) horizontal centerlineand (ii)

vertical centerline, in coiled flow inverter (one bend) with = 0.1, NDe = 6325,

NPr= 0.7at differentaxialplanes:(a) = 0,(b) = 15, (c) = 30,(d) = 60,

(e) = 90, (f) =120, (g) =180, (h) =270 and (i) =360.

Fig. 8. Development of axial velocity profile on: (i) horizontal centerline and

(ii) vertical line in coiled flow inverter (two bends) with = 0.1, NDe = 6325,

NPr = 0.7at differentaxialplanes:(a) = 0,(b) = 15,(c) = 30,(d) = 60,

(e) = 90, (f) =120, (g) =180, (h) =270 and (i) = 360.

almost asymmetrical at initial stage but as increased, the pro-

files became more and more symmetrical in nature. Fig. 7(ii)

shows the development of axial velocity profile on vertical cen-

terline at different axial planes in coiled tube with one bend.

With the increase of , the axial velocity shifted to the outside

of the pipe because of the unbalanced centrifugal force. Fig. 7

shows that the velocity profile at vertical and horizontal center-

line was interchanged from that of velocity profile in straight

coiled tube. This is because of reorientation of flow fields due

to 90 bend. Fig. 8 shows the axial velocity profiles at differentaxial planes in coiled tube with two bends. Same phenomenon

is obtained after second bend. The velocity profiles at vertical

and horizontal centerline were again interchanged.

4.2. Description of temperature fields

Figs. 911 represent the development of temperature field

at different axial positions in the coiled tube and coiled flow

inverter with one and two bends, respectively. Fig. 9 shows

the variation of computed temperature contours at various

cross-sectional planes in coiled tube. The developments of the

temperature fields agree with that of the axial velocity fields.

There was negligible effect of secondary flow near the tube

inlet. It can be seen that as increased, the secondary flow wasenhanced. Due to the secondary flow, the temperature fields with

lower values were pushed towards outer wall region. A com-

parison of dimensionless temperature contours at =270 and =360 reveals that there was not much change in temperaturedistribution. This indicates that the temperature boundary layer

has become fully developed, hence heat transfer is fully devel-

oped at = 270. When is small, the temperature fields were

symmetric to the centerline between the outer most point to the

innermost point. But at the later stage, rotation of temperature

contours was found, similar to the velocity fields. The tempera-

ture contours in the present study were found to be more uniform

as compared to the laminar flow [29]. In the laminar flow, two


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Fig. 9. Computed temperature contours in coiled tube with = 0.1, NDe = 6325,

NPr= 0.7 at different cross sections: (a) = 15, (b) = 30, (c) = 60, (d)

= 90, (e) =120, (f) =180, (g) =270, (h) =360 and (i) =720.

Dean roll cells were observed which were absent in this study.

Similar distinction between laminar and turbulent temperature

profiles was also made by Yang and Ebadian [1]. The absence

of Dean roll cells in turbulent flow may be due to the reason

that the secondary flow is not as prominent as in case of laminar

flow. Also the thermal diffusivity is high in case of turbulent

flow. Fluids at high thermal diffusivity rapidly adjust their tem-

perature to that of their surroundings, because they conduct heat

quickly. Hence there was no incursion or penetration in fluid as

was found in laminar flow. As a result the Dean Roll cells were

absent in fluid with turbulent flow. Fig. 10 shows the devel-opment of temperature contours at different axial positions in

coiled flow inverter with one bend. The direction of temperature

Fig. 10. Computed temperature contours in coiled flow inverter (one bend) with

= 0.1,NDe = 6325,NPr= 0.7at differentcrosssections: (a) = 15, (b) = 30,

(c) = 60, (d) = 90, (e) =120, (f) =180, (g) =270, (h) =360 and

(i) =720.

Fig. 11. Computed temperature contours in bent helix (two bends) with = 0.1,

NDe = 6325, NPr= 0.7 at different cross sections: (a) = 15, (b) = 30, (c) = 60, (d) = 90, (e) =120, (f) =180, (g) =270, (h) =360 and (i) =720.

contours gradually changed as was increased. It can be seen

that there was no appreciable change in distribution of tempera-

ture contours after =270. This further confirms that the flowwas fully developed. The maximum temperature was obtained

at the inner wall and minimum temperature at the outer wall

of the tube. Fig. 11 shows computed temperature fields at dif-

ferent cross sections in coiled flow inverter having two bends.

The same phenomenon was observed after the second bend as

observed after thefirst bend. It was also observed that thetemper-ature became more and more uniform as increased; indicating

a good mixing of fluids due to flow inversion. Further work is

being done considering the affect of variable properties on fluid

flowing through coiled flow inverter. The work will be included

in the part II of the paper.

Fig. 12 shows the temperature contours for fluids of vari-

ous Prandtl number at the outlet of torus and helical coil. It

was found that due to secondary flow, the contours were pushed

towards the outer wall of both the geometries. Slight distortion

of temperature contours were found in the helical coil similar

to the velocity contours. It was also observed that as the Prandtl

number was increased, the temperature fields became more and

more uniform. Reason for this can be because of the fact thatPrandtl number is the ratio of momentum diffusivity (kinematic

viscosity) to thermal diffusivity. Thermal diffusivity is the ratio

of heat conducted through the material to the heat stored per unit

volume. When Prandtl number is small, it means that the ther-

mal diffusivity is high and heat diffuses very quickly compared

to the velocity (momentum). If Prandtl number is high then the

thermal diffusivity is small. This means that a big part of the heat

is absorbed by the fluid and only a small portion is conducted

through it.

The development of computed temperature profile on hor-

izontal and vertical centerline at different axial planes in

coiled tube and coiled flow inverter at one and two bends, are


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Fig. 16. Fully developed axial velocity profile on: (i) horizontal centerline and

(ii) vertical centerline in coiled tube and different coiled flow inverter with

= 0.1, NDe = 6325, NPr= 0.7.

Fig.17. Fullydevelopedtemperature profile on: (i) horizontal centerline and (ii)

vertical centreline in coiled tube and different coiled flow inverter with = 0.1,

NDe = 6325, NPr= 0.7.

Fig. 18. Comparison of computed fullydeveloped Nusselt number forNPr=0.7

and = 0.1, with the data of Mori and Nakayama [14] and Schmidt [15].

The empirical formula presented by Schmidts experimental

work[15] is as following:

NNufd

NNus= 1.0 + 3.6(1 )0.8 (19)

where NNus = 0.023N0.8Re N0.4Pr .It can be observed from Fig. 18 that the present predictions

of Nusselt number were in good agreement with the available

results. The maximum deviationbetween the present predictions

and the empirical correlation is less than 5%.

The results of the numerical computations for enhancement

of heat transfer with Dean number (NDe) in coils with two bends

is shown in Fig. 19. From the Fig. 19, it can beseen thatthe Nus-selt number increased with the increase in Dean number. It can

also be observed that there was 413% heat transfer enhance-

ment in coiled flow inverter as compared to the straight helical

coil. The figure also shows 3545% increase of heat transfer

in coiled flow inverter as compared to straight tube. The heat

gain for straight helical coil over straight tube was found to be

Fig. 19. Nusselt number variation with Dean number at NPr= 0.7 and = 0.1;

in straight tube, coiled tube and coiled flow inverter with two bends.


12/13


Fig. 20. Nusselt number variation with Prandtl number at NDe = 6325 and

= 0.1; in straight tube, coiled tube and coiled flow inverter with two bends.

3235%. Computations were also carried out to study the effect

of Prandtl number. Fig. 20 shows 412% enhancement of heat

transfer in coiled flow inverter as compared to straight coiled

tube as Prandtl number was increased. It can also be observed

that the heat transfer in coiled flow inverter was 2443%

more as compared to straight tube with increase in Prandtl

number.

Itcanbeseenfrom Fig.21 thatthe increase of friction factorin

bent coil tube to that of straight coiled tube at low values of Dean

number was about 9%. The percentage increase of friction fac-

tor in coiled flow inverter to straight coiled tube decreased with

increase in value of Dean Number and it reduces to about 2% at

NDe = 9487. This may be due to the fact that the hydrodynamic

effect of both helical coil and coiled flow inverter becomes com-

parable at higher Dean number. So both configurations becomemore and more equivalent in performance. Similar observations

were found by Chagny et al. [26] who studied the chaotic flow

obtained by alternately turning the axis of curved tubes (half cir-

cles) by 90. The Fig. 21 also shows 2930% increase in frictionfactor in coiled flow inverter as compared to straight tube. The

increase in friction factor of straight helical coil over straight

tube is 2328%.

Fig. 21. Friction factor variation with Dean number at NPr = 0.7 and = 0.1; in

straight tube, coiled tube and coiled flow inverter with two bends.

6. Conclusion

In the present study, hydrodynamics and heat transfer of tur-

bulent forced convection in an innovative heat exchanger having

coils with one and two bends with circular cross section have

been investigated. The developments of velocity fields at differ-

ent axial positions in straight helical coil and coiled flow inverter

with one and two bends have been reported under turbulent flow

conditions. A slight rotation of velocity contours was observed

in the coiled tubes. This may be due to the torsion caused by

turbulent flow. It was found that buoyancy doesnt have major

role in the turbulent forced convection of fluid in coiled tubes.

The velocity fields were increasingly uniform as the numbers of

bends in the coiled tube were increased. This may be because

of increase in radial mixing. Similar results were found with the

temperature fields in coiled tube and coiled flow inverter. The

cold regions present in the straight coil are modified in bent coil

due to radial mixing. Eventually, the heat transfer was enhanced.

It was found that the enhancement of heat transfer in coiled flow

inverter as compared to straight coil and straight tube is morethan the increase in friction factor. The effect of heat transfer

in fluid of higher Prandtl number was also studied. The study

shows that the heat transfer increases with increase in Prandtl

number.

Acknowledgements

The authors gratefullyacknowledge the Ministry of Chemical

and Fertilizers, GOI, India for funding the project.

Appendix A. Nomenclature

A area (m2)

cf skin friction coefficient

C1, C2, C turbulent model constant

Cp specific heat (kJ/kg K)

dc diameter of the coil (m)

dh hydraulic diameter of the helical pipe (m)

dt diameter of the helical pipe (m)

D dispersion coefficient (m2/s)

H pitch (m)

k turbulent kinetic energy (m2/s2)

L length of reactor (m)

n coordinate direction perpendicular to a surfaceNDe Dean number [NDe = NRe

]

NNu Nusselt number [NNu, = qwdh/(Tw Tb)]NPr molecular Prandtl number [NPr = Cp/]NRe Reynolds number [NRe = u0dt/]p pressure (N/m2)

q heat flux (W/m2)

T temperature (K)

Tb fluid bulk temperature on one cross-section (K)

u velocity component in flow direction (m/s)

u0 inlet velocity (m/s)

ui velocity component in i-direction (i = 1, 2 and 3) (m/s)

u root-mean-square turbulent velocity fluctuation (m/s)


13/13


U axial velocity component (m/s)

Uavg average velocity over a cross-section (m/s)

x spatial position (m)

xi master Cartesian coordinate in i-direction (i = 1, 2 and

3) (m)

Greek symbols

T inverse effect Prandtl number for T

inverse effect Prandtl number for

inverse effect Prandtl number for

thermal conductivity (W/m K)

ij dirac delta function

dissipation ratio of turbulent kinetic energy (m2/s2)

axial angle () curvature ratio (= dt/dc) viscosity (kg/m s)

eff effective viscosity (kg/ms)

non-dimensional temperature ((Tb Tw)/(Tin Tw)) circumferential quantity

density of fluid (kg/m3) shear stress (N/m2)

Subscripts

0 inlet conditions

2nd secondary flow

b bulk quantity

m circumferential average quantity

t turbulent quantity

w wall condition

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