Fluid Flow and Heat Transfer Characteristics across an Internally Heated Finned … · ·...

J. Energy Power Sources Vol. 1, No. 6, 2014, pp. 296-303 Received: August 31, 2014, Published: December 30, 2014

Journal of Energy and Power Sources

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Fluid Flow and Heat Transfer Characteristics across an

Internally Heated Finned Duct

Sofiane Touahri and Toufik Boufendi

Energy Physics Laboratory, Physics Dept., Constantine1 University, Constantine, Algeria

Corresponding author: Toufik Boufendi ([email protected])

Abstract: We numerically study the fluid flow and the heat transfer characteristics in a horizontal pipe equipped by longitudinal and transversal fins attached on its internal surface used in several areas of thermal sciences. The longitudinal fins number is: Two vertical, four and eight, while the transversal fins number is eight. The considered heights fins are 0.12 and 0.24 cm. The convection in the fluid domain is conjugated to conduction in the pipe and in the fins thickness. The physical properties of the fluid are thermal dependant and the heat losses to the ambient are considered. The model equations are numerically solved by a finite volume method with a second order discretization. As expected for the longitudinal fins, the axial Nusselt number increase with increasing of number and height of fins. For Gr = 5.1 × 105, the Nusselt number without fins is equal to 13.144. The introduce of longitudinal fins gives a Nusselt number equal to 16.83, 22.52 and 32.06 for two vertical, four and eight fins respectively. The participation of fins located in the lower part of the tube on the improvement of heat transfer is higher than the upper fins. The longitudinal fins participate directly on increase the heat transfer; this is justified by the large local Nusselt number along the fins interface. This participation is moderate in the case of transverse fins, these latter are used to mix the fluid for increase the local Nusselt number in the axial sections following the transversal fins. Keywords: Conjugate heat transfer, mixed convection, internal fins, numerical simulation.

Nomenclature:

D Pipe diameter, (m)

Gr* Modified Grashof number, Gr* = gβGDi

5

Ksν2

G* Non-dimensional heat generation= Ks

Re0 Pr0

H Height of the fin, (m)

h Heat transfer coefficient, (W/m2K)

K Thermal conductivity, (W/mK)

K* Non-dimensional thermal conductivity= K

K0

L* Non-dimensional pipe length=

L

Di

Nu(θ,z) Local Nusselt number = K K0⁄

Nu Nusselt number (dimensionless)

P Pressure, (Pa)

P* Non-dimensional pressure = (P-P0) ρ0V02

Pr Prandtl number= ν α⁄

q Heat flux, (W/m2)

R Radial coordinate, (m)

Re Reynolds number = V0Di ν0⁄

r* Non-dimensional radial coordinate= r Di⁄

T Temperature, (°C)

T* Non-dimensional temperature= (T-T0) (GDi2 Ks )

V Velocity, (m/s)

V* Non-dimensional velocity= V V0⁄

Y+ Dimensionless wall distance

Z Axial coordinate, (m)

Z* Non-dimensional axial coordinate = z Di⁄

1. Introduction

Finned tubes are often used in many engineering

sectors for extend the contact surface between the tube

wall and the fluid and improve the heat transfer; the

researchers have studied the problem of optimizing the

shape and geometry of attached fins in order to increase

heat transfer effectiveness. Most of the studies

performed on this optimization consider longitudinal

fins which have symmetrical lateral profiles; this

assumption simplifies the treatment of the problem

with regard to the boundary conditions and gives

Fluid Flow and Heat Transfer Characteristics across an Internally Heated Finned Duct

297

symmetrical results concerning velocity and

temperature profiles. Many investigations, both

experimental and numerical, have been conducted for

different kinds of internally finned tubes. An analytical

model for fully developed turbulent air flow in

internally finned tubes and annuli was presented by [1].

In this study, the longitudinal fins are attached in the

inner wall, and the constant heat flux, is the thermal

boundary condition at the inner surface. The results

concern the heat transfer and the pressure drop

coefficients. A combined numerical and experimental

study of plate-fin and tube heat exchangers was

examined by [2]. The detailed numerical results of

pressure drop and heat transfer coefficient are

presented. A numerical study on hydrodynamically

fully developed, thermally developing flow inside

circular tubes with internal longitudinal fins having

tapered lateral profiles was conducted by [3]. The

results showed significant heat transfer enhancement

with the inclusion of internal fins. Water and engine oil

were assumed as fluids in their numerical studies, and

they concluded water to be a better coolant as

compared to engine oil. In Ref. [4], a numerical study

of thermally developing flow in an elliptical duct with

four longitudinal internal fins of zero thickness is

considered. A control volume based on finite

difference technique was used and an optimum value of

the local Nusselt number was obtained as a function of

the fin height. Ref. [5] performs an experimental

analysis of heat transfer in an internally finned tube, the

experimental results were compared with results from

the smooth channel tube, and a significant

improvement in heat transfer was observed for

internally finned cases. Similar studies were also

examined numerically and experimentally by [6-10].

In this work, we studied numerically the heat

transfer by mixed convection in horizontal pipe

equipped by longitudinal and transversal attached fins

on its internal wall. The mixed convection is conjugate

to thermal conduction in the pipe and fins walls. The

physical properties of the fluid are thermo- dependent

and the heat losses with the external environment are

considered. The objective of our work is study the

enhancement gives to the heat transfer by using

different shape of fins.

2. The Geometry and Mathematical Model

Fig. 1 illustrates the problem geometry. We consider

a long horizontal pipe having a length L = 1 m, an

outside diameter D0 = 1 cm and an inside diameter Di =

0.96 cm, this later is equipped by longitudinal and

transversal attached fins on its internal surface. The

longitudinal fins are fixed at (θ = 0, π/4, π/2, 3π/4, π,

5π/4, 3π/2 and 7π/4) while the transversal fins are

attached at eight axial stations from z* = 9.4404 to z* =

82.3594. The pipe and fins are made of Inconel having

a thermal conductivity Ks = 20 W m-1 K-1. An electric

current passing along the pipe (in the solid thickness)

produced a heat generation by Joulean effect. This heat

is transferred to distilled water flow in the pipe. At the

entrance the flow is of Poiseuille type with an average

axial velocity equal to 7.2 × 10-2 m s-1 and a constant

temperature of 15 °C. The density is a linear function of

temperature and the Boussinesq approximation is

adopted. The physical principles involved in this

problem are well modeled by the following non

dimensional conservation partial differential equations

with their initial and boundaries conditions.

(a)

(b)

Fig. 1 Pipe with (a) longitudinal fin; (b) transversal fin.

*eD*

iD

*eD

*iD


298

2.1 Modeling Equations

At 0* =t , 0**** ==== TVVV zr θ (1)

At 0* >t

Mass conservation equation:

( ) 011

*

**

***

**=

∂∂+

∂∂+

∂∂

z

VV

rVr

rrz

r θθ (2)

Radial momentum conservation equation:

( ) ( )

( )( ) ( ) ( )

∂∂+−

∂∂+

∂∂

++∂∂−=−

∂∂

+∂∂+

∂∂+

∂∂ ∗

***

**

***

**0

*20

*0

*

*

*

2***

*

***

*****

*

11

Re

1

)(cosRe

11

zrrrr

rz

rrrr

zrrr

rr

TGr

r

P

r

VVV

z

VVr

VVrrrt

V

τττθ

τ

θ

θ

θθθ

θ

θ

(3)

Angular momentum conservation equation:

( ) ( )( )

( ) ( ) ( )

∂∂+

∂∂+

∂∂

+−∂∂−=+

∂∂

+∂∂+

∂∂+

∂∂

∗

∗∗

**

**

*2**2*

0

*20

*0

**

****

*

**

******

11

Re

1

sinRe

1

11

zr

rz

r

zrr

rr

TGrP

rr

VVVV

z

VVr

VVrrrt

V

θθθθ

θθ

θθθθ

ττθ

τ

θθ

θ

(4)

Axial momentum conservation equation:

( ) ( )( )

( ) ( ) ( )

∂∂+

∂∂+

∂∂

+∂∂−=

∂∂

+∂∂+

∂∂+

∂∂

**

**

****

0

*

***

*

***

******

*

11

Re

1

11

zzzrz

zz

zzrz

zrr

rr

z

PVV

z

VVr

VVrrrt

V

ττθ

τ

θ

θ

θ

(5)

Energy Conservation Equation

( ) ( )( )

( ) ( ) ( )

∂∂+

∂∂+

∂∂

−=∂∂

+∂∂+

∂∂+

∂∂

∗

∗

zr

z

r

qz

qr

qrrrPrRe

GTVz

TVr

TVrrrt

T

**

***

**00

****

***

*****

*

111

11

θ

θ

θ

θ

(6)

where ( )

=fluidthein0

solidtheinPrRe/ 00*

* SKG

The viscous stress tensor components are:

*

*** 2

r

Vrrr ∂

∂= μτ ,

∂∂

+

∂∂==

θμττ θ

θθ

*

**

*

***** 1 r

rrV

rr

V

rr

+

∂∂

=*

**

*** 1

2r

VV

rr

θμτ θ

θθ ,

∂∂

+∂∂

==θ

μττ θθθ

*

**

**** 1 z

zzV

rz

V (7)

2∗

∗∗∗

∂

∂=z

Vzzz μτ ,

∂∂+

∂∂==

*

*

**

**** 1

z

V

rr

V rzzrrz μττ

The heat fluxes are:

*

***

*

*

*

**

*

*** and,

z

TKq

T

r

Kq

r

TKq Zr ∂

∂−=∂∂−=

∂∂−=

θθ (8)

2.2 The Boundaries Conditions

The previous differential equations are solved with

the following boundaries conditions:

(1) At the pipe entrance: z* = 0

• In the fluid domain: 5.0≤≤0 *r and πθ 2≤≤0

)4-1(2,0 2***** rVTVV zr ==== θ (9)

• In the solid domain: 5208.0≤≤5.0 *r and πθ 2≤≤0

0**** ==== TVVV zr θ (10)

(2) At the pipe exit: z* = 104.17

• In the fluid domain: 5.0≤≤0 *r and πθ 2≤≤0

0*

**

**

*

*

*

*

*

=

∂∂

∂∂=

∂∂=

∂∂=

∂∂

z

TK

zz

V

z

V

z

V zr θ (11)

• In the solid domain: 5208.0≤≤5.0 *r and

πθ 2≤≤0

0*

**

**** =

∂∂

∂∂===

z

TK

zVVV zr θ (12)

(3) At the outer wall of the pipe: r* = 0.5208

( )

+=∂∂−

===*

0*

**

*** 0

TK

Dhh

r

TK

VVV

icr

zr θ (13)

( )( )∞2∞

2 TTTThr ++= εσ (14)

The emissivity of the outer wall ε is arbitrarily

chosen to 0.9 while hc is derived from the correlation of

[11] valid for all Pr and for Rayleigh numbers in the

range 10−6 ≤ Ra ≤ 109.

[ ]

( )( )2

27/816/96/1 Pr/559.01/387.06.0

/

++=

=

air

airic

Ra

KDhNu

(15)

with

( )[ ]airairair

airair

oo PrDTzRTg

Ra αννα

θβ/,

-,, 3∞ == (16)

In Eq. (16) the thermophysical properties of the air

ambient are evaluated at the local film temperature

given as: [ ] 2),,( ∞TzRTT ofilm += θ .

In our calculations we have considered the solid as a

fluid with a dynamic viscosity equal to 1030. This very


299

large viscosity within the solid domain ensures that the

velocity of this part remains null and consequently the

heat transfer is only by conduction deducted from Eq.

(6).

2.3 The Nusselt Numbers

At the cylindrical solid-fluid interface, the local

Nusselt number is defined as:

( )

∂∂== =

)(-),,5.0(

),(),(

****

5.0***

0

**

*

zTzT

rTK

K

DzhzNu

b

ri

θθ

θ (17)

The axial Nusselt number for the cylindrical

interface is:

∫2

0

** ),(2

1)(

π

θθπ

dzNuzNu = (18)

Finally, we can define an average Nusselt number

for the whole cylindrical solid-fluid interface:

( )( ) ∫∫2

0

17.104

0

**),(17.1042

1π

θθπ

ddzzNuNuA = (19)

At the longitudinal fin interface, the local Nusselt

number is defined as

( )( )

∂∂==

=

)(),,(

),(),(

*****

***

0

****

zTzrT

TrK

K

DzrhzrNu

mfin

h fin

θ

θ θθ (20)

The axial Nusselt number for the longitudinal fin is:

( ) ( )*

*****

***

*

****

*

∫

∫*

**

*

**

)(),,(

1

),(1

)(

drzTzrT

TrK

H

drzrNuH

zNu

i

i

fin

i

i

R

HR mfin

R

HR

∂∂=

=

=

θ

θ θθ (21)

The average Nusselt number for the longitudinal fin

interface is defined as:

∫*

0

***

)(1

L

A dzzNuL

Nu =

(22)

At the transversal fin interface, the local Nusselt

number is

( )( )

∂∂==

=

)(),,(

),(),(

*****

**

0

**

zTzrT

zTK

K

DrhrNu

mfin

zzh fin

θθθ (23)

The axial Nusselt number for the transversal fin

interface is defined as:

( )( )( )

∫∫2

0

******

**

**

*

**

**

)(),,(2

1)(

π

θθπ

i

i

fin

R

HR mfin

zzddr

zTzrT

zTK

HzNu

∂∂=

= (24)

3. Numerical Resolution

For the numerical solution of modeling equations,

we used the finite volume method well described by

[12]; the using of this method involves the

discretization of the physical domain into a discrete

domain constituted of finite volumes where the

modeling equations are discretized in a typical volume.

We used a temporal discretization with a truncation

error of ∆t*2 order. The convective and nonlinear

terms have been discretized according to the

Adams-Bashforth numerical scheme, with a truncation

error of ∆t*2 order, the diffusive and pressure terms

are implicit. Regarding the spatial discretization, we

used the central differences pattern with a truncation

error of ∆r* 2, ∆θ 2 and ∆z* 2

order. So our

spatio-temporal discretization is second order. The

mesh used contains 52 × 88 × 162 points in the radial,

angular and axial directions. The considered time step

is ∆t*=5 × 10-4and the time marching is pursued until

the steady state is reached. The steady state is

controlled by the satisfaction of the global mass and

energy balances as well as the leveling off of the time

evolution of the hydrodynamic and thermal fields.

The accuracy of the results of our numerical code has

been tested by the comparison of our results with those

of other researchers. A comparison with the results of

[13] who studied the non conjugate and the conjugate

mixed convection heat transfer in a pipe with constant

physical properties of the fluid. Some of their results

concern the simultaneously developing heat transfer

and fluid flow in a uniformly heated inclined pipe

( 40=α ).The controlling parameters of the problem

are: Re = 500, Pr = 7.0, Gr = 104 and 106, 90=iDL ,

583.0=io DR , 700 =KKs . The used grid is 40 × 36

× 182 in the r*, θ and z* directions, respectively. We

reproduced the results of the cited reference with the


300

Fig. 2 Axial evolution of the circumferentially mean axial Nusselt number; A comparison with the result of [13].

first order calculus code concerning the conjugate and

non conjugate mixed convection. In Fig. 2, we

illustrate the axial evolution of the circumferentially

averaged Nusselt number. It is seen that there is a good

agreement between our results and theirs.

4. Results and Discussion

4.1 Development of the Secondary Flow

All the results presented in this paper were

calculated for Reynolds number, Re = 606.85, and the

Prandtl number, Pr = 8.082, while the Grashof number

is equal to 5.1 × 105. The obtained flow for the studied

cases is characterized by a main flow along the axial

direction and a secondary flow influenced by the

density variation with temperature, which occurs in the

plane (r* − θ), this flows are presented for eight fins in

the longitudinal case and four fins in the transversal

case. In the reference case (forced convection) the

transverse motion is nonexistent; the only flow is in

axial direction. In the presence of volumetric heating in

the pipe and fins wall, a transverse flow exists and

explained as follows: The hot fluid moves along the

hot wall from the bottom of the outer tube (θ=π)

upwards (θ=0) and moves down from the top to the

bottom along the centre of tube. The vertical plane

passing through the angles (θ=0) and (θ=π) is a plane

of symmetry. The transverse flow in the (r*, θ) plane is

represented by counter rotating cells; the cells number

is proportional to longitudinal fins number.

Fig. 3 The secondary flow vectors at the pipe exit for longitudinal fins.

Fig. 4 The secondary flow vectors at the fourth transversal fin section.

In Fig. 3, we present the secondary flow vectors at the

pipe exit for the case of longitudinal fins. For the case

of transversal fins, the position of fins is only in

selected axial sections. Far from these sections, the

secondary motion is similar to that of simple

cylindrical pipe. The Fig. 4, illustrates the secondary

flow vectors at the fourth transversal fin section (z* =

40.6914).

4.2 The Axial Flow Development

At the entrance, the axial flow is axisymmetric with

the maximum velocity in the center of the pipe. In the

presence of volumetric heating in the pipe and the fins

walls, the configuration of the axial flow completely

changes because the secondary flow causes an angular

Z*=104.17

Z*=40.6914

0,000 0,003 0,006 0,009 0,012 0,015 0,018 0,021 0,0240

5

10

15

20

25

30

35

40

45

50

0

5

10

15

20

25

30

35

40

45

50Ouzzane and Galanis [13]: Gr =104 non conjugate case

Gr =106 non conjugate case

Gr =104 conjugate case

Our results: Gr =104 non conjugate case

Gr =106 non conjugate case

Gr =104 conjugate case Nu

z*/(RePr)


301

Fig. 5 Axial velocity profiles at the pipe exit for longitudinal fins.

Fig. 6 Axial velocity profiles at the fourth transversal fin section.

variation which has a direct influence on the

distribution of axisymmetric axial flow. In Fig. 5, the

axial flow is presented at the exit of the pipe for the

case of longitudinal fins (height H* = 0.1875). In Fig. 6,

the axial flow is presented at the fourth transversal fin

section (z* = 40.6914). Through these figures, it is clear

that the axial velocity is null in the fins walls.

4.3 Development of the Temperature Field

In the presence of volumetric heating, a transverse

with flow exists and thus changes the axisymmetric

distribution of fluid and gives it an angular variation,

this variation explained as follows: The hot fluid near from

Fig. 7 The isotherms at the pipe exit for longitudinal fins.

Fig. 8 The isotherms at the fourth transversal fin section.

the pipe and fins walls moves upwards under the

buoyancy force effect, the relatively cold fluid

descends down in de centre of the pipe. A permanent

generation of heat in the pipe and the fins walls

imposes a continuous increase of the temperature of the

fluid up to the exit of the pipe.

The obtained results show that at given section, the

maximum fluid temperature T* is all the time located at

r* = 0.5 and θ = 0 (top of solid-fluid interface), because

the hot fluid is driven by the secondary motion towards

the top of the pipe. The isotherms are presented at the

exit of the pipe for the case of longitudinal fins having

height H* equal to 0.1875 in Fig. 7 and at the fourth

transversal fin section (z* = 40.6914) in Fig. 8.

Z*=104.17

2.3672.2682.1702.0711.9731.8741.7751.6771.5781.4791.3811.2821.1841.0850.9860.8880.7890.6900.5920.4930.3950.2960.1970.0990.000

Z*=40.6914

3.1002.9712.8422.7132.5832.4542.3252.1962.0671.9381.8081.6791.5501.4211.2921.1631.0330.9040.7750.6460.5170.3880.2580.1290.000

Z*=104.17

0.1710.1640.1570.1500.1430.1350.1280.1210.1140.1070.1000.0930.0860.0780.0710.0640.0570.0500.0430.0360.0290.0210.0140.0070.000

Z*=40.6914

0.1700.1630.1560.1490.1420.1350.1280.1210.1140.1060.0990.0920.0850.0780.0710.0640.0570.0500.0430.0350.0280.0210.0140.0070.000


302

4.4 The Heat Transfer

The phenomenon of heat transfer has been

characterized in terms of Nusselt numbers calculated at

the inner wall of the pipe, obtained by Eq. (17) and

those calculated at the fins wall, obtained by Eq. (20)

and Eq. (23). The variation of local Nusselt number at

cylindrical interface is presented in Fig. 9 for the case

of longitudinal fins and in Fig. 10 for the case of

transversal fins. The local Nusselt number of

longitudinal fins placed in the right side of the pipe at

(θ = 0, π/4, π/2, 3π/4, π) is shown in Fig. 11. The local

Fig. 9 The local Nusselt number variation at cylindrical interface for longitudinal fins case.

Fig. 10 The local Nusselt number variation at cylindrical interface for transversal fins case.

Nusselt number takes a maximum value equal to 63.56

on the fin placed at (θ = 3π/4), z* = 104.17 and r* =

0.3841. The comparison of axial Nusselt numbers

between finned tube and smooth tube is shown in Fig.

12. Quantitatively, there is a large increase in the axial

Nusselt when the number of fins is increased. At the

exit of the pipe, the axial Nusselt number is equal to

15.79, 23.47, 31.34 and 47.31 for the cases: Smooth tube,

(a)

(b)

(c)

(d)

48.73046.70044.66942.63940.60838.57836.54734.51732.48730.45628.42626.39524.36522.33520.30418.27416.24314.21312.18210.152

8.1226.0914.0612.0300.000

48.83046.79544.76142.72640.69238.65736.62334.58832.55330.51928.48426.45024.41522.38020.34618.31116.27714.24212.20810.1738.1386.1044.0692.0350.000

6 3 . 5 6

5 8 . 2 6

5 2 . 9 6

4 7 . 6 7

4 2 . 3 7

3 7 . 0 7

3 1 . 7 8

2 6 . 4 8

2 1 . 1 8

1 5 . 8 9

1 0 . 5 9

5 . 2 9

0 . 0 0

6 3 . 5 6

5 8 . 2 6

5 2 . 9 6

4 7 . 6 7

4 2 . 3 7

3 7 . 0 7

3 1 . 7 8

2 6 . 4 8

2 1 . 1 8

1 5 . 8 9

1 0 . 5 9

5 . 2 9

0 . 0 0

6 3 . 5 6

5 8 . 2 6

5 2 . 9 6

4 7 . 6 7

4 2 . 3 7

3 7 . 0 7

3 1 . 7 8

2 6 . 4 8

2 1 . 1 8

1 5 . 8 9

1 0 . 5 9

5 . 2 9

0 . 0 0

6 3 . 5 6

5 8 . 2 6

5 2 . 9 6

4 7 . 6 7

4 2 . 3 7

3 7 . 0 7

3 1 . 7 8

2 6 . 4 8

2 1 . 1 8

1 5 . 8 9

1 0 . 5 9

5 . 2 9

0 . 0 0


303

(e)

Fig. 11 The local Nusselt number variation at longitudinal fins interface: (a) fin placed at (θ = 0); (b) at (θ = π/4); (c) at (θ = π/2); (d) at (θ = 3π/4); (e) fin placed at (θ = π).

0 10 20 30 40 50 60 70 80 90 100

10

15

20

25

30

35

40

45

500 10 20 30 40 50 60 70 80 90 100

10

15

20

25

30

35

40

45

50

Axi

al N

usse

lt n

umbe

r

Z*

smooth tube

2 fins H*=0.125 4 fins H*=0.125 8 fins H*=0.125

Fig. 12 Variation of the axial Nusselt number for the different longitudinal fins studied cases.

two vertical fins, four fins and eight fins respectively.

The average Nusselt numbers for these cases are

13.144, 16.830, 22.523 and 32.066.

5. Conclusions

This study considers the numerical simulation of the

three dimensional mixed convection heat transfer in a

horizontal pipe equipped by internal, longitudinal and

transversal fins. The pipe and fins are heated by an

electrical intensity passing through its small thickness.

The obtained results show that the longitudinal fins

participate directly in improving the heat transfer; this

is justified by the high local Nusselt number at the

interface of longitudinal fins. By against, the transverse

fins participate in an indirect way in improving the heat

transfer their location facing the flow allowed to

rearrange the structure of the fluid for each passage

through the fins, which is used to mix the fluid and to

increase the heat transfer to the cylindrical interfaces.

The number and fins height are also important factors

in improving the heat transfer.

References

[1] S.V. Patankar, M. Ivanovic, E.M. Sparrow, Analysis of turbulent flow and heat transfer in internally finned tubes and annuli, Journal of Heat Transfer 101 (1979) 29-37.

[2] J.Y. Yang, M.C. Wu, W.J. Chang, Numerical and experimental studies of three-dimensional plate-fin and turbulent exchangers, Int. Journal of Heat and Mass Transfer 39 (1996) 3057-3066.

[3] I. Alam, P.S. Ghoshdastidar, A study of heat transfer effectiveness of circular tubes with internal longitudinal fins having tapered lateral profiles, Int. Journal of Heat and Mass Transfer 45 (6) (2002) 1371-1376.

[4] Z.F. Dong, M.A. Ebadian, A numerical analysis of thermally developing flow in elliptical duct with internal fins, Int. Journal of Heat and Fluid Flow 12 (2) (1991) 166-172.

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5 8 . 2 6

5 2 . 9 6

4 7 . 6 7

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