International Communications in Heat and Mass Transfer 32 (2005) 557–564

www.elsevier.com/locate/ichmt

Characteristics of forced convection heat transfer

in vertical internally finned tubeB

A. Al-Sarkhi*, E. Abu-Nada

Department of Mechanical Engineering, Hashemite University, Zarqa, 13115, Jordan

Available online 19 December 2004

Abstract

This work presents a numerical investigation of a vertical internally finned tube subjected to forced convection

heat transfer. The governing equations were solved numerically using the control volume technique. Nusselt

number, Nu, and friction factor multiplied by Reynolds number, fRe, are influenced greatly by the height and

number of the radial fins. The velocity and temperature distributions inside the tube depend on the number and

height of the radial fins. This paper suggests that for best heat transfer to be achieved there is an optimum

combination of fin numbers and height.

D 2004 Elsevier Ltd. All rights reserved.

Keywords: Forced convection heat transfer; Vertical internally finned tube; Control volume technique

1. Introduction

Internally finned tubes have received considerable attention because of the fact that they have been

used widely in industrial applications. Internally finned tube has found extensive use in heat exchangers.

When improvement in the process of heating or cooling is required, then better design of fin

compactness and spatial geometry is very essential. Several studies have been conducted to investigate

the effect of fin characteristics on heat transfer. Most of the relevant previous works have focused on

limited cases of the number and length of the internal fin shown in Fig. 1. Masliyah and Nandakumar

0735-1933/$ -

doi:10.1016/j.i

B Communic

* Correspond

E-mail add

see front matter D 2004 Elsevier Ltd. All rights reserved.

cheatmasstransfer.2004.03.015

ated by J.P. Hartnett and W.J. Minkowycz.

ing author. Tel.: +962 591 6600; fax: +962 591 6613.

ress: alsarkh@hu.edu.jo (A. Al-Sarkhi).

Fig. 1. Schematic of radial fins and the calculation domain.

A. Al-Sarkhi, E. Abu-Nada / Int. Commun. Heat and Mass Transf. 32 (2005) 557–564558

[1] have studied the heat transfer in internally finned tube. The internal fins were of triangular shape and

the number of fins was changed up to 24 fins and the length up to 0.8 of the tube radius. Finite element

method was used to analyze a laminar fully developed flow in an internally finned circular tube with

uniform axial heat flux around the wall. They conclude that the Nusselt number based on the inside

diameter was higher than that for a smooth tube without fins and also they found that for maximum heat

transfer there exists an optimum fin number for a given fin configuration. The influence of the buoyancy

force in combined free and forced convection in vertical tubes with internal fins was studied by Patankar

and Prakash [2]. A laminar fully developed flow was solved for the velocity and temperature using

finite difference technique. Straight radial fin configurations were analyzed for a range of Rayleigh

number and for various values of fin height up to 0.8 of the tube radius and number of fins up to 25 fins.

In their result they found that the buoyancy force increases the friction and heat transfer. The effect of

buoyancy is more significant when the number of fins is small and the fins are short. For smooth finless

circular tube, the fully developed combined forced and free convection has been solved analytically by

Morton [3] and also by Tao [4] and investigated numerically by Kemeny and Somers [5]. Numerous

studies have been focused on studying different shapes and arrangement of longitudinal shrouded fin

array [6–13].

The problem of laminar fully developed flow in circular tube with internal radial fins has been studied

numerically by Masiliyah and Nandakumar [1] and analytically by Hu and Chang [14]. However,

detailed studies for a wide range of fin number practically encountered in the industry for a wide range of

fin length are not available in literature. The present work is different from the available work in

literature because it uses a finite volume technique (in which the continuity, energy and momentum

equations are applied over each control volume) and a wide range for the fin number, up to 80 fins,

which is most likely the case in practical life and finally the complete possible fin’s length up to 0.9 of

the tube radius.

2. Analysis

2.1. Problem description

The problem to be considered is that of forced convection heat transfer for fully developed laminar

flow in a circular internally finned tube. The fins are radial, straight and equally distributed around the

A. Al-Sarkhi, E. Abu-Nada / Int. Commun. Heat and Mass Transf. 32 (2005) 557–564 559

circumference of the tube as shown in Fig. 1. The thickness of the fin is negligible. The flow is subjected

to a uniform heat input flux per unit axial length. Because of symmetry the calculation domain is

performed over a half sector (the complete sector is the area between the two consecutive fins) as shown

in Fig. 1. The dimensionless fin height, H=l/R (the fin length divided by the tube radius), was varied

from 0.1 to 0.9 and the number of fins was varied from 5 to 80.

2.2. Governing equation

Under the previously mentioned assumptions, the momentum and energy equations of the flow can be

written as follows.

Momentum equation in the axial (normal to the page) direction:

lr

B

BrrBw

Br

�þ l

r2B2w

Bh2¼ dP

dzþ qg

�ð1Þ

Energy equation:

ar

B

BrrBT

Br

�þ a

r2B2T

Bh2¼ w

BT

Bz

�ð2Þ

By using the same dimensionless variable, as in Patankar and Prakash [1]

h ¼ T � TwÞ= Qt=kÞðð ð3Þ

X ¼ wl=R2 � dP=dz� qwgÞð ð4Þ

X¯ ¼ wl=R2 � dP=dz� qwgÞð ð5Þ

r4 ¼ r=R ð6Þ

X¯ ¼Z Z

Xr4dr4dh=Z Z

r4dr4dh ð7Þ

In order to have the simple equation, as in Patankar and Prakash, a new variable is introduced as

U ¼ pd X¯d h ð8Þ

by substituting the dimensionless variables and the above variable we get the following simple form of the

governing equations.

Momentum equation:

j2X þ 1 ¼ 0 ð9Þ

Energy equation:

j2U � X ¼ 0 ð10Þ

Fig. 2. Variation of Nusselt number, Nu, with number of fins, N.

A. Al-Sarkhi, E. Abu-Nada / Int. Commun. Heat and Mass Transf. 32 (2005) 557–564560

The friction factor, f, in the tube can be calculated as

f ¼ Dh � dP=dz� qwgð Þ,

1

2qww

2

��ð11Þ

where Dh is the hydraulic diameter defined as

Dh ¼ 2R= NH=p þ 1ð Þ ð12Þwhere H is the dimensionless fin height, H=l/R.

Reynolds number Re is defined as

Re ¼ qwwDh=l ð13Þ

Finally, using the above dimensionless diameter the friction factor times the Reynolds number, fRe,

and Nusselt number, Nu,

f Re ¼ 2D2h=X

¯ R2 ð14Þ

Nu ¼ hT 2Rð Þ=K ð15Þ

Fig. 3. Maximum Nusselt number variation with fin height and number.

Fig. 4. Variation of Nusselt number with fin height for certain fin numbers.

A. Al-Sarkhi, E. Abu-Nada / Int. Commun. Heat and Mass Transf. 32 (2005) 557–564 561

where hT is heat transfer coefficient then Nu can be written as

Nu ¼ � 1=ðph¯Þ ð16Þwhere the average dimensionless temperature is defined as

ht ¼ Qt= p2R Tw � TbÞð Þð ð17Þ

h¯¼ � Tw � Tbð Þk=Qt ¼Z Z

Xhr4dr4dh=Z Z

Xr4dr4dh ð18Þ

where Tb is the bulk temperature of fluid defined as Tb=R R

wTrdrdh/wrdrdh, k is the thermal

conductivity of the fluid and Qt is the total heat input per unit axial length which is related to axial

gradient of temperature by

BT=Bz ¼ dTw=dz ¼ dTb=dz ¼ Qt=mmc ð19Þwhere m is the mass flow rate in the tube defined as m=qwpR2=q

R Rwrdrdd and c is the specific heat of

the fluid.

Fig. 5. Variation of fRe with number of fins for certain fin height.

Fig. 6. Variation of the parameter fRe with fin heights for certain fin numbers.

A. Al-Sarkhi, E. Abu-Nada / Int. Commun. Heat and Mass Transf. 32 (2005) 557–564562

2.3. Computational method

The above governing equations were solved simultaneously using finite volume technique. In this

method the computational domain is divided into a specified number of control volumes surrounding

each grid point. The number of grids used is 24�50 (24 in the d-direction and 50 in the r-direction). In

order to apply the governing equations for each grid point, the energy and momentum equations are

applied over each control volume. The resulting solution using this method implies that the integral

conservation (i.e., conservation of mass, momentum and energy) is exactly satisfied for all control

volumes (the whole calculation domain shown in Fig. 1). The convergence criterion was set on the

values of Nu and fRe. When the absolute error between two consequence iterations is less than 1�10�6

then the convergence is satisfied.

3. Results and discussion

The computations were performed for grid points of 24�50. Larger numbers of grids were tested and

similar results were achieved. A total of 110 tests were performed numerically for variable geometrical

configurations. The dimensionless fin height (H=l/R) was varied from 0.1 to 0.9 and the number of fins

Fig. 7. Normalized velocity and temperature distribution variation for the case of H=0.1.

Fig. 8. Normalized velocity and temperature distribution variation for the case of H=0.9.

A. Al-Sarkhi, E. Abu-Nada / Int. Commun. Heat and Mass Transf. 32 (2005) 557–564 563

N was varied from 5 to 80. Results present data comparison for average Nusselt number and fRe

parameter. The pressure gradient and the hydraulic diameter of the geometry are the main factors that

influence the fRe parameter. The variations of Nusselt Number, Nu, with number of fins, N, are shown in

Fig. 2. Nusselt number, Nu, increases with increasing number of fins and reaches a maximum value and

then decreases with increasing N.

Fig. 3 shows the number of fins, N, at which the maximum Nusselt number, Nu, occurs for a certain

fin height, H. This figure is very important such that someone can have a certain height and go vertically

to cross the value of the maximum Nusselt number that could be reached and the number of fins at which

this maximum value of Nusselt number occurred.

Fig. 4 shows Nu versus the dimensionless fin height, H, for certain N numbers. Nusselt increases with

increasing H. The increase of Nu is negligible up to H=0.6. Beyond H=0.6 the increase of Nu starts to be

significant. Fig. 5 shows the variation of the parameter fRe with the number of fins for a certain fin

height. fRe decreases with increasing N. After N=50 the slope of decreasing fRe with increasing N is

small and reaches an asymptotic value.

Fig. 6 shows the variation of fRe with H. fRe decreases slightly with increasing H until about H=0.3

and then starts to increase with steeper slope with increasing H up to a maximum point.

Figs. 7 and 8 show the normalized velocity and temperature distribution for the cases of H=0.1 and

0.9, respectively. For short fin length the velocity distribution is nearly parabolic. For the case of long fin

height the velocity distribution is not parabolic and the flow in the core of the tube at r/R less than about

0.15 is having higher velocity than the flow near the tube wall. For short fin height, the normalized

temperature distribution is nearly parabolic and as the fin height increases the distribution deviates from

the parabolic profile.

4. Conclusions

The present study has performed a numerical investigation of a vertical internally finned tube subjected

to forced convection heat transfer. For best heat transfer, there is an optimum number of fins and fin height.

Maximum Nu cannot be achieved at maximum height and fin number. There are certain fin numbers at

certain fin height to achieve maximum Nu, as shown in Fig. 3. In general Nusselt number increases with

increasing fin height. The parameter fRe decreases with increasing number of fins N. The velocity and

temperature distributions inside the tube are strongly a function of the height and number of fins.

A. Al-Sarkhi, E. Abu-Nada / Int. Commun. Heat and Mass Transf. 32 (2005) 557–564564

Nomenclature

c fluid specific heat

k thermal conductivity of the fluid

l fin height

N number of fins

Nu Nusselt number

f friction factor as in Eq. (11)

fRe parameter f times Re as in Eq. (14)

r radial coordinate

R tube radius

Re Reynolds number

r* dimensionless radial coordinate r/R

T fluid temperature

Tb fluid bulk temperature

Tw fin and tube wall temperature

m fluid mass flow rate

w fluid axial velocity

w the mean value of w

z axial coordinate normal to the page

d angular coordinate

h dimensionless temperature (Eq. (3))

U the variable pXh (Eq. (8)

X dimensionless axial velocity (Eq. (4))

V average dimensionless velocity (Eq. (5))

a fluid thermal diffusivity

q fluid density and qw fluid density at wall temperature

References

[1] J.H. Masliyah, K. Nandakumar, J. Heat Transfer 98 (1976) 257.

[2] S.V. Patankar, C. Prakash, Trans. ASME 103 (1981) 566.

[3] B.R. Morton, J. Fluid Mech. 12 (1959) 227.

[4] L.N. Tao, Appl. Sci. Res., Sect. A, Mech., Heat 9 (1960) 357.

[5] G.A. Kemeny, E.V. Somers, ASME J. Heat Transfer 84 (1962) 339.

[6] E.M. Sparrow, B.R. Baliga, S.V. Patankar, J. Heat Transfer 100 (1978) 572.

[7] L. Jin-Sheng, L. Min-Sheng, L. Jane-Sunn, W. Chi-Chuan, Int. J. Heat Mass Transfer 44 (2001) 4235.

[8] J.Y. Yun, K.S. Lee, Int. J. Heat Mass Transfer 42 (1999) 2375.

[9] E.M. Sparrow, D.S. Kadle, ASME J. Heat Transfer 108 (1986) 519.

[10] S. Acharya, S.V. Patankar, J. Heat Transfer 103 (1981) 559.

[11] Z. Zhang, S.V. Patankar, Int. J. Heat Mass Transfer 27 (1984) 137.

[12] I. Mutlu, T.T. Al-Shemmeri, Int. Comm. Heat Mass Transfer 20 (1992) 133.

[13] O.N. Sara, T. Pekdemir, S. Yapici, M. Yilmaz, Int. J. Heat Fluid Flow 22 (2001) 509.

[14] M.H. Hu, Y.P. Chang, ASME J. Heat Transfer 95 (1975) 332.