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 tube B A. Al-Sarkhi*, E. Abu-Nada Department of Mechanical Engineering, Hashemite University, arqa, 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 B Communicated by J.P. Hartnett and W.J. Minkowycz. * Corresponding author. Tel.: +962 591 6600; fax: +962 591 6613. E-mail address: alsarkh@hu.edu.jo (A. Al-Sarkhi). 0735-1933/$ - see front matter D 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2004.03.015
558 A. Al-Sarkhi, E. Abu-Nada / Int. Commun. Heat and Mass Transf. 32 (2005) 557 564 Fig. 1. Schematic of radial fins and the calculation domain. [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: l r B Br r Bw Br þ l r 2 B 2 w Bh 2 ¼ dp dz þ qg ð1þ Energy equation: a B r BT þ a B 2 T r Br Br r 2 Bh 2 ¼ w BT Bz ð2þ By using the same dimensionless variable, as in Patankar and Prakash [1] h ¼ ðt T w Þ= ðq t =kþ X ¼ wl=r 2 ð dp=dz q w gþ X ¼ w l=r 2 ð dp=dz q w gþ ð3þ ð4þ ð5þ r4 ¼ r=r X ¼ Xr4dr4dh= r4dr4dh ð6þ 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: j 2 X þ 1 ¼ 0 ð9þ Energy equation: j 2 U X ¼ 0 ð10þ
560 A. Al-Sarkhi, E. Abu-Nada / Int. Commun. Heat and Mass Transf. 32 (2005) 557 564 Fig. 2. Variation of Nusselt number, Nu, with number of fins, N. The friction factor, f, in the tube can be calculated as, 1 f ¼ D h ð dp=dz q w gþ 2 q ww 2 ð11þ where D h is the hydraulic diameter defined as D h ¼ 2R= ðnh=p þ 1Þ ð12þ where H is the dimensionless fin height, H=l/R. Reynolds number Re is defined as Re ¼ q w w D h =l ð13þ Finally, using the above dimensionless diameter the friction factor times the Reynolds number, fre, and Nusselt number, Nu, fre¼ 2D 2 h =X R 2 ð14þ Nu ¼ h T ð2rþ=k ð15þ Fig. 3. Maximum Nusselt number variation with fin height and number.
A. Al-Sarkhi, E. Abu-Nada / Int. Commun. Heat and Mass Transf. 32 (2005) 557 564 561 Fig. 4. Variation of Nusselt number with fin height for certain fin numbers. where h T is heat transfer coefficient then Nu can be written as Nu ¼ 1=ðph Þ where the average dimensionless temperature is defined as h t ¼ Q t = ðp2rðt w T b ÞÞ h ¼ T ð w T b Þk=Q t ¼ Xhr4dr4dh= Xr4dr4dh ð16þ ð17þ ð18þ where T b is the bulk temperature of fluid defined as T b = RR wtrdrdh/wrdrdh, k is the thermal conductivity of the fluid and Q t is the total heat input per unit axial length which is related to axial gradient of temperature by BT=Bz ¼ dt w =dz ¼ dt b =dz ¼ Q t =ṁmc ð19þ where ṁ is the mass flow rate in the tube defined as ṁ=qw pr 2 =q RR wrdrdd and c is the specific heat of the fluid. Fig. 5. Variation of fre with number of fins for certain fin height.
562 A. Al-Sarkhi, E. Abu-Nada / Int. Commun. Heat and Mass Transf. 32 (2005) 557 564 2.3. Computational method Fig. 6. Variation of the parameter fre with fin heights for certain fin numbers. 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 2450 (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 110 6 then the convergence is satisfied. 3. Results and discussion The computations were performed for grid points of 2450. 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.
A. Al-Sarkhi, E. Abu-Nada / Int. Commun. Heat and Mass Transf. 32 (2005) 557 564 563 Fig. 8. Normalized velocity and temperature distribution variation for the case of H=0.9. 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.
564 A. Al-Sarkhi, E. Abu-Nada / Int. Commun. Heat and Mass Transf. 32 (2005) 557 564 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 T b fluid bulk temperature T w fin and tube wall temperature ṁ 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 px h (Eq. (8) X dimensionless axial velocity (Eq. (4)) V average dimensionless velocity (Eq. (5)) a fluid thermal diffusivity q fluid density and q w 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]. hang, 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.