1. ISSN (Print): 2328-3777, ISSN (Online): 2328-3785, ISSN (CD-ROM): 2328-3793
American International Journal of
Research in Formal, Applied
& Natural Sciences
AIJRFANS 14-231; © 2014, AIJRFANS All Rights Reserved Page 63
Available online at http://www.iasir.net
AIJRFANS is a refereed, indexed, peer-reviewed, multidisciplinary and open access journal published by
International Association of Scientific Innovation and Research (IASIR), USA
(An Association Unifying the Sciences, Engineering, and Applied Research)
Effect of Fin Configuration on Heat Transfer of an Annulus Tube
1
Dipti Prasad Mishra, 2
Kailash Mohapatra
1
Department of Mechanical Engineering, Birla Institute of Technology, Mesra, Ranchi, India
2
Department of Mechanical Engineering, Raajdhani Engineering College, Bhubaneswar, India
NOMENCLATURE
C1: constant value of 1.44
C2: constant value of 1.92
D1: outer diameter of annulus tube
D2: inside diameter of inner tube
f: friction factor
H: height of the fin
h: heat transfer coefficient
k: turbulent kinetic energy
L: length of the tube
n: number of fins
Nu: Nusselt number
p: pressure
Prt: turbulent Prandtl number
q: heat flux
Re: Reynolds number
:ijS 1
2
ji
j i
UU
x x
 
    
T: temperature
t: time
U: velocity
:i ju u 2 t
ijS


w: width of the fin
x: axial location
Greek symbols
μ: dynamic viscosity
μt: turbulent viscosity
ε: rate of dissipation
σk: turbulent Prandtl number for k
σε: turbulent Prandtl number for ε
: a scalar variable either k or 
: thermal diffusivity
Subscripts
avg: average
B: bulk
x: local
wx: axial surface distance
inf: ambient
υ: kinematic viscosity
ρ: density
I. INTRODUCTION
Internally finned tubes are extensively used in many industrial applications to enhance heat transfer. Specifically
in heat exchangers annular tubes with fins plays a very vital role in transferring the heat from the heated wall to
fluid. In many industries such as power plants, chemical plants and petroleum industries attention is also
focussed to reduce the size of the heat exchanger without sacrificing the heat transfer rate which can be achieved
by changing the number, shape and size of the fins.
Experimental investigation performed by Braga and Saboya [1] with isothermal wall condition of inner tube and
insulated outer wall having 20 numbers of fins. It was found that Nusselt number was exclusive function of
Reynold number of the air flow and it was not depending on the thermal conductivity of the fin material. Heat
transfer and pressure drop characteristics of a double pipe structure have been investigated experimentally by
Yu et al. [2]. Friction factor and Nusselt number have been correlated with Reynolds number for blocked inner
tube and unblocked inner tube. Experimental investigations were performed for turbulent flow through annular
ducts with 560 internal pin fins to determine average heat transfer coefficients and friction factors where air was
made to flow through the annular channel and water through the inner circular tube [3]. For a fully developed
flow they concluded fin efficiency and thermal conductivity of fin material decreases with Reynolds number.
Abstract: Conservation equations of mass, momentum, energy have been solved numerically with two
equations based k- model to determine fluid flow and heat transfer characteristics of a finned annulus tube.
The outer wall of outside tube is subjected to a constant heat flux and the outer wall of the inner tube is
adiabatic. It is found from the numerical investigation that the heat transfer to the air increases with fin
number present between the annulus space and also the heat transfer increases with the fin height. It is also
obtained from the numerical experiment that trapezoidal shaped fin has a higher heat transfer rate compare
to other shape.
Key words: Nusselt number, annulus tube, friction factor, heated surface, wall temperature

2. Dipti Prasad Mishra et al., American International Journal of Research in Formal, Applied & Natural Sciences, 6(1), March-May 2014, pp.
63-69
AIJRFANS 14-231; © 2014, AIJRFANS All Rights Reserved Page 64
Numerical investigations were conducted by Li et al. [4] using three turbulent model namely mixing length
model, k- model and full stress model for fully developed flow in annular ducts with four apex angles and four
radius ratios. It was concluded numerical results agreed with the available correlation with maximum deviation
of 8.3%. Correlations were developed Dirker and Meyer [5] for wide range of annular diameter ratios to predict
Nusselt number for different Reynolds number based on hydraulic diameter. The deduced correlations predict
the Nusselt number within 3% of experimental values. Numerical investigation conducted by Qiuwang et al. [6]
shows there exists an optimal ratio inner core outside diameter to outer tube inner diameter for an annular wavy
finned tube for both constant wall temperature and constant heat flux. Experimental investigation on unfinned
and finned tube concentric passages have been performed by Kuvvet and Yavuz [7] shows the average Nusselt
number is augmented in the finned passage 3.6 times to 18 times compared to unfinned passage. Alijani and
Hamidi [8] performed experimental investigation on circular tube with core rod a honeycomb network inserts at
a high temperature to determine the heat transfer coefficient and concluded that at uniform wall temperature the
heat transfer coefficient of core rod was increased by 227% to 369% and 409% to 679% for honey comb
network insert compared with plain tube.
In the present work we intend to analyze the heat transfer and fluid flow characteristics of an annular internal
finned tube where the flow assumed to be turbulent and is developing in nature. The longitudinal fins are
attached inside annular space between the tubes and the thermal performance has been investigated by varying
the fin number, shape and size. We have also tried to match the present computation with the existing
experiment of Sobaya and Braga [1] for average Nusselt number. All the computations have been performed
using Fluent 14 for different flow conditions to determine the optimum fin condition for maximum thermal
performance.
II. MATHEMATICAL FORMULATION
The Computational investigation is carried out for an annulus finned tube of outer diameter, D1, inner surface
diameter, D2 and length, L as shown in Fig. 1. Air enters into the annulus space of the tube at one end where as
Fig.1 (a) Longitudinal section of an annular tube with boundary condition applied to it (b) Cross sectional view of the annular tube showing
the fins are attached inside the annular space
the other end is exposed to the surrounding atmosphere. Longitudinal fins are placed symmetrically within the
annulus space around inner periphery of the tube. The investigation is initiated with the rectangular fins of
length L, thickness t and height H as shown in Fig. 1 and subsequently the fin number, shape and sizes have
been changed while we keep the mass flow rate of air inside the tube to be constant. The flow field in the
domain would be computed by using three-dimensional, incompressible Navier-Stokes equations (2D axi-
symmetric model for fin less tube) along with the energy equations. The fluid used in the simulation is air, at
temperatures of 300 to 700 K, and is treated to be incompressible, at the inlet face of the tube with an inlet
velocity below 10 m/s.
A. GOVERNING EQUATIONS
The governing equations for the above analysis can be written as:
Continuity
( ) 0Ui
xi




(1)
Momentum
  ji i
i j
i j j i
UD U Up
u u
Dt x x x x

 
   
             
(2)
 is the fluid density anywhere in the domain and is the function of temperature and taken as per the ideal gas
equation and i ju u is expressed as
a
b

3. Dipti Prasad Mishra et al., American International Journal of Research in Formal, Applied & Natural Sciences, 6(1), March-May 2014, pp.
63-69
AIJRFANS 14-231; © 2014, AIJRFANS All Rights Reserved Page 65
1
2 ,
2
jt i
i j ij ij
j i
UU
u u S S
x x


 
       
(3)
Energy Equation
 
i t i
D T T
Dt x pr x
   
  
  
(4)
Conduction Equation
0
i i
T
k
x x
  
 
  
(5)
Turbulence kinetic energy - k
  2 1t t
j k j
D k k
S
Dt x x
  
 
  
   
     
    
(6)
Where S is the modulus of the mean rate-of-strain tensor, defined as
2 ij ijS S S (7)
Rate of dissipation of k
  2
1 2
1t t
j j
D
C S C
Dt k x x
 

   
 
  
     
      
      
(8)
t is expressed as
2
t
k
C 

 (9)
k and  are the Prandtl numbers for k and .
The constants used in the above k- equations are the following
1 21.44; 1.92; 0.09; 1.0; 1.3kC C C         , Prt =1
B. BOUNDARY CONDITIONS
Fig. 1 shows the boundary conditions for finned tube. The outer tube wall are solid and has been given a no-slip
and constant heat flux boundary condition where as the outer wall of inner tube has been applied with adiabatic
boundary condition. Pressure outlet boundary condition has been imposed at the outlet of the tube where as
velocity inlet boundary condition has been employed at the inlet of the tube. Axi-symmetric boundary
conditions have been used in case of the smooth tube. At the pressure outlet boundary, the velocity will be
computed from the local pressure field so as to satisfy continuity but all other scalar variables such as, T, k and 
are computed from the zero gradient condition, Dash [9]. The turbulent quantities, k and , on the first near wall
cell have been set from the equilibrium log law wall function as has been described by Jha and Dash [10, 11]
and Jha et al. [12]. The turbulent intensity at the inlet of the tube has been set to 2%, with the inlet velocity
being known, and the back flow turbulent intensity at all the pressure outlet boundary have been set to 5%. If
there is no back flow at a pressure outlet boundary then the values of k and  are computed from the zero
gradient condition at that location.
C. COMPUTATIONS OF SOME IMPORTANT HEAT TRANSFER AND FLUID FLOW PARAMETERS
The heat transfer and fluid flow parameters are computed as given in Yu et al. [2] and Islam and Mozumder[13].
Local Nusselt number for the finned heated tube
x
x
air
h D
Nu
k
 (10)
Local heat transfer coefficient has been computed as
x
sx Bx
q
h
T T


(11)
Where hx, D, kair, Tsx and TBx are respectively, the local heat transfer coefficient, internal nominal diameter of
tube, thermal conductivity of air, local tube wall temperature and local bulk temperature of fluid.
Bx iT T n
p
q P
x
mc
  (12)
Ti, q, Pn, x, Cp and ṁ are respectively, inlet fluid temperature, wall heat flux, nominal perimeter, axial position,
specific heat and mass flow rate of fluid.
m = inV A (13)

4. Dipti Prasad Mishra et al., American International Journal of Research in Formal, Applied & Natural Sciences, 6(1), March-May 2014, pp.
63-69
AIJRFANS 14-231; © 2014, AIJRFANS All Rights Reserved Page 66
, Vin and A are the density, inlet fluid velocity and area of tube at the inlet respectively.
The Reynolds number has been computed based on nominal diameter of the tube and is given by
Re in h
air
V D

 (14)
air is the kinematic viscosity of air based on inlet condition
avg
avg
air
h D
Nu
k
 (15)
avg
savg Bavg
q
h
T T


(16)
Where Nuavg, havg, Tsavg and TBavg are respectively, the average Nusselt number, average heat transfer coefficient,
average wall temperature and average bulk temperature.
2
1
2x
in
D
f P
x V
  (17)
Where fx, P,  and x are local friction factor, cumulative pressure drop, density and axial distance along the
tube respectively.
III. NUMERICAL SOLUTION PROCEDURE
Three-dimensional equations of mass, momentum, energy and turbulence have been solved by the algebraic
multi grid solver of Fluent 14 in an iterative manner by imposing proper boundary conditions. First order
upwind scheme (for convective variables) was considered for momentum as well as for the turbulent discretized
equations. As the density is taken as a function of temperature according to ideal gas equation therefore
SIMPLE algorithm with PRESTO (Pressure Staggered Option) scheme has been used for better convergence.
Under relaxation factors (0.3 for pressure, 0.7 for momentum and 0.8 for k and ) were used for the convergence
of all the variables. Convergence of the discretized equations were said to have been achieved when the whole
field residual for all the variables fell below 10-3
except energy equation and for energy equation residual was
set 10-6
.
IV. RESULTS AND DISCUSSION
A. MATCHING WITH OTHER COMPUTATION
We have tried to compare the existing experimental result of Sobaya and Braga [1] with the present CFD for
average Nusselt number. Here in the annulus section the working fluid is air and constant temperature boundary
condition is applied over the surface of the inner tube and adiabatic boundary condition is imposed over the
outside tube surface. It can be seen from the Fig. 2 the present CFD result matched reasonably well with the
existing experimental result with a maximum error of 6%.
Fig. 2 Average Nusselt number as a function of Reynolds number: A comparison between existing experiment and present CFD
B. GRID SENSITIVITY TEST
D1 = 70 mm, D2 = 50 mm, n = 4, t = 5 mm, q = 3000 W/m2
, ṁ = 0.0196 kg/s
Cell Number Outlet Temperature (To/Tinf)
16800 1.138
69930 1.13
237500 1.126
559500 1.1202
1791500 1.119
1 2 3 4 5
10
20
30
40
50
60
70
Nuavg
Rex10
-4
Experiment (Saboya and Braga, [1])
Present CFD

5. Dipti Prasad Mishra et al., American International Journal of Research in Formal, Applied & Natural Sciences, 6(1), March-May 2014, pp.
63-69
AIJRFANS 14-231; © 2014, AIJRFANS All Rights Reserved Page 67
Table 1 Outlet temperature as a function cell number
Grid sensitivity test has been shown in Table – 1 for a sample case of an annulus internal finned tube having
rectangular section. The tube length is 1 m, having internal diameter of outer tube of 0.07 m and outer diameter
of internal tube of 0.05 m which is subjected to a constant heat flux of 3000 W/m2
and other operating parameter
can be seen from the table – 1 itself. Initially coarse grids have been taken and subsequently cells are refined
and it is seen that after refining the entire meshes inside the tube from 16, 800 to 2,37, 500, the outlet
temperature decreases by 1.5%. After that we refined the meshed near walls of tube and fins twice (a cross
sectional view of the pipe can be seen from the Fig. 3) which increases the mesh number to 17,91,500. It was
found that after increasing the cell number more than 5,59,500 does not have a significant change in outlet
temperature. So finally we have considered 5,59,500 meshes to perform all the computations. However when
the shape and of the internal fins are changed we perform the grid test before we report any results.
Fig. 3 Cross sectional view of the internal finned annulus tube showing the meshes inside the tube and around the fins
C. FIN NUMBER
Effect of fin number on wall temperature can be seen from Fig. 4. In the present numerical investigation
longitudinal fins having rectangular section have been used and other parameters can be seen from the plot
itself. It can be seen from the Fig. 4 as the fin number increases the wall temperature reduces and the outlet air
temperature increases (can be seen from Fig. 5). Fig. 6 shows that the Nusselt number increases with fin number
which suggests more heat transfer to air as the fin number increases. It is also visualised from Fig. 6 the Nusselt
number is higher at the entrance of the pipe and it is continuously decreases towards the end. This implies
initially at the entrance convection heat transfer is dominated and towards the end conduction heat transfer is
more prominent because of growth of boundary layer.
0.0 0.2 0.4 0.6 0.8 1.0
1.04
1.06
1.08
1.10
1.12
1.14
1.16
1.18
Tw
/Tinf
x/L
n = 4
n = 8
n = 12
n = 16
n = 20
q = 3000 W/m
2
t = 5 mm
m = 0.0196 kg/s
D1
= 70 mm, D2
= 50 mm
.
2 4 6 8 10 12 14 16 18 20 22
1.1200
1.1205
1.1210
1.1215
1.1220
1.1225
1.1230 q = 3000 W/m
2
m = 0.0196 kg/s, t = 5 mm
D1
= 70 mm, D2
= 50 mm
To
/Tinf
Fin Number
.
Fig. 4 Effect of fin number on wall temperature Fig. 5 Outlet temperature as a function of fin number
0.0 0.2 0.4 0.6 0.8 1.0
60
90
120
150
180
210
240
Nux
x/L
n = 4
n = 8
n = 12
n = 16
n = 20
q = 3000 W/m
2
H = 10 mm, t = 5 mm
D1
= 70 mm, D2
= 50 mm
m = 0.0196 kg/s
.
Fig. 6 Nusselt number as a function of fin number

6. Dipti Prasad Mishra et al., American International Journal of Research in Formal, Applied & Natural Sciences, 6(1), March-May 2014, pp.
63-69
AIJRFANS 14-231; © 2014, AIJRFANS All Rights Reserved Page 68
D. FIN HEIGHT
Fin height plays a crucial role in transferring the heat in internal finned annular tube. In the annular tube the
outer tube diameter is kept constant with 70 mm where as inside tube diameter varies with fin height. It can be
seen from the Fig. 7, increasing the fin height or reducing the inner tube diameter, the wall temperature
decreases where as average Nusselt number increases as seen from Fig. 8. This clearly indicates the heat transfer
to air increases with the fin height. But it has an adverse effect on air flow as the pressure drop is high and due
to that the friction factor becoming more (can be seen from Fig. 9).
E. FIN SHAPE
Fig. 10 shows the wall temperature distribution for different shape fin having same inlet mass flow rate and fin
height. It can be seen from the plot the wall temperature is minimum in trapezoidal shaped fin compare to other
0.0 0.2 0.4 0.6 0.8 1.0
1.04
1.06
1.08
1.10
1.12
1.14
1.16
1.18
1.20
Tw
/Tinf
x/L
D2
= 60 mm
D2
= 50 mm
D2
= 40 mm
D2
= 30 mm
q = 3000 W/m
2
, n = 4
t = 5 mm, D1
= 70 mm
m = 0.0196 kg/s
.
0.05 0.10 0.15 0.20 0.25 0.30
0
50
100
150
200
250
300
q = 3000 W/m
2
m = 0.0196 kg/s
D1
= 70 mm, n = 4
t = 5 mm
Nuavg
H/D1
.
Fig. 7 Effect of fin height on wall temperature Fig. 8 Average Nusselt number as a function fin height
0.0 0.2 0.4 0.6 0.8 1.0
0.03
0.04
0.05
0.06
0.07
0.08
fx
x/L
D2
= 50 mm
D2
= 40 mm
D2
= 30 mm
q = 3000 W/m
2
n = 4, H = 10 mm
m = 0.0196 kg/s
D1
= 70 mm, t = 5 mm
.
Fig. 9 Friction factor as a function of fin height
0.0 0.2 0.4 0.6 0.8 1.0
1.05
1.10
1.15
1.20
1.25
1.30
1.35
Twx
/Tinf
x/L
Rectangular
+ shape
Triangular
Trapezium
Plane tube
q = 3000 W/m
2
, t = 5 mm
m = 0.0196 kg/s, n = 4
D1
= 70 mm, D2
= 50 mm
.
+ - shape Rectangular Triangular Trapezium
72
74
76
78
80
82
84 q = 3000 W/m
2
, n = 4
m = 0.0196 kg/s
D1
= 70 mm, D2
= 50 mm
Nuavg
Fin shape
.
Fig. 10 Effect of fin shape on wall temperature Fig. 11 Nusselt number as a function of fin shape

7. Dipti Prasad Mishra et al., American International Journal of Research in Formal, Applied & Natural Sciences, 6(1), March-May 2014, pp.
63-69
AIJRFANS 14-231; © 2014, AIJRFANS All Rights Reserved Page 69
shape which depicts the heat transfer is highest in trapezoidal shaped fin. The same result is also visualised from
Fig. 11, which shows higher value of Nusselt number for trapezoidal shaped fin compare to other.
V. CONCLUSIONS
The wall temperature, tube outlet temperature and friction factor for an internal annulus finned tube have been
computed by solving the conservation equations for mass, momentum and energy. The average Nusselt number
of the existing experimental observation matched well with the present CFD result. The heat transfer to the air
is increases with fin height and fin number. The heat transfer is maximum in trapezoidal fin compare other
shape.
REFERENCES
[1] Braga C V M and Saboya F E M (1999) Turbulent heat transfer, pressure drop and fin efficiency in annular regions with
continuous longitudinal rectangular fins. Experimental thermal and fluid science 20:55-65.
[2] Yu B, Nie J H, Wang Q W, and Tao W Q (1999) Experimental study on the pressure drop and heat transfer characteristics of
tubes with internal wave-like longitudinal fins Heat and mass transfer 35:65-73.
[3] Nieckele A O and Mouraosaboya F E (2000) Turbulent heat transfer and pressure drop in pinned annular regions. Journal of
Brazilian Society of Mechanical Science 22:119-132.
[4] Li Z Y, Hung T C and Tao W Q (2001) Numerical simulation of fully developed turbulent flow and heat transfer in annular-
sector duct. Heat and mass transfer, 38:369-377.
[5] Dirker J and Meyer JP (2005) Convective heat transfer coefficients in concentric annuli. Heat transfer engineering 26(2):38-44.
[6] Qiuwang W, Mei L and Min Z (2008) Effect of blocked core-tube diameter on heat transfer performance of internally
longitudinal finned tubes. Heat transfer Engineering 29(1):107-115.
[7] Kuvvet K and Yavuz T (2011) Effect of fin pitches on heat transfer and fluid flow characteristics in the entrance region of a
finned concentric passage. Journal of thermal science and technology 31(2):109-122.
[8] Alijani M R and Hamidi A A (2013) Convection and radiation heat transfer in a tube with core rod and honeycomb network
inserts at high temperature. Journal of mechanical science and technology 27(11):3487-3493.
[9] Dash S K (1996) Heatline visualization in turbulent flow. Int. J. Numerical Methods for Heat and Fluid Flow 6:37-46.
[10] Jha P K, Dash S K (2002) Effect of outlet positions and various turbulence models on mixing in a single and multi strand tundish.
Int. J. Num. Method for Heat and Fluid Flow 12:560-584.
[11] Jha P K, Dash S K (2004) Employment of different turbulence models to the design of optimum steel flows in a tundish. Int. J.
Numerical Methods for Heat and Fluid Flow 14:953-979.
[12] Jha P K, Ranjan R, Mondal S S, Dash S K, (2003) Mixing in a tundish and a choice of turbulence model for its prediction. Int. J.
Numerical Methods for Heat and Fluid Flow 13: 964-996.
[13] Islam M A, Mozumder A K (2009) Forced convection heat transfer performance of an internally finned tube. Journal of
Mechanical engineering ME 40:54-62.