Applied Mechanics and Materials
ISSN: 1662-7482, Vol. 786, pp 193-198
doi:10.4028/www.scientific.net/AMM.786.193
© 2015 Trans Tech Publications, Switzerland
Submitted: 2014-12-07
Revised: 2015-01-07
Accepted: 2015-01-10
Online: 2015-08-26
Finite Element Analysis of Heater Length in a Porous Annulus - Part A
QUADIR G.A.1,a*, AHMED N.J.S.2,b, AL-RASHED A.A.A.A.3,c,
BADRUDDIN I.A.4,d , KHALEED H.M.T.5,e, KAMANGAR S.4,f
1
School of Mechatronic Engineering, University Malaysia Perlis (UniMAP), Pauh Putra,
02600 Arau, Perlis, Malaysia
2
Faculty of Engineering & Technology, Multimedia University, Bukit Beruang, 75450 Malacca,
Malaysia
3
Public Authority for Applied Education and Training, Industrial Training Institute, 13092 Kuwait
4
Dept. of Mechanical Engineering, University of Malaya, Kuala Lumpur, 50603, Malaysia
5
Faculty of Mechanical Engineering, Islamic University, Madinah Munawwarra, Kingdom of Saudi
Arabia.
*[email protected], [email protected], [email protected],
[email protected], [email protected], [email protected]
d
Keywords: Porous Annulus, FEM, Discrete Heating, Darcy Law.
Abstract. The focus of present study is to investigate the influence of discrete heating by an
isothermal heater placed at the inner radius of a vertical annular cylinder containing porous medium
between its inner and the outer radius. Finite element method is used to solve the governing partial
differential equations by employing 3 noded triangular elements. Darcy model is used to represent
the flow behavior inside the porous medium. It is assumed that the thermal non-equilibrium
condition exists between the fluid and solid phases of the porous medium. The study is conducted
for different lengths of heater corresponding to 20%, 35% and 50% of the total height of the
cylinder. It is found that the Nusselt number for fluid, solid phases as well as total Nusselt number
initially decreases and the increases along the length of heater.
Introduction
It is well known that the porous medium plays a significant role in many applications such as
geothermal heat extractions, nuclear reactor waste disposal, heat exchangers, electronic
components, solar energy storage technology, exothermic reactions in packed bed reactors, storage
of grains, food processing, and the spread of pollutants underground etc, thus making itself as an
important topic for scientific research. There has been enormous amount of research dedicated to
understand the different issues related to porous medium leading to substantial literature. The heat
transfer in porous medium is generally studied either employing a thermal equilibrium model [1-10]
where the temperature of fluid and solid phases are presumed to be equal or applying the more
accurate approach of thermal non-equilibrium that takes into account the discrepancy among solid
and fluid phases of porous medium[11-18]. The heat transfer in porous medium is predominantly
studied with respect to regular geometries such as cylinders, square or rectangular cavities and
vertical plates with heat being applied to whole of the surface but recent years have seen an interest
in segmental heating of the porous region where a portion of the porous medium is subjected to
heating [19-27]. The current work is undertaken to understand the heat transfer behavior inside the
porous annulus when the central portion of inner radius in maintained at an isothermal temperature.
Mathematical Model
Consider an annular porous medium sandwiched between the inner and our radii ri and ro of vertical
annular cylinder as shown in Fig.1. A central portion of the cylinder at inner radius is maintained at
an isothermal temperature Th and the outer surface is subjected to temperature T∞. The coordinate
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194
Trends and Applications in Mechanical Engineering
system is chosen in such a way that the r and z axis points towards the radial and vertical direction
of the annulus.
Fig 1. Annular Porous Media
The equations that govern the fluid flow and heat transfer in porous annulus can be given as:
(ru ) (rw)

0
r
z
w u gK T


r z
 r
c 
p f
T f
 T f
u

w
 r
z

(1)
(2)
 1   T f

  k f 
r

 r

r

r




  Ts   Ts
r
 2
 r r  r  z
2
1   k s  1
2
  Tf

 z 2


  hTs  T f


  h T T
s
f




(3)

(4)
The non-dimensional form of above equations can be shown to be
 2
z
2
r
T
  1  

  r Ra f
 r  r  r 
r
1   T f  T f   1 



r   r  z
 z  r   r r
 1   Ts

r
 r r   r


  2 Ts

2

 z
(5)
 T f   T f
r

 r   z 2


2

  HKrTs  T f



  H T  T
s
f




(6)
(7)
The corresponding boundary conditions in the non-dimensional form are:
At
r  ri and L1  z  L2 ,
  0,
T f  Ts 
1
2
(8a)
Applied Mechanics and Materials Vol. 786
At
r  ro ,
 0
At
z  Ht ,
  0,
T f  Ts  
1
2
195
(8b)
T
0
z
(8c)
Nusselt number is given by:
For fluid
For solid
 T f
N u f  
 r




 r ri
 T 
N u s   s 
 r  r ri
(9)
(10)
 Ts  
1 
  T f 


 
(11)
Kr

 
Kr  1   r  r r  r  r r 
i 
i

Here H and Kr represent the interphase heat transfer coefficient and conductivity ratio respectively
total Nusselt number is: N u t  
Results and Discussion
The governing equations 5-8 subjected to boundary conditions 8 are solved using finite element
method to predict the heat and fluid flow behavior inside the porous medium. 3 noded triangular
elements are used to divide the domain under consideration. The results are presented in terms of
local Nusselt number along the heated surface of porous medium as shown if Fig.2. This figure is
obtained at Ra  100, H  1, Kr  1, Rr  1, where Ra and Rr are Rayleigh number and radius ratio
respectively. It be noted that the length of the heater is varied in 3 steps such that the length
remains at 20%, 35% and 50% of the height of the annulus. Figure 1 shows the variation of Nusselt
number for 3 lengths of the heater and 2 values of aspect ratio i.e. Ar = 0.5 and 1. It is observed
that the local Nusselt number is higher towards the bottom and top edge of heater as compared to
middle section. The fluid Nusselt number is higher towards bottom side and solid Nusselt number is
higher near the top edge of heater. This behavior is consistent for all lengths and aspect ratio being
investigated. It is found that the local Nusselt for fluid and solid is equal at certain point of heater
indicating that the thermal gradient for fluid and solid phases is equal at that particular point as
indicated by crossing of fluid and solid Nusselt number lines in Fig. 2. This crossing point of fluid
and solid Nusselt number shifts towards the upper edge of heater with increase in the heater length.
The increase in aspect ratio from 0.5 to 1 leads to increase in the discrepancy of fluid and solid
Nusselt number towards the edges of heater. This illustrates that the thermal non-equilibrium effect
is stronger in case of Ar = 1 as compared to that of Ar = 0.5.
a)
Fig. 2 : Local Nusselt number variation for heater length of a) 20% b) 35% and c) 50%
Ar = 0.5 (left column), Ar = 1 (right column)
196
Trends and Applications in Mechanical Engineering
b)
c)
Fig. 2 (Contd): Local Nusselt number variation for heater length of a) 20% b) 35% and c) 50%
Ar = 0.5 (left column), Ar = 1 (right column)
Conclusion
The current study is carried out to investigate the effect of heater length placed at mid of annular
porous annulus and aspect ratio. Finite element method is used to solve the governing equations. It
is found that the Nusselt number is higher at two edges of heater for all the 20%, 35% and 50%
heater length. It is also found that the thermal non equilibrium is stronger for Ar = 1 as compared to
that of Ar = 0.5
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N.H. Saeid, and I. Pop, Maximum density effects on natural convection from a discrete heater
in a cavity filled with a porous medium, Acta Mech. 171(3-4) (2004) 203-212.
G. Saha, Finite element simulation of magnetoconvection inside a sinusoidal corrugated
enclosure with discrete isoflux heating from below, Int. Commun. Heat Mass Transfer. 37(4)
(2010) 393-400.
M. Sankar, and Y. Do, Numerical simulation of free convection heat transfer in a vertical
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198
Trends and Applications in Mechanical Engineering
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Trends and Applications in Mechanical Engineering
10.4028/www.scientific.net/AMM.786
Finite Element Analysis of Heater Length in a Porous Annulus - Part A
10.4028/www.scientific.net/AMM.786.193
DOI References
[1] V. Prasad, F. A Kulacki, Natural convection in a vertical porous annulus, Int. J. Heat Mass Transfer. 27
(1984) 207-219.
10.1016/0017-9310(84)90212-6
[2] R. C Rajamani, C. Srinivas, P. Nithiarasu, K.N. Seetharamu, Convective Heat-Transfer in Axisymmetrical
Porous Bodies, Int. J of Numer Methods Heat Fluid Flow. 5(9) (1995) 829837.
10.1108/eum0000000004126
[3] I.A. Badruddin, A.A.A.A. Abdullah, N. J. S. Ahmed, S. Kamangar, Investigation of heat transfer in square
porous-annulus, Int. J Heat Mass Transfer. 55 (7-8) (2012) 2184-2192.
10.1016/j.ijheatmasstransfer.2011.12.023
[4] N.J.S. Ahmed, I.A. Badruddin, Z.A. Zainal, H.M.T. Khaleed, J. Kanesan, Heat transfer in a conical
cylinder with porous medium, Int. J. Heat Mass Transfer. 52(13-14) (2009) 30703078. b) c).
10.1016/j.ijheatmasstransfer.2008.12.030
[5] I.A. Badruddin, Z.A. Zainal, P.A. Narayana, K.N. Seetharamu, L.W. Siew, Free convection and radiation
for a vertical wall with varying temperature embedded in a porous medium, Int. J. Therm. Sci. 45(5) (2006)
487-493.
10.1016/j.ijthermalsci.2005.05.008
[6] I.A. Badruddin, Z.A. Zainal, P.A. Narayana, K.N. Seetharamu, Heat transfer in porous cavity under the
influence of radiation and viscous dissipation, Int. commun. Heat Mass Transfer. 33(4) (2006) 491-499.
10.1016/j.icheatmasstransfer.2006.01.015
[7] I.A. Badruddin, Z.A. Zainal, Z. A Khan, Z. Mallick, Effect of viscous dissipation and radiation on natural
convection in a porous medium embedded within vertical annulus, Int. J. Therm. Sci. 46 (3) (2007) 221-227.
10.1016/j.ijthermalsci.2010.04.021
[8] A. Raptis, Radiation and free convection flow through a porous medium, Int. Commun. Heat Mass
Transfer. 25(2) (1998) 289-295.
10.1016/s0735-1933(98)00016-5
[9] I.A. Badruddin, Z.A. Zainal, P.A. Narayana, K.N. Seetharamu, Heat transfer by radiation and natural
convection through a vertical annulus embedded in porous medium, Int. Commun. Heat Mass Transfer. 33(4)
(2006) 500-507.
10.1016/j.icheatmasstransfer.2006.01.008
[10] N.J.S. Ahmed, S. Kamangar, I.A. Badruddin, A.A.A.A. Al-Rashed, G.A. Quadir, H.M.T. Khaleed,
T.M.Y. Khan, Conjugate heat transfer in porous annulus, J. Porous Media. 17(12) (2014) 1109-1119.
10.1615/jpormedia.v17.i12.70
[11] N.H. Saeid, Analysis of free convection about a horizontal cylinder in a porous media using a thermal
non-equilibrium model, Int. Commun. Heat Mass Transfer. 33(2) (2006) 158-165.
10.1016/j.icheatmasstransfer.2005.09.009
[12] T.W. Ting, Y.M. Hung, N. Guo, Entropy generation of viscous dissipative nanofluid flow in thermal
non-equilibrium porous media embedded in microchannels, Int. J. Heat Mass Transfer. 81, (2015) 862-877.
10.1016/j.ijheatmasstransfer.2014.11.006
[13] Y. Mahmoudi, Effect of thermal radiation on temperature differential in a porous medium under local
thermal non-equilibrium condition, Int. J. Heat Mass Transfer. 76 (2014) 105-121.
10.1016/j.ijheatmasstransfer.2014.04.024
[14] I.A. Badruddin, Z. A Zainal, P.A. Narayana, K.N. Seetharamu, Thermal non-equilibrium modeling of
heat transfer through vertical annulus embedded with porous medium, Int. J. Heat Mass Transfer. 49(25-26)
(2006) 4955-4965.
10.1016/j.ijheatmasstransfer.2006.05.043
[15] N.J.S. Ahmed, I.A. Badruddin, J. Kanesan, Z.A. Zainal, K.S.N. Ahamed, Study of mixed convection in
an annular vertical cylinder filled with saturated porous medium, using thermal non-equilibrium model, Int. J.
Heat Mass Transfer. 54(17-18) (2011).
10.1016/j.ijheatmasstransfer.2011.05.001
[16] I.A. Badruddin, Z.A. Zainal, P.A. Narayana, K.N. Seetharamu, Numerical analysis of convection
conduction and radiation using a non-equilibrium model in a square porous cavity, Int. J. Therm. Sci. 46(1)
(2007) 20-29.
10.1016/j.ijthermalsci.2006.03.006
[17] P. Bera, S. Pippal, A.K. Sharma, A thermal non-equilibrium approach on double-diffusive natural
convection in a square porous-medium cavity, Int. J. Heat Mass Transfer. 78 (2014) 1080-1094.
10.1016/j.ijheatmasstransfer.2014.07.041
[18] I.A. Badruddin, A.A.A.A. Al-Rashed, N.J.S. Ahmed, S. Kamangar, K. Jeevan, Natural convection in a
square porous annulus, Int. J. Heat Mass Transfer. 55(23-24) ( 2012) 71757187.
10.1016/j.ijheatmasstransfer.2012.07.034
[19] U. Akdag, Numerical investigation of pulsating flow around a discrete heater in a channel, Int. Commun.
Heat Mass Transfer. 37(7) (2010) 881-889.
10.1016/j.icheatmasstransfer.2010.04.001
[20] E. Bilgen, and A. Muftuoglu, Conjugate heat transfer in open cavities with a discrete heater at its
optimized position, Int. J. Heat Mass Transfer. 51(3-4) (2008) 779-788.
10.1016/j.ijheatmasstransfer.2007.04.017
[21] F.Y. Zhao, D. Liu, G.F. Tang, Resonant response of fluid flow subjected to discrete heating elements,
Energy Convers. Manage. 48(9) (2007) 2461-2472.
10.1016/j.enconman.2007.04.008
[22] N.H. Saeid, and I. Pop, Maximum density effects on natural convection from a discrete heater in a cavity
filled with a porous medium, Acta Mech. 171(3-4) (2004) 203-212.
10.1007/s00707-004-0142-x
[23] G. Saha, Finite element simulation of magnetoconvection inside a sinusoidal corrugated enclosure with
discrete isoflux heating from below, Int. Commun. Heat Mass Transfer. 37(4) (2010) 393-400.
10.1016/j.icheatmasstransfer.2009.12.001
[24] M. Sankar, and Y. Do, Numerical simulation of free convection heat transfer in a vertical annular cavity
with discrete heating, Int. Commun. Heat Mass Transfer. 37(6) (2010) 600-606.
10.1016/j.icheatmasstransfer.2010.02.009
[25] G.F. Tang, F.Y. Zhao, D. Liu, Natural convection in a porous enclosure with a partial heating and salting
element, Int. J. Therm. Sci. 47(5) (2008) 569-583.
10.1016/j.ijthermalsci.2007.04.006
[26] S.K.W. Tou, and X.F. Zhang, Three-dimensional numerical simulation of natural convection in an
inclined liquid-filled enclosure with an array of discrete heaters, Int. J. Heat Mass Transfer. 46(1) (2003) 127138.
10.1016/s0017-9310(02)00253-3
[27] W.M. Yan, and T.F. Lin, Natural convection heat transfer in vertical open channel flows with discrete
heating, Int. Commun. Heat Mass Transfer. 14(2) (1987) 187-200.
10.1016/s0735-1933(87)81009-6