International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 5- May2013
DIMENSIONLESS NUMBERS EFFECT ON FORCED CONVECTION
IN HORIZONTAL PIPE FILLED WITH POROUS MATERIAL
1
1
Arpan Manchanda, 2Mohit Taneja, 3Sandeep Nandal, 4Ajay Kumar Agarwal
Asstt. Prof. & H.O.D., Mechanical Engineering Department, RIMT, Chidana, Sonipat, Haryana, India
2,4
Asstt. Prof., Mechanical Engineering Department, RIMT Chidana, Sonipat, Haryana, India
3
Asstt. Prof., Mechanical Engineering Department, DVIET Karnal, Haryana, India
Abstract: Laminar forced convective flow through a
horizontal pipe partially filled with a porous material is
investigated numerically. The porous material has a
cylindrical shape placed at z=0.05L from the pipe inlet.
The momentum equations are used for describing the
fluid flow in the clear region. The Darcy-ForcheimerBrinkman model is adopted to describe the fluid transport
in the porous region. The mathematical model for energy
transport is based on the one equation model which
assumes that there is a local thermal equilibrium between
the fluid and the solid phases. The study covers a wide
range of the dimensionless outer radius of the porous
material 0 ≤ Rp ≤ 1 and the effect of Darcy number, 2 X
-4
-1
10 ≤ Da ≤ 2 X 10 . In addition, the Reynolds number
has values of 200, 400 and 600 while Prandtl number has
values of 0.7, 5, 10 and 20. Through the study the ratio
between pipe length to outer diameter and porosity were
kept constant at 25 and 0.9 respectively.
Keywords: Heat transfer; Numerical simulation; porous
material; partially filled; flow inside pipe; Dimensionless
numbers.
I.
INTRODUCTION
The application of fluid flow and heat transfer phenomena
through porous media in industrial and natural fields is
multiple, extensive and important. For these reasons
numerous investigators are interested in studying and
understanding the fluid mechanics and heat transfer in
this complex phenomenon at several and various states.
The wide applications available have led to many
investigations in this area such as solar receiver devices,
ground water-hydrology building thermal insulation, heat
exchangers, energy storage units, drying processes, petroleum
reservoir and geothermal operations. The forced convective
flow through porous media was studied by many
investigators in several cases with various procedures:
analytically, numerically and experimentally. Mahmoudi
[ 1 ] investigated analytically the forced convection through
a channel partially filled with porous medium. Thermally
developed condition is considered and the local thermal
non-equilibrium model is utilized to obtain the exact
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solutions of both fluid and solid temperature fields for flow
inside the porous material as well as for flow in the clear
region. Basak et al. [2] to analyze the influence of various
walls thermal boundary conditions on mixed convection lid
driven flows in a square cavity filled with porous medium.
Varol [ 3 ] performed a numerical work to examine the heat
transfer and fluid flow due to natural convection in a
porous triangular enclosure with a centered conducting body.
The Darcy law model was used to write the governing
equations and they were solved using a finite difference
method. Singh et al. [4] studied the transient as well as nonDarcian effects on laminar natural convection flow in a
vertical channel partially filled with porous medium.
Forcheimer–Brinkman extended Darcy model is assumed to
simulate momentum transfer within the porous medium.
Bagchi and Kulacki [5] based on the one-domain formulation
of the conservation equations. It was found that Nusselt
numbers increase with a decrease in the heater length and
height ratio, and increase with the Darcy number. Yang and
Hwang [6] carried out Numerical simulations to investigate
the turbulent heat transfer enhancement in the pipe filled with
porous media. Akansu [7] presented a numerical heat-transfer
and pressure-drop for porous rings inserted in a pipe with
constant applied heat-flux to the outer surface of the pipe.
Angirasa [8] performed experiments that proved the
augmentation of heat transfer by using metallic fibrous
materials with two different porosities, namely 97% and 93%.
Yang et al. [9] concerned with fluid flow in a fluid-saturated
porous medium, accounting for the boundary and inertial
effects in the momentum equation. Mohamad [10] studied
numerically the flow in a pipe or channel fully or partially
filled with a porous medium. An air stream with a uniform
velocity and temperature is considered at the inlet to the
conduit.
II.
MATHEMATICAL MODEL
The porous material has a cylindrical shape placed at
z=0.005L & r pe varies from 0 < rp < ro, Fig.1 The analysis is
carried out for an incompressible steady state, laminar and
two-dimensional flow in a pipe flow in a pipe partially or fully
filled with porous medium, where the porous medium is
saturated with a single phase Newtonian fluid. The fluid
enters the pipe with uniform velocity u0 & and uniform
temperature T0 temperature at the wall Tw is assumed to be
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International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 5- May2013
constant. In addition, the inserted porous material in the pipe
is considered homogenous and isotropic with a uniform
porosity and uniform permeability magnitude. Furthermore,
the variation of the thermo-physical properties of the working
fluid and of the solid matrix with temperature is considered
negligible. Also viscous dissipation, gravitational effects,
natural convection and thermal radiation heat transfer are all
assumed to have negligible effect on the velocity and
temperature fields and the fluid and the solid matrix are
considered in local thermal equilibrium.
Where:
In the fluid region Tkr = 1, and in the porous region the value
of Tkr is dependent on the thermal conductivity of fluid and
solid matrix.
2.1 Boundary Conditions
,
(4)
&
2.2 Dimensionless form of the Governing Equations
The following dimensionless parameters are:
After introducing the dimensionless parameters, the set of
governing equations and boundary conditions have the
following dimensionless form:
Continuity equation:
Momentum equation:
Energy equation:
Fig 1: Schematic for Problem
The governing equations based on the one-domain approach
can be written as follows:
Continuity Equation:
2.3 Dimensionless form of the Boundary Conditions
,
&
Momentum equation:
Energy equation:
(9)
3. Nusselt Number Calculations
In tubular flow the local convective heat transfer coefficient is
usually calculated from
Where Tw is the wall temperature and Tb is the so-called fluid
bulk temperature inside the pipe.
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3.1. For Pipe Partially Filled with Porous Material (r p< ro)
For the case of fluid flow inside pipe partially filled with
porous material, the convective heat transfer coefficient is
calculated from the following equation:
Writing the above equation in dimensionless form, the local
Nusselt number for pipe partially filled with porous material:
And the dimensionless bulk fluid temperature can be
calculated as follow:
Fig 2: Variation of Nusselt Number with Number of Iterations
for Re = 100, Pr = 0.7, Da = 2x10-3 and Rp= 0.8
3.2 For Pipe Completely Filled with Porous Material
(rp= ro)
The convective heat transfer coefficient is calculated from the
following equation:
Writing the equation in dimensionless form we get the
local Nusselt number for pipe completely filled with
porous material:
The effective thermal conductivity of the porous
medium
is presented in two different correlations as
follow:
The average Nusselt number Nu over the length of the
pipe is expressed as
4. Solution Procedure
The governing equations were solved using the finite
volume technique developed by Patankar. This technique
was based on the discretization of the governing equations
using the central difference in space. The iteration method
used in this program is a line-by-line procedure, is a
combination of the direct method and the resulting Tri
Diagonal Matrix Algorithm (TDMA). The convergence of
the iteration is determined by the change in the average
Nusselt number as well as other dependent variables through
one hundred iterations to be less than 0.01% from its initial
value, Figs. 2 & 3 showing the convergence and stability of
the solution.
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Fig 3: Variation of Axial Mean Velocity at Exit Section with
Number of Iterations for Re= 100, Pr = 0.7, Da = 2x10-3 and
Rp = 0.8
4.1 Grid Size Selection
To verify the independence between the grid and the
numerical results and choosing the grid of the
computational domain over which we will solve the governing
equations of the problem, we tried three different mesh grids
181x151, 161x131, and 141x111 in Z and R-directions,
respectively. The three meshes were tested by comparing the
dimensionless axial velocity profiles for laminar flow in pipe
filled with porous media at half length of the pipe, z=0.5L.
The comparisons of velocity profiles are presented in Figs. 4
and 5 showing that the difference between the results for
different meshes is not noticeable; the maximum difference
between grid, 181x151, and grids of 141x111 equals to 1.05%
at the same location with the same flow conditions, and the
maximum difference between grid (181x151) and grid
(161x131) equal 0.59% The results obtained using the code is
independent of the used grid sizes. From this conclusion, the
grid (181x151) is sufficient enough for the calculations.
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5.1 Effect of Reynold Number on the Average Heat
Transfer
In order to study the influence of Reynolds number on the
average Nusselt number, three different Re numbers are
considered 200, 400 and 600. The results are computed for
different values of (a) Da=2x10-1 (b) Da=2x10-2 (c) Da=2x10-3
(d) Da=2x10-4 and presented in Fig.7 (a), (b), (c) and (d),
respectively.
Fig. 4: Variation of Axial Velocity at (Z=0.5 L) with Number
of Iterations for Re=100, Pr=0.7, Da=2x10-4 and Rp=1.0
Fig. 5: Variation of Local Nusselt Number for Re=100,
Pr=0.7, Da=2x10-4 and Rp=1.0
4.2 Checking the Validity of the Program
Fig. 6 represents a comparison between the solution obtained
by the present and the numerical solution of Bogdan et al. [11]
the fully developed pressure drop for laminar flow in a pipe
partially filled with porous material the comparison shows that
the difference between the two solutions is not noticeable and
the results of this work are very close to those of previous
work.
Fig. 6: Comparison of The Pressure Drop for Present Code
and Bogdan et al for Re=100, Pr=0.7, Da=2x10-3
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International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 5- May2013
at Da = 2x10-4 and for: Pr=0.7, 5, the critical thickness
Rpcr=0.8; Pr =20, the critical thickness Rpcr=0.9.
Fig. 7: Effect of Reynolds Number on the Average Nusselt
Number for Pr=0.7 (a) Da = 2x10-1 (b) Da = 2x10-2 (c) Da =
2x10-3 (d) Da = 2x10-4
From the obtained results, we can note the following remarks:
1. As expected, the average Nusselt number increases when
Re increases, this is due to the fact that the increase in Re
increases the thermal entrance length, like it is known, the
heat transfer in the entrance is high.
2. The influence of increasing Re on the average Nusselt
number for different Rpe and different Da is very close to
the influence of increasing Re on the average Nusselt
number for flow in empty pipe (Rp=0).
3. The critical radius (Rpcr at which the heat transfer begin to
decrease) may be affected by changing Reynolds number,
where at Da = 2x10-2
 Re=200, the critical radius Rpcr=0.6;
 Re=600, the critical radius Rpcr=0.7
5.2 Effect of Prandtl Number on the Average Heat
Transfer
The examination of the effect of Prandtl number on the
average Nusselt number is investigated for Pr = 0.7, 5, 10, and
20. The results are computed for different values of Da 2x10-1,
2x10-2, 2x10-3, and 2x10-4 and shown in Fig. 8 (a), (b), (c) and
(d) respectively. From the obtained results, we can observe
that:
1. When Pr increases the thermal entrance length increases,
like it is known the heat transfer in the entrance region is
high, this performs to increase the average Nusselt
number.
2.
3.
For Da = 2x10-1, 2x10-2 and at any Rp, the effect of Pr on
the average heat transfer is nearly identical to the effect of
Pr on the average heat transfer of flow in empty pipe
(Rp=0).
The critical radius may be changed by changing Pr, where
At Da = 2x10-2 and for: Pr = 0.7, the critical thickness
Rpcr=0.6; Pr=10, 20, the critical thickness Rpcr=0.7.
At Da = 2x10-3 and for: Pr = 0.7, the critical thickness
Rpcr=0.7; Pr = 10, 20, the critical thickness Rpcr=0.8. Also
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International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 5- May2013
Figure 8: Effect of Prandtl Number on the Average Nusselt
Number for Re=100 (a) Da = 2x10-1 (b) Da = 2x10-2 (c) Da =
2x10-3 (d) Da = 2x10-4
1.
2.
3.
4.
5.
III.
CONCLUSIONS
For different Rp and for different Da, the insertion of
porous material inside the pipe leads to a higher average
Nusselt number than that of fluid flow in clear pipe.
From the results of average Nusselt number and the
pressure drop, it is readily seen that any increase in
the radius of porous material over the critical radius
has a negative impact on the thermal performance and
on the pumping power.
The insertion of porous material inside the pipe at
different Rp and at different Da decreases the
hydrodynamic entrance length in comparison with the
case of flow in clear pipe.
The results of the average Nusselt number show that
the critical radius may be influenced by some
operating parameters such as Da, Re, & Pr.
As Re increased the heat transfer increased.
Brinkman extended Darcy model" International Journal
of Heat and Mass Transfer, 54, pp 1111–1120.
[5] Bagchi, F.A. Kulacki, "Natural convection in fluid–
superposed porous layers heated locally from below"
International Journal of Heat and Mass Transfer. Vol54, Pages 3672-3682, (2011).
[6] Yang Y and Hwang M, 2009 "Numerical simulation of
turbulent fluid flow and heat transfer characteristics in
heat exchangers fitted with porous media" International
Journal of Heat and Mass Transfer, 52, pp 2956–2965.
[7] S. Akansu, Heat transfer and pressure drops for porousring urbulators in a circular pipe, Applied Energy, 83
(2006), pp 280–298.
[8] D. Angirasa, Experimental investigation of forced
convection heat transfer augmentation with metallic
fibrous materials. Int. J. Heat Mass Transfer, 45(2002),
pp 919-922.
[9] D. Yang, Y. Yang, V.A.F .Costa, Numerical simulation
of non-Darcian flow through a porous medium.
Particuology 7(2009) 193–198.
[10] A. Mohamad, Heat transfer enhancements in heat
exchangers fitted with porous media Part 1: constant
wall temperature. Int. J. Thermal Sciences 42 (2003)
385-395.
[11] Bogdan Ionel Pavel, Experimental and numerical
investigation of heat transfer enhancement using porous
media, M.Sc. Thesis , Departement of Mechanical and
Manufacturing Engineering, University of Calgary
,2004.
[12] S.V. Patankar, Numerical Heat Transfer and Fluid
Flow. 1980, Mc Graw-Hill, New York.
IV.
REFERENCES
[1] Maerefat M, and Mahmoudi Y., "Analytical
investigation of heat transfer enhancement in a channel
partially filled with a porous material under local
thermal non-equilibrium condition" International
Journal of Thermal Sciences, 50, pp 2386-2401, (2011).
[2] Pradeep P.V. K, Basak T, Roy S and Pop I, "Finite
element based heat line approach to study mixed
convection in a porous square cavity with various
wall thermal boundary conditions" International
Journal of Heat and Mass Transfer, vol. 54, Pages
1706-1727, (2011).
[3] Varol Y, "Natural convection in porous triangular
enclosure with a centered conducting body.
International Communications in Heat and Mass
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[4] Singh K, Agnihotri P, Singh N.P and Singh A. K, 2011
"Transient and non-Darcian effects on natural
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with porous medium: Analysis with Forchheimer–
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