Numerical Study of Forced Convection in a Rectangular Channel Original Research Article Journal of Chemistry and Materials Research Vol. 1 (1), 2014, 7–11 Salim Gareh

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© (2014) Copyright ORIC Publications
Journal of Chemistry and Materials Research
Vol. 1 (1), 2014, 7–11
JCMR
Journal of Chemistry and
Materials Research
ORICPublications
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Original Research
Numerical Study of Forced Convection in a Rectangular Channel
Salim Gareh*
Department of Mechanical Engineering, University of Biskra, Biskra.07000, Algeria
Received 18 May 2014; received in revised form 08 Jun 2014; accepted 19 Jun 2014
Abstract
We studied the phenomenon in this section of the laminar flow in a rectangular channel. Or utilization this channel in a rectangular solar
collector. To solve this problem we used the finite volume method. There are four such that the continuity equation, the Navier- stocks and
thermal energy equation. A program to solve the Navier-stocks with the schematic SIMPLE is used. The results show that this problem
connects with Re whenever the number of the boundary layer increases Re decreases.
Keywords: Fluid mechanics, Heat transfer, Finite volume simulation.
1. Introduction
The process of free convective heat transfer from the
surface of a body to the surrounding fluid is of great
importance in the field of thermo-fluid mechanics. This
process is of technological importance in the design of solar
collectors.
Although in heat exchangers the forced convection is
dominant, the free con-vection becomes the dominant mode of
heat transfer in case of power failure. Moreover, rectangular
channel geometry is the large cases of which enable academic
researchers [1–3].
He also has many research, where he studied the simulation
of the flow of laminated by ending volumes. For example,
Sangi et al. [4] are stated the modeling and numeri-cal
simulation of solar chimney power plants. Mahfouz and
Kocabiyik [5] have treated problem of the transient numerical
simulation of buoyancy driven flow adja-cent to an elliptic
tube.
In the present work, we became interested in the study of
coolant flow in flat plate collectors, the numerical solution of
the equations governing the flow with simplifying hypotheses
* Corresponding author. Tel.: +213796651689.
E-mail address: garehsalim@gmail.com (S. Gareh).
All rights reserved. No part of contents of this paper may be reproduced or
transmitted in any form or by any means without the written permission of
ORIC Publications, www.oricpub.com.
appropriate in a rectangular enclosure subjected to a heat flux
imposed by the top wall and the side and rear walls are
adiabatic.
2. Mathematical modeling
2.1.Statement of the problem
We will consider the fluid flow sensor of a plane between
two parallel planes: the upper absorber which is transmitted by
the heat flow to be recovered by the fluid. And then the lower
one as insulating (adiabatic wall).
2.1.1. Simplifying Hypotheses
The system of equations governing the flow that is
established in the configuration (Fig.1) in question must take
into account the nature of the flow (rolling or turbulent), the
type of fluid used (perfect viscous compressible ... etc). As
well as simplifying hypotheses selected. In other words;
Writing the governing equations must adapt to the type of
proposed study as well as additional simplifying hypotheses
made. To simplify the problem, the equations adopted to
describe the flow are formulated assuming the validity of the
following assumptions:
- The system is stationary.
- The fluid is incompressible.
- The flow is laminar
- The configuration studied contains no source or energy or

2. 8 S. Gareh / Journal of Chemistry and Materials Research 1 (2014) 7–11
material well and is the seat of any chemical reaction.
- Viscous dissipation is negligible.
Conservation equations as follows:
0





Y
V
X
U
(1)

























2
2
2
2
Y
U
X
U
X
P
Y
U
V
X
U
U 
(2)
g
Y
V
X
V
Y
P
Y
V
V
X
V
U  
























2
2
2
2
(3)






















2
2
2
2
Y
T
X
T
k
Y
T
V
X
T
Ucp
(4)
where,
X: (x, y, z): Spatial Coordinates
Δx: step next x
Δy: step next it
Cp: Specific heats at constant pressure
k: Thermal conductivity
h: Height
l: Length of channel
U: Velocity in the x direction
P: Pressure
T: Temperature
μ: Dynamic viscosity
ρ: Density
Pr: Prandtl number Pr = (μ Cp / k)
Re: Reynolds number Re = (ρνd / μ)
Gravity forces are negligible in horizontal flow [5].
2.1.2. The boundary conditions
Modelizing the equations is solved with the following
boundary conditions:
- To the collector inlet and for y = 0, x = 0: U = Uo, V =
0, T = To.
- To the top wall (absorber) y = H: U = V = 0, T = Tc.
- To the bottom wall (insulating) y = 0: U = V =0,
Dimensionless form of the equations:
0*.*  v (5)































2
2
2
2
*
*
*
*
Re
1
*
*
*
*
*
*
*
*
Y
U
X
U
X
P
Y
U
V
X
U
U (6)































2
2
2
2
*
*
*
*
Re
1
*
*
*
*
*
*
*
*
Y
V
X
V
Y
P
Y
V
V
X
V
U
(7)




























2
2
2
2
*
*
*
*
Pr
1
*
*
*
*
*
*
Y
T
X
T
Y
T
V
X
T
U
(8)
where
U *: Dimensionless velocity
V *: Dimensionless velocity
P *: Dimensionless pressure
T *: Standard Temperature
U ** Estimate of U
V ** Estimate of V
P **: Estimate
P: Pressure correction
Conditions to the dimensional boundaries:
- To the input, x* =0: U* = 1, V* = 0, T* = 0
- To top wall y* =1: U* = V*=0, T* = 1,
- To the bottom wall, y* = 0: U* = V* = 0,
 
,0
*
*
0**,









Y
T
yx
Fig. 1. Geometry and boundaries conditions dimensionless
2.2.Algorithm
At present, the SIMPLE algorithm [6] to solve the
conservation equations in primitive variables is explained.
This algorithm consists of four main steps which are listed be-
low:
Step.1 estimate or guess the pressure field P ** U ** V **
Step.2 solve the Navier-Stokes equations for the velocity
components U **, V **.
Step. 3 solve the equation pressure correction.
Step. 4 calculating parameters (U, V, P).
Step .5 calculate the pressure P = P + P **.
Step. 6 calculate the velocity components from the values U **
V **, using the rela-tionship:
Step. 7 solving other conservation equations for calculating the
temperature (or con-centration) if coupled to the equations of
movements Equation.
Step. 8 Pressure P is a new estimate of the pressure P **, then
return to step 2 and continue until a converged solution.
Our programs and drawing results with software (Matlab .7.1).

3. S. Gareh / Journal of Chemistry and Materials Research 1 (2014) 7–11 9
0
5
10
15
20
25
0 5 10 15 20 25
la vitesse V(u,v)
y
x
Fig. 2. Directions velocities (Re = 100)
3. Results and discussion
Isothermal environment temperature is the same at any
point of the study area, as well as all the physical properties
present in the transport equations are constant. If more of the
velocity of sound, which corresponds to a low Mach number,
then the fluid may be regarded as incompressible and the
density ρ does not vary along the flow knowing the
dependency of it is negligible when Ma is less than 0.2 [7].
This is our case, the model adopted is that of an
incompressible fluid and indilatable.
In our equations, a single dimensionless parameter, the
Reynolds number Re and the velocity field is completely
determined by Re Flow we will study is that of a confined
fluid, because of the effect the wall and we can not neglect the
viscosity. For if we neglect the viscosity of the flow equations
are those of Euler.
3.1.Dynamic Study
In internal flows, the velocity profiles whose knowledge is
essential convection heat and mechanical energy dissipation
(pressure drop) determine the dimensioning of fluids or
exchange circuits and to study the energy balance. The
influence of the walls is very important, the diversity of
possible geometries obviously has a huge variety in the
structure of flows, but they are often boundary layers such as
in channels.
But the laminar or turbulent nature of the flow is not
sufficient to fully characterize the field velocities, because the
structure of the flow is also modeled by the forms walls.
The boundary layer phenomenon occurs at the entrance to
the close to the wall but beyond a certain distance as the wall
completely enveloping the fluid in the direction perpendicular
to the flow, the boundary layer thus had an with itself to give
rise to a new dynamic structure:
0
5
10
15
20
25
0
10
20
30
0
0.5
1
1.5
Re =100
x
Velocity
0.2
0.4
0.6
0.8
1
1.2
1.4
0
5
10
15
20
25
0
10
20
30
0
0.5
1
1.5
Re =200
x
Velocity
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
0
5
10
15
20
25
0
10
20
30
0.2
0.4
0.6
0.8
1
1.2
Re =300
x
Velocity
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
0
5
10
15
20
25
0
10
20
30
0.2
0.4
0.6
0.8
1
1.2
Re = 400
x
Velocity
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 3. Velocity distribution V (u, v) in the channel Re = 100, 200,
300 and 400

4. 10 S. Gareh / Journal of Chemistry and Materials Research 1 (2014) 7–11
0
10
20
30
0
5
10
15
20
25
0
0.2
0.4
0.6
0.8
1
Re = 100
x
Pressure
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0
10
20
30
0
5
10
15
20
25
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Re = 200
x
Pressure
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0
10
20
30
0
5
10
15
20
25
-0.1
0
0.1
0.2
0.3
0.4
Re = 300
x
Pressure
0
0.05
0.1
0.15
0.2
0.25
0.3
0
10
20
30
0
5
10
15
20
25
-0.05
0
0.05
0.1
0.15
0.2
Re = 400
x
Pressure
0
0.05
0.1
0.15
Fig. 4. Pressure distribution in the channel Re = 100, 200,300 and
400
The flow set still has the characteristics of a boundary layer
flow but it is ordered in a particular way.
For uniform distribution of velocity in the inlet section, the
laminar boundary layer thickens regularly until occupy the
whole of the fluid stream, which gives rise to the steady state.
In the central part also called central core, the fluid is
subject to acceleration which compensates for the braking
underwent in the boundary layer and the velocity in this case
out of the boundary layer depends both on the distance of the
input and the dis-tance to the wall. The length setting (L)
depends on the Reynolds number Fig. 2 and Fig. 3.
At the entry, the velocity of the flow is uniform by Y. Next,
the velocity profile of Fig. 2 Shows the effect of the edges
which results in the development of boundary layers (top and
bottom).
The pressure field is directly dependent on the velocity.
Sees the Navier-Stokes as the flow is not established, thus is
changing the pressure along the flow and reduces the input
pressure to the output value. This decrease is due to dissipation
of the fact that a viscous force, the fluid loses some of its
energy which translates the decrease in pressure. In these areas
the velocity profile changes along the channel then:
  0 xu 0v   cstexp 
The presence of walls has the effect of imposing very
conditions similar to those en-countered in an external
boundary layer as follows:
     0;; 2222
 xpyuxuuv
And this is only the case if there is a drive after a constant
section and of sufficient length so that the flow will become
established. This length is called length of estab-lishment of
the dynamic system (Fig. 4).
At the entrance, the effect of boundary layer is visible, the
pressure profile in the form of a curve which from a certain
distance where the inlet system is determined, the two
boundary layers meet, see Fig. 4.
3.2.Thermal Study
The exchange of heat between the absorber and the fluid is
favored close to it (see Fig. (5)). It is noted that the
temperature ranges of the dimensionless value T = 1 to T = 0.1
over the top half of the fluid axis of nearly symmetry.
-There is the appearance of a thermal boundary layer which
undertakes to the en-trance.
Large-temperature values are noticed adjacent the
absorber. While, the fluid near the wall is not achieved by
these high temperatures.
4. Conclusion
The study is related to the assessment by a numerical method
different thermal and dynamic parameters that can help

5. S. Gareh / Journal of Chemistry and Materials Research 1 (2014) 7–11 11
optimize performance collector plans.
This study was conducted for the case of a configuration of
a collector sleek driving. The velocity profiles and calculated
temperatures show the side effect of on the input speed limits
for two developing layers extend over a more or less large
length accord-ing to the value of the Reynolds number.
Secondly, there is the appearance of which is established
adjacent the thermal boundary layer absorber.
References
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vortex roll structures in a mixed convective air flow through a
horizontal plane channel a numerical study. lnternational. Journal.
Heat Mass Transfer, 40, 505-518.
[2] Ryan, D. and Burek, S.A.M. (2010). Experimental study of the
influence of collector height on the steady state performance of a
passive solar air heater, Solar Energy, 84, 1676–1684.
[3] Zhang, L., Wang, W., Yu, Z., Fan, L., Hu, Y., Fan Y. J. and Cen, K.
(2012). An expe-rimental investigation of a natural circulation heat pipe
system applied to a parabolic trough solar collector steam generation
system, Solar Energy, 86, 911–919.
[4] Sangi, R., Amidpour, M. and Hosseinizadeh, B. (2011). Modeling and
numerical simulation of solar chimney power plants. Solar Energy, 85,
pp. 829–838
[5] Mahfouz, F.M., and Kocabiyik, S. (2003). Transient numerical
simulation of buoyancy driven flow adjacent to an elliptic tube.
International Journal of Heat and Fluid Flow, 24, pp. 864–873.
[6] Versteeg, H.K. and Malaskera, W. (1995). An introduction of
computational fluid dynamics, Longman Group Ltd England.
[7] Candel, S. (1995). mécanique des fluides, Paris :Dunod,.
[8] Munson, R.M. and Young, D.F. (2002). Fundamentals of Fluid
Mechanics, department of Mechanical Engineering. USA: Iowa State
University Ames, Iowa.
[9] Patankar, S.V. (1980). Numerical Heat transfer and Fluid Flow, series in
computational methods in mechanics and thermal sciences. New York:
Hemisphere Publishing.
0.10.10.1
0.1
0.20.20.2
0.2
0.30.30.3
0.3
0.40.40.4
0.4
0.50.50.5 0.60.60.6 0.70.70.7 0.80.80.8 0.90.90.9
x
Temperature
Re = 100
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.10.10.1
0.1
0.20.20.2
0.2
0.30.30.3
0.3
0.40.40.4
0.4
0.50.50.5
0.5
0.60.60.6
0.6
0.70.70.7
0.80.80.8
0.90.90.9
x
Temperature
Re = 200
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.10.10.1
0.1
0.20.20.2
0.2
0.30.30.3
0.3
0.40.40.4
0.4
0.50.50.5
0.5 0.60.60.6
0.70.70.7
0.80.80.8
0.90.90.9
x
Tepmerature
Re = 400
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Fig. 5. Temperature distribution in the channel Re = 100,200 and
400, Pr = 0.71.