Environmental Engineering Reference
In-Depth Information
ʲ
to gravity and
is the thermal expansion coefficient. Equations (
1
)-(
3
)havetobe
solvedwith the following boundary conditions. Uniformflow at the channel entrance,
with
ˈ
varying from 0 to 1 and
=
0for
H
≤
Y
≤
L
; no slip at the walls with
ˈ
=
0
at the left, bottom and right walls and
ˈ
=
1 at the top wall; and relaxed parallel flow
2
conditions at the outlet:
∂ˈ/∂
X
=
∂
ˈ/∂
X
∂
Y
=
0. Wall vorticities are evaluated
n
2
is the grid
space normal to the wall (Thom
1933
). The boundary conditions for the temperature
are the following: fixed temperature,
n
2
, where
using Thom's classical formula,
ʩ
W
=
2
(ˈ
W
+
1
−
ˈ
W
)
/
ʸ
=
1, 0 at the left and right walls, respectively;
fixed fluid inlet temperature,
ʸ
=
0
.
5; adiabatic top and bottom walls,
∂ʸ/∂
Y
=
0;
the condition at the exit is the normally assumed relaxed condition
0.
The local Nusselt numbers at the left and right isothermal walls are computed as
Nu
(
Y
)
∂ʸ/∂
X
=
|
X
=
0
,
1
.
5
. The space averaged Nussel
t n
umber is then computed by
integrating the local Nusselt number along each wall,
Nu
=±
∂ʸ/∂
X
)
0
=
(
1
/
H
Nu
(
Y
)
dY
.
3 Numerical Results and Model Validation
The equations were discretized using a second-order central difference formulation
for all the spatial derivatives and solved using the ADI scheme. The system of nonlin-
ear equations (
1
)-(
3
) along with their corresponding boundary conditions are solved
numerically using a uniform grid of 60
60 that yielded independent results in terms
of the average Nusselt number and the maximum value of the stream function. A
nondimensional time step
×
10
−
3
has been used for all com-
τ
=
u
0
t
/
L
=
1
×
10
−
7
was used to measure
the convergence of the dependent variables. The code has been validated by com-
paring the results of the simulations against those of Singh and Sharif (
2003
)for
mixed convective cooling of a rectangular cavity with inlet and exit opening. The
computed results for the average Nusselt number at the left wall for different values
of the Richardson number are shown in Table
1
. Clearly, good agreement between
the present predictions is found with a maximum discrepancy of 2.37%, justifying
the numerical method used in this study.
putations for fixed
Pr
=
3,060. A tolerance value of 1
×
Ta b l e 1
Comparison of variation of the average Nusselt number at the left wall
Average Nusselt number at
Re
=
50 with
Pr
=
0.71
Ri
0.001
0.01
0.1
1
10
Singh and Sharif (
2003
)
9.2
9
8.8
10.5
13.98
Present work
9.1432
9.2136
8.6343
10.7212
13.7654
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