Page 229 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

Vertical distance CONVECTION HEAT TRANSFER 221
1
0.8
0.6
0.4
0.2
0
6
8
2
4
0
Temperature 10 12 14 16
Figure 7.21 Forced convection flow through a two-dimensional rectangular channel. Tem-
perature profiles at various distances
Hot sphere

Cold air stream

Figure 7.22 Forced convection flow past a sphere

profile distribution is as shown in Figure 7.21. As may be seen, a parabolic temperature
profile is achieved at around the same distance from the entrance as that for the parabolic
velocity profile. It should also be noted that as the length of the channel increases, the
average temperature of the fluid also increases and approaches that of the wall temperature.
The second problem considered is a three-dimensional flow over a hot sphere. The heat
transfer aspects of the hot sphere are studied as it is exposed to a cold air stream. The
problem definition is different from that of the channel flow, which is an internal flow, for
in this case the flow past a sphere is an external flow problem as shown in Figure 7.22.
As shown, the sphere is in an unbounded space, and an outer boundary needs defining in
order to carry out the computation. The boundary conditions on the boundary walls should
be fixed in such a way that they do not affect the heat transfer and flow properties close to
the sphere. The best way to minimize the influence of these outer boundary conditions on
the heat transfer and flow around the sphere is to place the boundaries far from the sphere.
In the problem discussed here, an outer boundary is fixed in such a way that the inlet is
at a distance of five diameters from the centre of the sphere, and the exit is at 20 diameters
downstream of the centre of the sphere (Nithiarasu et al. 2004). The side boundaries are
also at a distance of five diameters away from the centre of the sphere. It is possible to
imagine the sphere being placed inside a three-dimensional channel, which is 25 diameters
long having 10 diameter sides. However, the difference from the previous channel problem
is that there is no solid outer wall in this case.
The boundary conditions are simple as in the previous problem. The inlet has a non-
dimensional velocity of unity and a non-dimensional temperature of zero. The surface of
the sphere is subjected to a no-slip velocity boundary condition and a non-dimensional