Optimization and Computational Fluid Dynamics - Department of ...

8 Multi-objective Optimization in Convective Heat Transfer 225
lows:
and, in general:
uin = uout
(8.6)
u(x, y, z)=u(x + L,y,z) (8.7)
where L is the length of the repeating module. After the split of the total
pressure, the newly introduced periodic pressure, ˜p, can be handled as the
velocity field:
(8.8)
and, in general:
˜pin =˜pout
˜p(x, y, z)=˜p(x + L,y,z) . (8.9)
Finally, standard no-slip conditions are imposed at the walls.
8.3.3 Temperature Boundary Conditions
The temperature field in a heat exchanger or in a regenerator is not periodic
since it changes continuously along the channel in the mean flow direction.
However, a region of fully developed thermal condition can be identified and
appropriate boundary conditions can be imposed for a single module. The
thermal boundary conditions are different for the two cases presented in this
work so they are treated separately. For the wavy channel case a constanttemperature
boundary condition is used as representative of e.g., automotive
radiators with negligible wall thermal resistance at high liquid flow rates. On
the other hand, the CC channel is used in gas-gas recuperators where the
wall temperature is not uniform. For this case, a flux boundary condition has
been developed.
8.3.3.1 Wavy Channel
Shah and London [49] discuss fully developed flow in parallel plates for many
wall-boundary condition cases. By fully developed thermal field, itismeant
that the Nusselt number is constant along the flow.
As in Patankar et al. [37], the concept of the thermally developed regime
is generalized to the periodic thermally developed one. The condition for a
constant cross-sectional channel writes as follows:
∂θp
∂x
where θp is the local non-dimensional temperature
= 0 (8.10)