Optimization and Computational Fluid Dynamics - Department of ...

8 Multi-objective Optimization in Convective Heat Transfer 231
been added as a source term to the momentum equation and updated using
Eq. (8.26) by means of CCL expression to obtain the desired velocity. To
solve the thermal field, Eq. (8.19) is computed at each time step using a CCL
expression, and the gradient γ is introduced in the last term of Eq. (8.18) to
compute the source term of the energy equation for both hot and cold fluid
streams.
The average Nusselt number can be easily obtained by inspecting Eq. (8.21).
The global heat transfer coefficient, neglecting wall resistance, can be written
as
�
1
U =
¯hh
+ 1
�−1 ¯hc
(8.33)
where ¯ hh and ¯ hc are the mean heat transfer coefficients for the hot and cold
side, respectively. Since the fluids, the flow heat capacities and the geometry
for the hot and cold ducts are equal, a unique mean heat transfer coefficient
can be introduced ¯ hh ≡ ¯ hc ≡ h, and Eq. (8.21) can be simplified as:
φ = h
· A · ∆T (8.34)
2
from which the mean Nusselt number can be obtained as:
Nu = φDh
∆T Ah k
(8.35)
where Dh =2b is the hydraulic diameter of the channel [26]. The local Nusselt
number can be derived from (8.35) as:
Nu = 2 φ′′ (x) Dh
∆T k
where φ ′′ (x) is the local specific heat flux at the wall.
8.5 Geometry Parametrization
(8.36)
Different geometry parametrizations have been used for the wavy channels
and the CC module and are described next.
8.5.1 Wavy Channels
The shape of two-dimensional convective channel is represented either by
linear-piecewise profiles, or by NURBS. The two different geometric profiles,