Optimization and Computational Fluid Dynamics - Department of ...

8 Multi-objective Optimization in Convective Heat Transfer 229
β ∝ U m av
(8.25)
to hold, with a value of the exponent m close to 1. In these conditions,
evaluating the flow field with a test value for β, takingintoaccountthat
the desired average velocity is unitary, and updating iteratively the pressure
gradient as follows:
βn+1 = βn
Uav,n
(8.26)
it is expected to reach the exact value of β in few steps. This has been verified
during this study, where only 3 to 6 steps are required to reach a value of β
which gives an error on Re below 0.1%.
8.4.2 Thermal Field Iterative Solution
The thermal field solution for the two cases reported here is quite different
since for the wavy channel a uniform temperature boundary condition has
been adopted while for the CC channel a constant flux has been considered.
Therefore, the algorithms used in the two cases will be described independently.
8.4.2.1 Wavy Channels
After the velocity field has been obtained, the thermal field is computed with
an iterative approach. This is based on the fact that the heat flux at the wall
has to be balanced by the enthalpy difference between inlet and outlet. The
task is to find a value of σ, Eq. (8.15), that ensures this balance:
˙mcp (Tb,in − Tb,out)=
�
w
−k ∂T
ds. (8.27)
∂n
Assembling the dimensional terms and remembering Eq. (8.15), one is left
with
1 − 1
�
2
=
σ Re Pr
− ∂θ
ds.
∂n
(8.28)
It can be recognized that there is a relation between σ and the heat flux
at the wall and in particular, an incorrect value of σ leads to a generation
term on the outlet boundary which has no physical meaning. So, starting
from a tentative value of σ and updating it iteratively by means of (8.28), it
converges towards the correct solution.
Once the correct thermal field has been calculated, the mean Nusselt number,
Nu, is obtained as:
w