Optimization and Computational Fluid Dynamics - Department of ...

8 Multi-objective Optimization in Convective Heat Transfer 227
periodicity
wall
periodicity periodicity
(a) (b) (c)
Fig. 8.4 Periodicity conditions for the CC heat exchanger: (a) assembled domain periodicity;
(b) hot domain periodicity; (c) cold domain periodicity
Equation (8.16) can be substituted into the energy equation, Eq. (8.3), to
obtain the transport equation for the periodic component:
�
∇· ρcpu ˜ � �
T = ∇· λ ∇ ˜ �
T + γuρcp . (8.18)
Equation (8.18) can be solved with appropriate boundary conditions as in [51]
where a uniform wall flux was imposed and to satisfy the energy balance, the
temperature gradient was computed as:
γ = φ
˙mcpL
(8.19)
where L is the length of the periodic module in the mean flow direction,
and φ is the overall heat flux specified at the walls. While the uniform flux
is a practical solution for imposing a thermal boundary condition, it does
not correctly describe the heat transfer in a recuperator module. Indeed the
fluid flows in the furrows of the CC ducts with complicated three-dimensional
patterns, strongly affecting the local heat transfer rates.
In this work the elementary periodic module is doubled to take into account
both hot and cold fluids as shown in Fig. 8.3, while the periodicity conditions
for both fluids are reported in Fig. 8.4. Equation (8.18) is applied to both
domains and no boundary condition has to be imposed on the interface wall.
A similar approach has been used by Morimoto et al. [29] to solve the heat
transfer in a periodic channel, but with a different geometry. To satisfy the
global energy balance the same flux has been applied to the hot and cold
domains:
φh = −φc
(8.20)
using Eq. (8.19) to compute the last term of Eq. (8.18). Temperatures of
the hot and cold domains are automatically adjusted during the iterative
computation to satisfy automatically the overall energy balance:
φ = UA∆T (8.21)
where ∆T is the temperature difference between hot and cold fluids: