Optimization and Computational Fluid Dynamics - Department of ...

226 Marco Manzan, Enrico Nobile, Stefano Pieri and Francesco Pinto
θp(x, y)=
T(x, y) − Tw
Tb(x) − Tw
(8.11)
and Tb(x) is the streamwise varying bulk temperature. In the case of periodic
wavy channel, a weak condition, between two sections a period length apart,
can be imposed:
θp(x, y)=θp(x + L,y) . (8.12)
The use of the periodic temperature field, defined in Eq. (8.11), leads to a volume
force term, in the energy equation, which depends on the x streamwise
coordinate. So, another equation must be introduced [37], but this is cumbersome
to implement on the triangular unstructured grids that COMSOL,
the FE package employed for wavy channels, uses. For this reason, another
strategy has been used to tackle the problem.
A fixed arbitrary reference value has been adopted to make the temperature
non-dimensional. The value chosen, for convenience, is the bulk temperature
at the inlet boundary. So the periodicity condition, Eq. (8.12), changes
into:
θ(xin,y)=θ(xout,y) · σ (8.13)
where
T(x, y) − Tw
θ(x, y)=
Tb,in − Tw
and σ is the ratio between the inlet and outlet temperature differences:
σ = Tb,in − Tw
Tb,out − Tw
(8.14)
. (8.15)
In place of the adjoint equation introduced by Patankar [37], an iterative
numerical procedure based on an energy balance has been introduced to
reach fully developed conditions. This will be explained in Sect. 8.4.
8.3.3.2 CC Channel
In gas microturbine regenerators, the hot and cold streams are in countercurrent
flow arrangement and are characterized by similar heat flow capacities.
With these conditions the heat transferred between the fluids remains
almost constant from repetitive module to module, so temperature and pressure
can be treated in a similar way. The temperature is expressed as the
sum of a periodic and a linear component driven by the gradient in the mean
flow direction γ:
T(x, y, z)=γx+ ˜ T(x, y, z) (8.16)
˜T(x, y, z)= ˜ T(x + L,y,z) . (8.17)