Optimization of high performance small-scale plate-fin heat exchangers.

This work treats the macro-scale description of flow and heat transfer in systems with spatially periodic solid structures, like compact heat exchangers and heat sinks with fin arrays or tube bundles.
The presented macro-scale description allows us to extract the physically meaningful overall characteristics of the very detailed flow velocity, pressure and temperature fields in a heat transfer device.
In that regard, it enables data reduction for the huge amount of detailed information which results after direct numerical simulation (DNS) of the Navier-Stokes flow and temperature equations for the device.

The presented macro-scale description also allows us to determine the macro-scale characteristics of the flow and temperature fields in the laminar periodically developed regime by solving a simple closure model on a unit cell of the solid structures.
As the closure model for the flow and heat transfer within the unit cell is in exact agreement with DNS results for the entire array of solid structures, model reduction is achieved in a consistent manner with respect to DNS.

The model equations for the macro-scale flow and heat transfer are obtained by weighted spatial averaging, or filtering, of the Navier-Stokes equations.
It is shown that the macro-scale flow velocity and macro-scale pressure in the periodically developed flow regime must be defined through a double volume-average filter to ensure that the macro-scale flow velocity and macro-scale pressure gradient are spatially constant.
In that case, also the interfacial force and the momentum dispersion source, which appear in the macro-scale flow equations, are spatially constant, so that they can be easily governed from the developed flow equations on a unit cell.

Furthermore, it is shown that the macro-scale temperature in the periodically developed heat transfer regime in isothermal solid structures is best defined through a filter that is matched to the temperature decay rate.
With the matched filter, the macro-scale fluid temperature varies exponentially in the main flow direction and plays a role similar to that of the bulk temperature in developed duct flow.
The matched filter also ensures that the macro-scale interfacial heat transfer between the flow and the solid structures can be represented by a spatially constant heat transfer coefficient.
In addition, the matched filter defines a spatially constant thermal dispersion coefficient to represent the thermal dispersion source in the macro-scale temperature equation.
Both coefficients are easily governed from the rescaled temperature on a unit cell.

Lastly, it is demonstrated that the appropriate definition of the macro-scale temperature in the periodically developed conjugate heat transfer regime again requires a double volume-average filter.
The double volume-average filter yields a linearly changing macro-scale temperature and a constant macro-scale interfacial heat flux in the periodically developed conjugate heat transfer regime.
Moreover, with the double volume average, the thermal dispersion source, the thermal tortuosity and the interfacial heat transfer coefficient all become spatially constant, so that they are easily governed from the periodic temperature part on a unit cell.