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Development of Closure for Heat Exchangers Based on Volume Averaging Theory

  • Author(s): Zhou, Feng
  • Advisor(s): Catton, Ivan
  • et al.
Abstract

The design of heat exchangers, which has always been important, has become increasingly critical in today's world and has also become closely coupled with optimization. The commonly used optimization methods, such as design of experiment (DOE), genetic algorithm (GA), etc., require a large population of design points/individuals to be evaluated. To shorten the optimization process, the evaluation of these large number of design points/individuals should be fast, which has precluded the use of time-consuming evaluation methods, such as direct numerical simulations (DNS) due to the significant computational costs of performing the flow and heat transfer in such heterogeneous (and porous), hierarchical devices with conjugate effects included, nor experimental evaluation that requires a large number of different heat exchangers be fabricated which is not only time-consuming but also expensive. This makes population-based optimization quite difficult if combined with system level CFD or experimental evaluation.

To enable these optimization methods, a fast-running solver is necessary to perform an evaluation of the many needed design points/individuals. Breakthroughs in the modeling of transport phenomena in heterogeneous media with Volume Averaging Theory (VAT) have allowed engineers to fully simulate flow and heat transfer in thermal devices in seconds, in comparison with hours or days it takes to do so with system level CFD simulation over one single design of a heat exchanger.

VAT is a hierarchical modeling method that includes mathematical description on two different levels. The lower level is flow and heat transfer at the pore scale for individual subscale elementary volume, which is described by the point-wise Navier-Stokes and thermal energy equations. The upper level is for the whole heat transfer device, described by the VAT-based mass, momentum, and thermal energy transport equations. The two levels are rigorously connected by mathematical scaling procedures, yielding additional integral and differential terms which need to be evaluated.

Proper evaluation of these extra terms is called the closure problem of the VAT based model, which has been the primary measure of advancement and for measuring success in research on transport in porous media. A method to obtain closures of the VAT based governing equations by CFD evaluation is developed. The procedure of the closure evaluation consists of eight main steps: 1) select the representative elementary volume correctly, which is simply the periodic unit cell in the case of periodic media; 2) define a proper characteristic length scale; 3) select a proper numerical method according to the flow condition; 4) discretize the computational domain carefully and conduct validation and verification of the adopted numerical method; 5) determine the number of REVs needed to obtain reasonable fully developed local values; 6) conduct numerical simulation over the selected REVs; 7) extract the macroscopic hydrodynamic and thermal characteristics from the microscopic results by evaluating the closures over the REV; 8) collect the evaluated results for friction factor and heat transfer coefficient and develop the corresponding correlations. Since the closure evaluation is conducted by carrying out direct numerical experiments at the pore scale of the representative elementary volume (REV) which usually has a quite small computational domain, therefore the computation load is much lower than simulation at the device scale. With the closure correlations, the upper-level governing equation set is relatively simple and allows a nonlocal description of transport phenomena in heterogeneous thermal devices, with the morphology directly incorporated into the field equations and conjugate effects fully treated. The simplicity of the VAT based governing equations makes it possible for them to be solved discretely and a rapid evaluation of the design points to be performed.

With a fast-running solver being developed, the population based optimization methods can be exploited to guide the design to its optimal configuration. To demonstrate how the VAT based solver can be combined with commonly used optimization methods, a fin-and-tube heat exchanger is optimized using genetic algorithm and a heat sink with scale-shaped surface roughness is optimized using design of experiment. It should be noted that the development of the optimization algorithms is not the focus of the present work. The algorithms adopted in the present study are quite basic ones, which were borrowed from other researchers.

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