Multi-scale analytical methods for complex flows in process engineering: Retrospect and prospect

The difficulties of scaling in process engineering lie in the multi-scaling of dynamic structure, the behavior of which becomes progressively involved with increasing scales. Multi-scale methodology looms naturally in practice that reveals the complexity of nature. Traditional hierarchical coarse-graining in continuum approaches has faced fundamental closure problems for the highly non-equilibrium multiphase flows in process engineering. An alternative approach called analytical (variational) multi-scale methodology has been proposed in [1]. It was based on the pio- neering work on gas-solid multiphase flow [2, 3]. The later development can be found in [4, 5]. The inter-scale correlation was considered as a result of the compromise among the dominant mechanisms underlying the complex flow behavior. Mathematically, the dominant mechanisms can be expressed as extremum tendencies. And the compromise can be expressed as a mutually constrained extremum that constitutes the stability condition. This leads to a closed multi-scale model with dynamic equations. This methodology is presented in this work. The coverage is by no means complete, but it aims to be informative to those with different backgrounds. Implications of the approach to the general area of complex systems and nonlinear science are emphasized and presented to encourage the discovery of new possibilities that otherwise may have been hidden and escape the attention of other researchers. The review in [6] elucidates the methodology and its applications in more details.