^{1}, Yohan Davit

^{1,2}, Michel Quintard

^{1,2}

^{1}Université de Toulouse ; INPT, UPS ; IMFT (Institut de Mécanique des Fluides de Toulouse) ; Allée Camille Soula, F-31400 Toulouse, France

^{1,2}CNRS; IMFT ; F-31400 Toulouse, France

**Abstract**

*CO*

_{2}absorption. The packings often consist of corrugated plates that significantly increase the exchange surface between gas and liquid phases in the column. The columns are generally operated in a counter-current flow mode: a thin liquid film is driven by gravity and is sheared by the upward gas flow. The structure of the packings may be seen as a porous medium with a large porosity and multi-scale features, with a locally repeating elementary pattern. A pore-scale, associated to the elementary pattern, and a macro-scale assimilated to the packing scale, define the two-scale description. An upscaling approach was used previously to develop a macro-scale law for the gas-phase flow at relatively large Reynolds numbers [2]. At this scale the flow is governed by averaged equations, and information concerning the dynamic of the phase as well as its interactions with the solid structure are embedded in effective parameters. In this model, it was assumed, as a first approximation, that the liquid film is sufficiently thin so that its impact on the flow and the average gas pressure drop can be neglected.

_{eff}. This approach is valid in the limit where the fluctuations induced by the rough wall on the flow are confined into the boundary layer of the gas phase and vanish sufficiently far from the wall. The wall-law is used on Γ

_{eff}and consists in a combination of the normal derivatives of the velocity field at the interface. At first order, it is a Navier slip condition. The coefficients present in the wall law are computed by the resolution of closure problems over an elementary pattern of the rough wall, which is refered as Ω

_{1i}in Fig. 1.

*Re*<1, the macro-scale equation reduces to the classic Darcy's law, where the intrinsic permeability captures the rough geometry of the medium via the concept of effective condition. The method is first applied to a channel flow for which we compare the pressure drop computed from a direct numerical simulation of the wavy wall with the case of effective conditions prescribed at smooth walls. The method is also applied to the elementary pattern of structured packings, where the wavy pattern on the walls represents the wavy liquid-gas interface. This model allows us to describe how the roughness impacts the apparent permeability at the packing-scale via the concept of effective boundary condition.

*Θ*

^{∗}. Following the idea suggested by Mahr et al. [3] and extended by Soulaine et al. [5], the liquid film is separated into two distinct phases in order to capture the radial spreading of the liquid phase into the column. The two resulting films are not and exchange mass at the contact points between the corrugated sheets. This yields a two equations system for the liquid phases, with multiphase permeability tensors [5]. The resulting macro-scale system for the liquid-gas system is then a three equations model that involves effective tensors, relating the superficial velocities to the pressure gradients.

- We have proposed a method to evaluate the potential pressure drop in distillation process induced by the surface waves at the liquid-gas interface.
- We have developed a three-scale analysis of the problem allowing us to compute numerically the effective parameters of the model. This is essentially done using the concept of effective surface that greatly reduces the mesh element number.

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[2] . On the use of a Darcy-Forchheimer like model for a macro-scale description of turbulence in porous media and its application to structured packings. *International Journal of Heat and Mass Transfer *, 74(0):88 - 100, 2014. URL http://www.sciencedirect.com/science/article/pii/S0017931014001975.

[3] . Two-phase flow in structured packings: Modeling and calculation on a macroscopic scale. *AIChE Journal*, 54(3):614-626, 2008. URL http://dx.doi.org/10.1002/aic.11400.

[4] . Primary instabilities of liquid film flow sheared by turbulent gas stream. *International Journal of Multiphase Flow *, 35(7):617 - 627, 2009. URL http://www.sciencedirect.com/science/article/pii/S0301932209000573.

[5] . Gas-liquid flow modeling in columns equipped with structured packing. *AIChE Journal*, 60:3665-3674, 2014. URL http://dx.doi.org/10.1002/aic.14550.