**1. Introduction**

Metal foam is a class of materials with a porous structure. A typical property of metal foams is high porosity, and therefore a low density. The thermal and mechanical properties of metal foams remain those of their base metals and sufficiently meet the requirements of light weight, low pressure drop, malleability, improved mixing, and heat transfer. Therefore, metal foams have been applied in many industries involved in enhanced fluid mixing and heat transfer. Metal foams are also used as physical support for catalysts or even as catalyst substrate in chemical processes, such as fuel cells and micro-reactors [1,2]. Metal foams have been used to improve the conversion efficiency in micro-reactors because it can enhance the mixing of the liquid flow [3] and emulsify the immiscible two-phase fluid [4]. However, the flow field and droplet behaviors (e.g., breakup and deformation) are still poorly understood in metal foam reactors.

Various experiments [5] and simulations [6–11] have been carried out to understand the effects of porous structure on fluid hydrodynamics in immiscible binary fluids. However, most of them in this topic rely on empirical correlations of experimental measurements and traditional computational fluid dynamics (CFD). The multiphase flows in the CFD approach are simulated by solving the macroscopic Navier–Stokes equations. Among the approaches of tracking interfaces, the front-tracking method, the volume of fluid (VOF) method, and the level set method are widely used [8–13]. Because the interface must be manually ruptured, the front-tracking methods are not suitable for simulating interface breaking and coalescing [14]. The VOF and level set methods can simulate interface breaking and coalescing; however, to determine the interfacial tension, the force and the flux across the interface is required in the VOF. The level set method uses a signed distance function to represent the interface, which requires a re-initialization procedure to keep the distance property when large topological changes occur around the interface. This process can be time-consuming and not always physically consistent [15]. In addition, the VOF and level set methods will suffer from numerical instability at the interface region when the interfacial tension becomes a dominant factor in complex geometries [16]. For example, it is a challenge to apply the VOF or level set methods to simulate capillary displacement in porous media. Microscopically, the phase segregation and the interfacial dynamics between different phases are due to inter-particle forces or inter-actions [17]. Thus, mesoscopic level models are expected to accurately describe the complex dynamic behavior of multiphase flows [18].

With the advance of computational physics and image technology, the simulation of droplet breakup in metal foams has become feasible. For porous reconstructing, there are two ways of representing the pore scale geometry: an idealized structure and micro X-ray computed tomography (CT). The idealized structure can reconstruct simplified pore geometries by taking into account the structural complexity of the medium [19,20]. A main drawback of the idealized structure is that the pore structure has to be mathematically reasonably simplified to fit the model. This simplification can cause a substantial error in describing the real structure. Micro-CT can accurately regenerate a porous structure. Hundreds of images from various angular views are acquired while the porous object rotates. Then the hundreds of images of virtual cross-section slices are synthesized by a computer to regenerate the porous structure. Montminy et al. [21] reconstructed a 3D metal foam using micro-CT. Carvalho et al. [22] followed their approach to investigate the pressure drop of the single-phase fluid flow passing the porous media.

In recent years, the lattice Boltzmann method (LBM) has emerged as an attractive numerical tool for simulating multiphase flow because of its advantage in dealing with complicated geometry and interfacial dynamics [23,24]. Unlike the methods mentioned before, LBM is based on the solution of macroscopic variables such as velocity and density, and built upon microscopic models and mesoscopic kinetic equations. Compared with traditional numerical methods, LBM is more efficient in dealing with complex boundaries [25–27]. LBM has been successfully applied to investigate the multiphase fluid flow in a porous medium.

Several lattice Boltzmann models such as the Shan-Chen (SC) model, the free energy-based model, and the color gradient-based model [28–30] have been proposed for simulating multiphase flow. Among these models, the SC multiphase model is the simplest. In this model, hydrophobic interaction between fluid phases and additional interaction between the fluid and solid surfaces are taken into account [31–33]. The SC model is capable of simulating the complete range of contact angles and the equilibrium distribution of the phase in a porous medium. Because of these advantages, the SC model has been widely used to study the hydrodynamics of single and multiple phase flows where the interaction between the fluid–fluid and fluid–solid are considered. Li et al. [34] applied the SC model to study the deformation and breakup behavior of liquid droplets past a circular cylinder. Park et al. [35] successfully simulated the motion of liquid droplet flow in a porous medium using the SC model and a reconstructed method. However, the works by Li et al. and Park et al. are limited to two-dimensional simulations. Frank et al. [36] simulated the droplet spreading on a porous surface. Jonas et al. [37] simulated an immiscible binary fluid flow in a porous medium. It should be noted that an ideal porous structure which randomly distributes in the matrix was taken for the simulations by Li et al. and Jonas et al. This ideal porous structure leads to an obvious deviation of simulation results from real results.

In spite of great progress in the simulation of multiphase fluid in porous media using LBM, these porous media are either rocks in geological reservoirs or matrixes of soils. Unlike rock or soil, metal materials have their own properties. The porosity of metal foams is usually larger than rock or soil; therefore, metal foams have a stronger circulation capacity and less capillary action. A few 3D simulations on single phase fluids have been carried out in metal foams using X-ray reconstructed 3D porous structures [38]. However, to the best of our knowledge, no works have been reported on the LBM simulation of droplet breakup and deformation in a metal foam against experimental measurements. Regarding the wide application of metal foams in industries, it is necessary to deepen our understanding of the mechanism of droplet behavior in metal foams.

A primary objective of this paper is to simulate the processes of droplet breakup and deformation in a metal foam generated by X-ray CT using the lattice Boltzmann model with an SC model of multiphase. The Green function (*Gw,k*) is obtained by comparing the measured contact angles and the simulation results. The simulation results of the hydrodynamics of droplets passing through metal foam were verified by the results recorded using high-speed video. The effects of several non-dimensional parameters on the hydrodynamics of droplets passing through metal foam are discussed. It should be mentioned that no original point in the numerical method applied, but the hydrodynamics of droplets passing through metal foam revealed by the simulation is crucial to the design of metal foam reactors or mixers.

#### **2. Experimental**

#### *2.1. Experimental Setup*

The experimental setup for measurements of the hydrodynamics of a droplet passing through a metal foam is illustrated in Figure 1. The metal foams employed in this study were supplied by SiPing AKS Metal Material Technology Co., Ltd. (Siping, China); The width, length, pore density, and porosity of the metal foams were 10 mm, 25 mm, 60 pores per inch (PPI), and 95% porosity, respectively. The height ranged from 2 to 4 mm. The metal foam was placed into a Plexiglass tube with a rectangular shape. The Plexiglas tube was filled with silicon oil (Haishi Co., Shanghai, China). One water droplet above the metal foam was formed by injecting distilled water into the silicon oil with a micro-syringe (HAMILTON Co., Bonaduz, Switzerland). The water droplet diameter was controlled by adjusting the injected water volume. When the water droplet held a steady suspension in silicon oil, the valve connected with the outlet of the Plexiglass tube was opened, and the droplet went through the metal foam. The droplet deformation and breakup after the droplet left the metal foam were recorded by a high-speed video (Integrated Device Technology, Longmont, CO, USA). A ruler was attached to the Plexiglass tube for the calculation of the droplet velocity based on the droplet movement recorded by the high-speed video. All the experiments were carried out at room temperature.

**Figure 1.** Setup for measurements on the hydrodynamics of a droplet passing through a metal foam.

#### *2.2. Metal Foam Micro-Tomography*

The morphology of the metallic foam was generated using the Skyscan high-resolution desktop micro-CT system (Micro Photonics Inc., Allentown, PA, USA). The metal foam sample dimensions were 20 × 20 × 2 mm. The metal foam sample was illuminated by a micro-focus X-ray source at 40 kV with a beam current of 250 μA. A planar X-ray detector collected the magnified projection images with a pixel of 36 μm. The 2D cross-section images acquired from various angular views and the morphology of the metal foam were reconstructed. The Matlab Image Processing Toolbox (Matlab R2006a, MathWorks, Natick, MA, USA, 2007) were used to binarized the 2D cross-section images.

A raw section image of the metal foam structure is shown in Figure 2a. A 3D structure of the restructured metal foam sample is illustrated in Figure 2b. Its porosity was computed to be 93.6%, which is very close to that of 95.0% provided by the metal foam supplier.

**Figure 2.** (**a**) Raw section image of the metal foam structure. (**b**) 3D reconstruction of a part of the foam sample of 3.6 × 3.6 × 1.4 mm (mesh number 100 × 100 × 40 of the matrix).
