Solid-fluid equilibrium computer simulations

II. COMPUTER SIMULATION METHODS FOR SOLID-FLUID EQUILIBRIUM... 

Figure 4.8 Phase diagram for a pure substance composed of hard spheres. The fluid-phase Z was computed from the Carnahan-Starling equation (4.5.4) the solid-phase Z was taken from the computer simulation data of Alder et al. [14]. The broken horizontal line at Zt = 6.124 connects fluid (T = 0.494) and solid (t = 0.545) phases that can coexist in equilibrium, as computed by Hoover and Ree [12].

The numerical solution method for the above fluid-solid coupling model is an iterative computation process. To reduce the computational complexity, the solid deformation and fluid flow are regarded as two coupled equation systems, solved by FEM. The equilibrium in solid matrix is solved using Eq.(6) with an added coupling item apS j and the pore pressure is treated as an equivalent initial stress term. The flow equation (5) is solved with an added term of volume strain, reflecting the effect of solid deformation on fluid flow. It can be treated as a source or converge. In each iterative loop, the solid matrix deformation is solved firstly. The stress and strain results are then taken as inputs for the flow calculation with modified hydraulic parameters. After flow model is solved, the pore pressure values are transferred into solid matrix deformation model and begins next iterative loop. In this way, the flow and deformation of oil reservoir can be simulated. 

With the development of improved numerical methods for solution of differential equations and faster computers it has recently become possible to extend the numerical simulation to more complex systems involving more than one adsorbable species. Such a solution for two adsorbable species in an inert carrier was presented by Harwell et al. The mathematical model, which is based on the assumptions of plug flow, constant fluid velocity, a linear solid film rate expression, and Langmuir equilibrium is identical with the model of Cooney (Table 9.6) except that the mass transfer rate and fluid phase mass balance equations are written for both adsorbable components, and the multicomponent extension of the Langmuir equation is used to represent the equilibrium. The solution was obtained by the method of orthogonal collocation. 

Initially, the void space between the particles is completely filled with liquid ( = 0 for all voxels). Evaporation from the liquid-gas interface and liquid relaxation into capillary equilibrium are then computed in an alternating sequence. For this simulation we assume a scale separation in time, i.e., that the evaporation occurs on a much slower time scale than the liquid motion. We resolve only the evaporation time scale, which yields a quasi-static approach in each evaporation step, liquid is removed according to the local evaporation rates computed from the solution of the vapor diffusion problem in the gas phase. Then the liquid is relaxed to the capillary equilibrium by volume-preserving mean curvature flow. This quasi-static approach is in contrast to a fully dynamic simulation (via computational fluid dynamics), but may come with considerably lower computational cost. Evaporation is modeled by vapor diffusion in the gas phase, with a no-flux condition at solid-gas interfaces and equilibrium vapor pressure imposed on liquid-gas interfaces (for more details, see [15]). The equilibrium liquid disttibution... 

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