Problems and Application Limits of Numerical Simulations

1 Problems of Existing Impedance Formulas Used in Circuit Theory-Based Approaches 

---

Equation 1.3 represents a system of usually several thousand coupled differential equations of second order. It can be solved only numerically in small time steps At via finite-difference methods [16]. There always the situation at t + At is calculated from the situation at t. Considering the very fast oscillations of covalent bonds, At must not be longer than about 1 fs to avoid numerical breakdown connected with problems with energy conservation. This condition imposes a limit of the typical maximum simulation time that for the above-mentioned system sizes is of the order of several ns. The limited possible size of atomistic polymer packing models (cf. above) together with this simulation time limitation also set certain limits for the structures and processes that can be reasonably simulated. Furthermore, the limited model size demands the application of periodic boundary conditions to avoid extreme surface effects. 

Typical applications of the boundary element method in the context of adhesion technology are commonly found for the modeling of cracks (fi-acture mechanics) and other types of stress singularities, cf the bibliography of (Mackerle 1995a). The article by (Vable and Maddi 2010) addresses the specific problems (i.e., numerical modeling considerations which limited the application of BEM in the past) related to bonded joints and boundary element simulation. In addition, numerical results of lap joints, cf. O Fig. 26.18, with several spew angles were presented which demonstrate the potential of the boundary element method in analysis of bonded joints. 

Despite highly developed computer technologies and numerical methods, the application of new-generation rate-based models requires a high computational effort, which is often related to numerical difficulties. This is a reason for the relatively limited application of modeling methods described above to industrial problems. Therefore, a further study in this field - as well as in the area of model parameter estimation - is required in order to bridge a gap and to provide process engineers with reliable, consistent, robust and user-friendly simulation tools for reactive absorption operations. 

Computational Fluid Dynamics (CFD) and Process Simulation are important tools for the design and optimization of chemical processes (Bezzo et al., 2000). CFD is a particularly powerful tool for the study of fluid dynamics and heat transfer with detailed account of complex equipment geometry. Nonetheless, despite many recent improvements, CFD s ability to describe the physics or solve the underlying numerical problems in several application areas is still limited. Adsorption technology is an application area where the numerical methods employed in most CFD packages are inadequate to solve the strongly coupled nonlinearities introduced by the presence of the adsorbed phase. 

---