Computer simulation algebraic properties

Unsteady-state or dynamic simulation accounts for process transients, from an initial state to a final state. Dynamic models for complex chemical processes typically consist of large systems of ordinary differential equations and algebraic equations. Therefore, dynamic process simulation is computationally intensive. Dynamic simulators typically contain three units (i) thermodynamic and physical properties packages, (ii) unit operation models, (hi) numerical solvers. Dynamic simulation is used for batch process design and development, control strategy development, control system check-out, the optimization of plant operations, process reliability/availability/safety studies, process improvement, process start-up and shutdown. There are countless dynamic process simulators available on the market. One of them has the commercial name Hysis [2.3]. 

Throughout this book, we have seen that when more than one species is involved in a process or when energy balances are required, several balance equations must be derived and solved simultaneously. For steady-state systems the equations are algebraic, but when the systems are transient, simultaneous differential equations must be solved. For the simplest systems, analytical solutions may be obtained by hand, but more commonly numerical solutions are required. Software packages that solve general systems of ordinary differential equations— such as Mathematica , Maple , Matlab , TK-Solver , Polymath , and EZ-Solve —are readily obtained for most computers. Other software packages have been designed specifically to simulate transient chemical processes. Some of these dynamic process simulators run in conjunction with the steady-state flowsheet simulators mentioned in Chapter 10 (e.g.. SPEEDUP, which runs with Aspen Plus, and a dynamic component of HYSYS ) and so have access to physical property databases and thermodynamic correlations. 

An algebraic equation relating the fundamental state variables of a fluid P, V and T is known as an equation of state, abbreviated here by EOS. The simplest EOS is the ideal gas law PV=RT. The models based on equations of state are widespread in simulation because allow a comprehensive computation of both thermodynamic properties and phase equilibrium with a minimum of data. EOS models are applied not only to hydrocarbon mixtures, as traditionally, but also to mixtures containing species of the most various chemical structures, including water and polar components, or even to solutions of polymers. The most important equations of state are presented briefly below, but they will be examined in more detail in other sections. 

The singular value decomposition (SVD) method, and the similar principal component analysis method, are powerful computational tools for parametric sensitivity analysis of the collective effects of a group of model parameters on a group of simulated properties. The SVD method is based on an elegant theorem of linear algebra. The theorem states that one can represent an w X n matrix M by a product of three matrices ... 

Despite their highly successful record, MD or MC simulations are still hardly extended to the direct interpretation of complex set-ups, typical of most rheological experiments. In such cases it is preferable to employ mean-field or continuum descriptions, based of the numerical solution of the constitutive equations describing hydrodynamic properties. Such techniques were for instance applied to the prediction of transient director patterns of liquid crystalline nematic samples [11-14]. Hydrodynamic treatments are algebraically complex and computationally intensive, and their implementation is limited mostly to nematic phases. 

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