Equation-based simulation programs

In an equation based simulators the executive program sets up the flow-sheet and the set of equations that describe the unit operations, and then solves the equations taking data from the unit operations library and physical property data bank and the file of thermodynamic sub-routines. 

In equation-based simulators, the mathematical equations that describe the physical process are entered into an equation solver that then uses appropriate techniques to solve them. In modular-based process simulators, the mathematical equations that describe the physical process are coded into modules that the user "flow sheets" together. Modular-based process simulators are preferred over equation-based simulators because it is easier for the user to "map" the real world into the virtual one, and programming and debugging of the modules are easier than analyzing sets of equations (Popovic and Bhatkar, 1997). However, equation-based simulators have proved highly successful in the field of optimum process control. Equation-based models handle instrument error and incorrect or errors from modeling simplifications better than modular-based simulators. Modular-based simulators invariably have a data reconciliation step, where the model is nm against values obtained from the instrumentation system and then a least-squares fit is performed to fit the model to the process. 

A simulation was performed for an experiment with [O3] = 3.16 X 10-5 M and [OH-] = 7.17 X 10-3 M. The kinetic curve calculated by the kinsim program form this model is depicted in Fig. 5-4. Also shown is an experimental curve calculated from an empirical equation, based on the equation that applies at this OH- concentration ... 

Equation based programs in which the entire process is described by a set of differential equations, and the equations solved simultaneously not stepwise, as in the sequential approach. Equation based programs can simulate the unsteady-state operation of processes and equipment. 

In the past, most simulation programs available to designers were of the sequential-modular type. They were simpler to develop than the equation based programs, and required only moderate computing power. The modules are processed sequentially, so essentially only the equations for a particular unit are in the computer memory at one time. Also, the process conditions, temperature, pressure, flow-rate, are fixed in time. 

Design of extraction processes and equipment is based on mass transfer and thermodynamic data. Among such thermodynamic data, phase equilibrium data for mixtures, that is, the distribution of components between different phases, are among the most important. Equations for the calculations of phase equilibria can be used in process simulation programs like PROCESS and ASPEN. 

Commonly used simulation programs are based on a numerical solution of the onedimensional classical fluid mechanics equation by the method of characteristics [1-2]. For numerical simulation the piping system is devided in sections with constant cross-sectional areas which are connected by knot-elements. 

In the equation-based approach, the equations for all units are collected and solved simultaneously. The natural decomposition of the system into its constituent unit operations is therefore lost. Moreover, the simultaneous solution of large numbers of equations, some of which may be nonlinear, can be a cumbersome and time-consuming problem, even for a powerful computer. For all these reasons, most commercial simulation programs were still based on the sequential modular approach when this text was written. 

Ihese difficulties vanish if the system equations are simply collected and solved for all unknown variables. Several powerful equation-solving algorithms are available in commercial programs like Maple , Mathematica , Matlab , Mathcad , and E-Z Solve that make the equation-based approach competitive with the sequential modular approach. Many researchers in the field believe that as this trend continues, the former approach will replace the latter one as the standard method for flowsheet simulation. (Engineers are also working on simultaneous modular methods, which combine features of both sequential modular and equation-based approaches. We will not deal with these refinements here, however.)... 

The concentrations of mercury in the landfill gas were estimated with the aid of a simulation program EandGEM that was developed by the office of Research and Development United States Environmental Protection Agency (US EPA). Land GEM is based on a fust-order decomposition rate equation for quantifying emissions from the decomposition of landfilled waste in municipal solid waste (MSW) landfills. The software provides a relatively simple approach to estimating landfill gas emissions. Model defaults are based on empirical data from US landfills. Field test data can also be used in place of model defaults when available. This software was also useful to forecast the future levels of mercury generation from LEG. 

Hughes and Hart (23) have developed an analog simulation program, BASE, which was used for the prediction of a TG curve. The calculation involved the plotting of the TG curve from the equation. 

The short-distance, perturbative part Dq(x, fip) models the evolution of a quark produced off-shell at the scale /x/7 via gluon emissions to a quark on its mass shell. This is what is usually implemented in the parton shower algorithms of the Monte Carlo simulation programs. A parton shower develops through successive splitting until the perturbative approach becomes unreliable ( Aqcd)- The parton shower represents an approximative perturbative treatment of QCD dynamics based on the DGLAP evolution equations. It improves the fixed order pQCD calculation by taking into account soft and collinear enhanced terms to all orders. 

Mathematical models are particularly useful because of the large body of mathematical and computational theory that exists for the study and solution of equations. Based on this theory, a wide range of techniques has been developed. In recent years, computer programs have been written that implement virtually all of these techniques. Computer software packages are now widely available for both simulation and computational assistance in the analysis and design of control systems (Chapter 5). 

---