Solution Methodology for Dynamic Simulation

In the simulation, the first step is to develop mathematical modeling. The modeling, based on first principles, is done by applying a standard input/ output approach for time-dependent systems with one or multiple inputs x t) and one or multiple outputs y t). The mathematical descriptions of components and hardware are formulated in the form of ordinary differential equations with the time t as the independent variable. The system description is represented mathematically by a system of coupled, nonlinear, first-order differential (or integral) equations  

Time invariant input-output system for one input x t) and one output y(f) and a first-order differential equation relating output and input function. 

The equations used to model the different elements are collected together to form a set of differential equations 

In the block diagram, each block represents multiple equations. These equations are represented as block methods. These block methods are evaluated (executed) during the execution of a block diagram. The evaluation of these block methods is performed within a simulation loop, where each cycle through the simulation loop represents the evaluation of the block diagram at a given point in time. There are several tools to handle the time-dependent variable equation sets in each block such as fixed step solvers and variable step solvers. 

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. 

---