Equation-based simulation

The sequential modular approach to process simulation solves system equations in blocks corresponding to the unit operations that make up the process. The block diagram for the process looks very much like the traditional process flowchart. Since engineers are accustomed to viewing chemical processes as sequences of unit operations, they lend to feel comfortable with this approach. 

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 following example illustrates the equation-based approach. 

EXAMPLE 10.3-Ii Simulation and Design of a Two-Column Separation Process 

---

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. 

Finally, we should mention that in addition to solving an optimization problem with the aid of a process simulator, you frequently need to find the sensitivity of the variables and functions at the optimal solution to changes in fixed parameters, such as thermodynamic, transport and kinetic coefficients, and changes in variables such as feed rates, and in costs and prices used in the objective function. Fiacco in 1976 showed how to develop the sensitivity relations based on the Kuhn-Tucker conditions (refer to Chapter 8). For optimization using equation-based simulators, the sensitivity coefficients such as (dhi/dxi) and (dxi/dxj) can be obtained directly from the equations in the process model. For optimization based on modular process simulators, refer to Section 15.3. In general, sensitivity analysis relies on linearization of functions, and the sensitivity coefficients may not be valid for large changes in parameters or variables from the optimal solution. 

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. 

Computational fluid dynamics (CFD) is the analysis of systems involving fluid flow, energy transfer, and associated phenomena such as combustion and chemical reactions by means of computer-based simulation. CFD codes numerically solve the mass-continuity equation over a specific domain set by the user. The technique is very powerful and covers a wide range of industrial applications. Examples in the field of chemical engineering are ... 

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 ... 

Since the middle of the 1990s, another computation method, direct simulation Monte Carlo (DSMC), has been employed in analysis of ultra-thin film gas lubrication problems [13-15]. DSMC is a particle-based simulation scheme suitable to treat rarefied gas flow problems. It was introduced by Bird [16] in the 1970s. It has been proven that a DSMC solution is an equivalent solution of the Boltzmann equation, and the method has been effectively used to solve gas flow problems in aerospace engineering. However, a disadvantageous feature of DSMC is heavy time consumption in computing, compared with the approach by solving the slip-flow or F-K models. This limits its application to two- or three-dimensional gas flow problems in microscale. In the... 

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. 

The principal advantage of equation based, dynamic, simulators is their ability to model the unsteady-state conditions that occur at start-up and during fault conditions. Dynamic simulators are being increasingly used for safety studies and in the design of control systems. 

Optimization Using Equation-Based Process Simulators.525... 

Simultaneous modular. The process simulator is composed of modules, but simplified, approximate, or partial representation of the modules enables solution techniques used in equation-based methods to be employed. 

OPTIMIZATION USING EQUATION-BASED PROCESS SIMULATORS... 

In this section we consider general process simulator codes rather than specialized codes that apply only to one plant. To fnesh equation-based process simulators with optimization codes, a number of special features not mentioned in Chapter 8 must be implemented. 

Effective computer codes for the optimization of plants using process simulators require accurate values for first-order partial derivatives. In equation-based codes, getting analytical derivatives is straightforward, but may be complicated and subject to error. Analytic differentiation ameliorates error but yields results that may involve excessive computation time. Finite-difference substitutes for analytical derivatives are simple for the user to implement, but also can involve excessive computation time. 

Simulation Sciences, Inc. Documentation for ROMEO (Rigorous On-line Modeling with Equation-based Optimization. Brea, CA (1999). 

P. J. Mulquiney and P. W. Kuchel, Model of 2,3 bisphosphoglycerate metabolism in the human erythrocyte based on detailed enzyme kinetic equations Computer simulation and metabolic control analysis. Biochem. J. 342 (3), 597 604 (1999). 

Equations (8.1)-(8.13) can be solved to provide transient- or steady-state profiles of O2 and CH4 concentration, reaction rates and surface fluxes for any combination of the controlling variables 9q,0], v,k, a, Vm,Vq and Vr. Where, as is usual, one or more of the controlling variables may be further simplified, approximated or neglected, process-based simulation of CH4 emission becomes possible using a relatively limited set of input data. 

Gratzel et al. demonstrated a wide variety of organophosphorus pesticides, including paraoxon, parathion, and malathion, which possess aromatic side chains and/or sulfur atoms can be totally decomposed into carbon dioxide and the corresponding mineral acids by irradiation in a solar simulator in the presence of anatase TiO2 suspensions [41]. The balanced equation based on carbon dioxide evolution for the mineralization of parathion is represented by Eq. (6) ... 

Figures 23 and 24 show the liquid-phase compositions for, respectively, the reboiler and condenser as functions of time. After column startup, the concentration of methanol decreases continuously whereas the distillate mole fraction of methyl acetate reaches about 90%. A comparison of the rate-based simulation (with the Maxwell-Stefan diffusion equations) and experimental results for the liquid-phase composition at the column top and in the column reboiler demonstrates their satisfactory agreement (Figures 23 and 24). Figure 25 shows the simulation...

---