Equation-based flowsheeting

Presently, several equation-based flowsheeting systems exist perhaps the best known of these is SPEED-UP, which continues to be developed at Imperial College by Sargent and Perkins and their students. Now commercially available, its strongest attraction in industry seems to be as a dynamic simulator. TISFLO at Dutch State Mines is also often mentioned in the literature. As a complete flowsheeting system, no equation-based approach is yet very popular. One might wonder why. 

Kuru, S., Dynamic Simulation with an Equation Based Flowsheeting System. Ph.D. thesis, Carnegie Mellon University, Pittsburgh, 1981. 

Clearly define, in your own words, the terms design variables and state variables, sequential modular flowsheet simulation, equation-based flowsheet simulation, tear stream, convergence block, and design specification. 

Equation-based flowsheet simulators have not reached the level of commercialization of their sequential modular counterparts as of the time of this writing, and the development of efficient equation-solving algorithms is the subject of much continuing research. 

Sets of linear and/or nonlinear equations can be solved simultaneously using an appropriate computer code (see Table L.l) by one of the methods described in Appendix L. Equation-based flowsheeting codes pertaining to chemical engineering can be used for the same purpose. The latter have some advantages in that the physical property data needed for the coefficients in the equations are transparently transmitted from a data base at the proper time in the sequence of calculations. 

Four codes currently under development for equation-based flowsheeting are ASCEND, Quasilin, Speedup, and TISFLO-II. An equation-based code should include the following characteristics ... 

When the modules are connected to one another as represented in a flowsheet, a long train of units may become coupled together for calculations. Thus, a set of modules may require a fixed precedence order of solution so that convergence of the calculations may be slower than in equation-based codes. 

The application of simultaneous optimization to reactor-based flowsheets leads us to consider the more general problem of differentiable/algebraic optimization problems. Again, the optimization problem needs to be reconsidered and reformulated to allow the application of efficient nonlinear programming algorithms. As with flowsheet optimization, older conventional approaches require the repeated execution of the differential/algebraic equation (DAE) model. Instead, we briefly describe these conventional methods and then consider the application and advantages of a simultaneous approach. Here, similar benefits are realized with these problems as with flowsheet optimization. 

An alternative to the sequential modular approach is to solve the equations modeling all of the units in a process flowsheet simultaneously this is known as the equation-based approach. Advantages to the sequential modular approach include (1) specialized numerical techniques tailored to each unit operation can be used, and (2) the numerical failure of one unit operation may still yield usable flowsheet information. Advantages to the equation-based... 

Since pure equation-based models involving as many as a few hundred thousand equations arise and require solution, ways are needed to ensure that this activity can occur with some degree of success. Perhaps the oldest of such approaches comes with flowsheeting programs. 

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.)... 

Stadtherr, M. a., H. S. Chen, and C. M. Hilton, Development of a New Equation-Based Process Flowsheeting System Numerical Studies, Selected Topics on Computer-Aided Process Design and Analysis, R. S. H. Mah and G. V. Reklaitis, eds., AIChE Symposium Series, v. 78, no. 214, p. 12 (1982). 

The two basic flowsheet software architectures are sequential modular and equation-based. In sequential modular, we write each unit model so that it calculates output(s), given feed(s), and unit parameters. This is the most commonly used flowsheeting architecture at present, and examples include Aspen+ plus Hysys (AspenTech), ChemCAD, and PROll (SimSci). In equation-based (or open-system) architectures, all equations are written describing material and energy balances as algebraic equations in the form/(x) = 0. This is the preferred architecture for new simulators and optimization, and examples include Speedup (AspenTech) and gPROMS (PSE pic). Each is discussed in turn. 

The most important part of process optimization is linking the process flowsheeting tool to the optimization algorithm. With an equation-based architecture, the unit equations (material and energy balances, operating constraints, and specifications) are constraints in a general nonlinear programming formulation. The main problems are... 

It is interesting to compare the performance of interval methods on modular and equation based formulations of flowsheeting problems. Efficiency and convenience both need to be analysed when making comparisons. The equation based formulation will be more computationally efficient so it is important to know what the price to pay is for having the convenience of a modular formulation. The following sections will demonstrate modular and equation based optimisation for three case studies. 

This flowsheet contains the same units as case study I plus a reactor. The reactor model has a single input stream, F2,i and three parameters, A, Bj, and the extent of reaction, v. The equation based formulation is given in Floudas (1995). 

Basically, DESIGNER can use different physical property packages that are easy to interchange with commercial flowsheet simulators. For the case considered, the vapor-liquid equilibrium description is based on the UNIQUAC model. The liquid-phase binary diffusivities are determined using the method of Tyn and Calus (see Ref. 72) for the diluted mixtures, corrected by the Vignes equation (57), to account for finite concentrations. The vapor-phase diffusion coefficients are assumed constant. The reaction kinetics parameters taken from Ref. 202 are implemented directly in the DESIGNER code. 

Process-scale models represent the behavior of reaction, separation and mass, heat, and momentum transfer at the process flowsheet level, or for a network of process flowsheets. Whether based on first-principles or empirical relations, the model equations for these systems typically consist of conservation laws (based on mass, heat, and momentum), physical and chemical equilibrium among species and phases, and additional constitutive equations that describe the rates of chemical transformation or transport of mass and energy. These process models are often represented by a collection of individual unit models (the so-called unit operations) that usually correspond to major pieces of process equipment, which, in turn, are captured by device-level models. These unit models are assembled within a process flowsheet that describes the interaction of equipment either for steady state or dynamic behavior. As a result, models can be described by algebraic or differential equations. As illustrated in Figure 3 for a PEFC-base power plant, steady-state process flowsheets are usually described by lumped parameter models described by algebraic equations. Similarly, dynamic process flowsheets are described by lumped parameter models comprising differential-algebraic equations. Models that deal with spatially distributed models are frequently considered at the device... 

Linear. Since mass and energy are linearly related between modules, purely linear flowsheet calculations can be formulated as a solution to a set of linear equations once linear models for the modules can be constructed. Linear systems, especially for material balance calculations can be very useful (16). Two general systems, based on linear models, SYMBOL (77) and MPB II (7 ) are indicated in Table 1. MPB II is based on a thesis by Kniele (79). If Y is the vector of stream outputs and the module stream inputs are X, then as discussed by Mahalec, Kluzik and Evans (80)... 

Equation (8) is of the form of the Newton-Raphson method. The A(X) matrix, however, is not necessarily the Jacobian, J(X). Just how the A(X) is set up depends on the application. Bending and Hutchison (88) developed the method for pipe flow networks. Hutchison and Shewchuk (89) applied the method to multiple distillation towers. Gorczynski and Hutchison (90) detail the method for flowsheeting systems. Quasilin (91) is a flowsheeting system based on this approach. MULTICOL (92) appears to solve interconnected columns by means of this approach as well. 

Equation-oriented approaches are based on sets of equations that are written for the units in a particalar flowsheet.36 Unlike the sequential-modular systems, which often contain the necemery information for a veriety of process units, equaiion-orianted synthesizers require the pinctirioner to develop the mode] eqon-lions. These are solved through iterative tecbaiques with staadatd numerical methods. 

In the period 1970-80 s the major engineering bureaus, as well as some large manufacturing companies in refining and petrochemical industries developed in-house flowsheeting programs. Mostly adopted the modular sequential architecture. However, some were based on the equations oriented approach, as SPEEDUP at Imperial College in London (UK) and TISFLO at DSM in The Netherlands. 

In Simultaneous-Modular approach the solution strategy is a combination of Sequential-Modular and Equation-Oriented approaches. Rigorous models are used at units level, which are solved sequentially, while linear models are used at flowsheet level, solved globally. The linear models are updated based on results obtained with rigorous models. This architecture has been experimented in some academic products. 

Dynamic flowsheeting is based on the unsteady state balance of mass and energy, and may be formulated in general terms by the following equation ... 

With the feasible path approach the optimization algorithm automatically performs case studies by variing input data. There are several drawbacks the process equations (32c) have to be solved every time the performance index is evaluated. Efficient gradient-based optimization techniques can only be used with great difficulties because derivatives can only be evaluated by perturbing the entire flowsheet with respect to the decision variables. This is very time consuming. Second, process units are often described by discrete and discontinuous relations or by functions that may be nondifferentiable at certain points. To overcome these problems quadratic module models can be... 

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