Solution of material balances for flowsheets modelled with elementary modules: The unconstrained case

Abstract

A study is made of the structure and properties of material balance simulation problems, and a technique is developed for their efficient solution. The method requires neither simultaneous solution of all balance equations nor iterative convergence methods. Instead, for each stream mixing point in the flow sheet, a vector balance equation is developed which contains as unknowns only mixer output streams. This unique set of vector equations is sequenced for solution by using precedence ordering and substitution techniques. It is shown that only as many vector equations need to be solved simultaneously as there are streams which would require iteration in the conventional sequential approach. SCOPEThe solution of material balances for large scale process flow sheets without the incorporation of detailed unit operations models is a basic chemical engineering problem of considerable practical importance both in preliminary design and in the study of existing plants. Three computer oriented strategies for solving flow sheet material balances have been reported or are in use at the present time: the completely simultaneous approach, the recycle stream iteration approach, and the split-stream approach of Rosen-Nagiev (Rojen, 1962). All of these approaches employ a representation of flow sheets in terms of a fixed repertoire of standard models which are linear in the stream flows. The simultaneous solution method necessitates assembly and storage of very large equation sets, requires the use of sparse matrix techniques, and does not directly exploit the inherent network structure of flow sheets in which species flows are aggregated into streams. The recycle iteration approach requires the selection of streams to be estimated and iterated upon and necessitates time consuming iterations with uncertain convergence. The Rosen-Nagiev technique uses simultaneous solution of the sets of material balances written around each stream mixing point aggregated by species. In general, in the presence of reactions, the species balances become coupled, and the entire system of sets of equations must be solved in iterative passes.Recognizing that the material balance simulation problem is inherently a highly structured linear problem, the goal of this paper is to analyze the general problem structure and to develop a technique which requires no iterations and a minimum of simultaneous equation solving. The key to the method is the development of a vector balance equation for each stream mixing point in the flow sheet. The equation for each mixing point is constructed by tracing upstream the input streams to that mixer until either another mixer output or a process input stream is reached and aggregating the stream transformation operators of the flow sheet units encountered along the way. These equations can be constructed very efficieutly from the specification of the flow sheet stream connections using symbolic manipulation. Again, employing only symbolic manipulation, the equations can be sequenced for solution using equation orderin...

Tearing and Precedence Ordering in Flowsheeting-an Effective Method

“…They have been picked up from the works of Zhou Li et at. (1988), Gundersen and Hertzberg (1983) and Sood et at. (1979).…”

Reformulation of Data Reconciliation Problem with Unknown-but-Bounded Errors

“…5 5.5] x ˆ3 ∈ [7. 5 In Figure 1b, this set of solutions is thus defined by a box (in gray color) in the plane (x 2 , x 3 ). For each solution inside this box, x ˆ1 can be easily defined by direct substitution x ˆ1 ) x ˆ2 + x ˆ3.…”

“…The first studies of Ripps (1962), Vaclavek (1969), and Smith and Ichiyen (1973) were concerned with data reconciliation using the now classical technique of equilibration of production balances. In subsequent stages, this data reconciliation principle was generalized to processes that are described by algebraic equations that are either linear in the case of total flow rates or nonlinear in the case of chemical concentrations. , …”

“…In subsequent stages, this data reconciliation principle was generalized to processes that are described by algebraic equations that are either linear in the case of total flow rates 4 or nonlinear in the case of chemical concentrations. 5,6 At the same time, data reconciliation was employed for more general applications than establishing statistically coherent balances. It was then applied to more fundamental problems such as detection, localization, and estimation of gross errors; 7,8 diagnosis and observability of systems; 9,10 optimization of sensor locations; 11,12 and the study of the reliability of a measurement system.…”

A simultaneous‐modular approach to process flowsheeting and optimization. Part I: Theory and implementation

“…Various techniques employing an approximate Jacobian involving only tear stream variables have also been suggested (Mahalec et al, 1979;Sood et al, 1979a;McLane et al, 1979;Perkins, 1979;Metcalfe and Perkins, 1978). Sood et al (1979b) and Sood and Reklaitis (1979) have shown this sort of approach to be very effective in solving linear flowsheeting problems. One may also put in this category the methods sometimes referred to as sequential-modular with a "different" convergence block, different in the sense that something other than the usual direct or accelerated substitution is used, typically a Newton or quasi-Newton approach.…”