Flow networks

The time constants associated with the settling of flow in a network are often very short compared with the other time constants of the plant being modelled, and in such cases it is justifiable to assume that the flows adjust instantaneously to the slower changes in the conditions (pressure, specific volume) occurring at the boundaries of the network. Further discussion of and justification for the assumption of instantaneous adjustment is given in Section 18.10 of this chapter. 

The assumption that the flows in the network are in a continuously evolving steady state brings the benefit that the model is rendered much less stiff thereby. Furthermore, the resulting simultaneous equations may be solved explicity in simple flow networks. In more complex networks, however, the resulting set of nonlinear, simultaneous equations requires a more sophisticated approach in order to bring about the desired savings in model execution time. This chapter considers both 

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The ventilation model is a simple flow network with one zone and the different openings modeled as airflow links from the hall to outside Fig. 11.52). For the flow through the roof hood, two additional nodes were considered between the different cross-sections through which the air flows (Fig, 11.53). 

S.5.2 Transient two-phase-flow pressure drop. Calculation of transient behavior in a complex flow network containing a compressible fluid in two-phase states was... 

A. Description and Characterization of Flow Networks 1. Graphs and Digraphs... 

It should be clear from Eq. (7) that since mass conservation applies to the flow network as a whole, there can be no net inflow or outflow of material. This requirement is obviously met for an isolated network. However, most applications of practical interest will include external inputs and outputs,... 

Every link in the product flow network can have a validity interval. Validity intervals may be automatically extended to add missing validity intervals that are necessary to cover the whole planning horizon. 

Tokeshi M, Minagawa T, Uchiyama K, Hibara A, Sato K, Hisamoto H, Ki-tamori T (2002) Continuous-flow chemical processing on a microchip by combining microunit operations and a multiphase flow network. Anal Chem 74 1565-1571... 

A similar approach was undertaken by Mah et al. (1976) in their attempt to organize the analysis of process data and to systematize the estimation and measurement correction problem. In this work, a simple graph-theoretic procedure for single component flow networks was developed. They then extended their treatment to multicomponent flow networks (Kretsovalis and Mah, 1987), and to generalized process networks, including bilinear energy balances and chemical reactions (Kretsovalis and Mah, 1988a,b). 

Badal, M.Y., Wong, M., Chiem, N., Salimi-Moosavi, H., Harrison, D.J., Developing a routine coating method for multichannel flow networks on a chip using pyrolyzed poly(dimethylsiloxane). Micro Total Analysis Systems, Proceedings 5th pZAS Symposium, Monterey, CA, Oct. 21-25, 2001, 535-536. 

Ulanowicz RE University of Maryland, Chesapeake Biological Laboratory. 1987. NET-WRK4 a package of computer algorithms to analyse ecological flow networks [computer program], Solomons (MD) http //www.cbl.cees.edu/ ulan/ntwk/network.html (accessed December 28, 2007). 

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. 

The search for an optimal design may be considerably challenging, since one may contemplate a very large number of geometric possibilities, and boundary and initial conditions in a fluid flow network. In practice, one may examine a number of alternative configurations, optimize their performance, and compare the optimized alternatives. Finally, one should select a suitable configuration with the least irreversibility or cost. 

Systemization. The Distribution Manager must relate the creative, kaleidoscopic activities of chemicals physical distribution with information concepts, systems, and hardware to achieve appropriate control. The Distribution Manager who masters this very complex simultaneous equation will achieve for his company a continuous flow network with the end costs of the delivered material progressively improving. The most complex and challenging system which the Distribution Manager will need to develop will be himself and his people organization. 

Consider the system of Figure 2.3, which represents a flow network with six nodes. Liquid flows from an upstream accumulator, at pressure pt, to three downstream accumulators, at pressures pj, p and p. The flow passes through a pipeline network with line conductances Cn, C23, C24, C45 and C46. Let us assume that the network forms part of a larger model. 

Figure 2.3 Schematic of liquid flow network, discharging to three accumulators.

It is a characteristic of pumped liquid systems and of the steam or gas flow networks with turbines and compressors that no explicit solution is generally available. It is clear from the form of equations (2.91) to (2.92) that no explicit solution can be expected even for... 

A good subroutine for solving nonlinear simultaneous equations may be provided within the overall simulation package, or it may be necessary for the modeller to introduce such a routine himself. Commercial software is available if not already provided within the simulation package. Further detail on iterative methods for solving implicit equations is given in Chapter 18, Section 18.5, which includes a discussion on how to speed up convergence in flow networks. 

As an example, taken the flow network of Figure 2.3, described by equations (2.91) and (2.92). Differentiating the two equations with respect to time gives ... 

Figure 17.9 represents a complex flow network, which may be solved iteratively at each timestep by guessing and then successively refining the estimates of the pressures and temperatures at the compressor nodes, po.i. Tqj, f = 1,2.N +. The correct val-... 

It will always be worth attempting to reduce the complexity of any extensive flow network by applying the rules for parallel and series networks as outlined in the previous two sections. However, there are many plant arrangements where this approach can only proceed... 

An example of such an irreducible set is given in the flow network of Figure 18.3, assumed for simplicity to carry liquid at a constant temperature. This network can hardly be described as complicated, but it cannot be reduced further using either parallel or serial transformations, and we shall need to solve an implicit equation in order to find the intermediate pressure, P2, and the flows, Wc, Wjj and W24, as will now be demonstrated. 

Strategy for solving flow networks using iterative methods... 

Invariably the flow network will form part of a larger model, which will define the boundary pressures. These will remain fixed during the process of iteration. The basic strategy is to use initial guesses of the pressures at the intermediate nodes and, in some cases, of the nodal temperatures or specific volumes, in order to calculate the flows and then to see if they balance. The guesses will then be refined successively using an algorithm such as Newton-Raphson until the flows balance to a predetermined tolerance. 

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