Heat stream splitting

Thus loops, utility paths, and stream splits offer the degrees of freedom for manipulating the network cost. The problem is one of multivariable nonlinear optimization. The constraints are only those of feasible heat transfer positive temperature difference and nonnegative heat duty for each exchanger. Furthermore, if stream splits exist, then positive bremch flow rates are additional constraints. 

Figure B.l shows a pair of composite curves divided into vertical enthalpy intervals. Also shown in Fig. B.l is a heat exchanger network for one of the enthalpy intervals which will satisfy all the heating and cooling requirements. The network shown in Fig. B.l for the enthalpy interval is in grid diagram form. The network arrangement in Fig. B.l has been placed such that each match experiences the ATlm of the interval. The network also uses the minimum number of matches (S - 1). Such a network can be developed for any interval, providing each match within the interval (1) satisfies completely the enthalpy change of a strearh in the interval and (2) achieves the same ratio of CP values as exists between the composite curves (by stream splitting if necessary).

Once the initial network structure has been defined, then loops, utility paths and stream splits offer the degrees of freedom for manipulating network cost in multivariable continuous optimization. When the design is optimized, any constraint that temperature differences should be larger than A Tmin or that there should not be heat transfer across the pinch no longer applies. The objective is simply to design for minimum total cost. 

Neglecting flow nonuniformities, the contributions of molecular diffusion ana turbulent mixing arising from stream splitting and recombination around the sorbent particles can be considered additive [Langer et ah, Int. J. Heat and Mass Transfer, 21, 751 (1978)] thus, the axial dispersion coefficient DL is given by ... 

Zamora, J. M. and I. E. Grossmann. A Global MINLP Optimization Algorithm for the Synthesis of Heat Exchanger Networks with No Stream Splits. Comput Chem Eng 22 367-384(1998). 

Combinatorial. Combinatorial methods express the synthesis problem as a traditional optimization problem which can only be solved using powerful techniques that have been known for some time. These may use total network cost diiecdy as an objective function but do not exploit the special characteristics of heat-exchange networks in obtaining a solution. Much of the early work in heat-exchange network synthesis was based on exhaustive search or combinatorial development of networks. This work has not proven useful because for only a typical ten-process-stream example problem the alternative sets of feasible matches are cal.55 x 1025 without stream splitting. 

An alternative starting network is one without stream splits. The networks from the TI method maximize energy recovery and may introduce heat-load loops. Stream splits are not made in the initial steps of network invention. The ED method is proposed to be one in which heuristic rules and strategies would be used to improve the networks developed by the TI method. The importance of a thermodynamic base for evolutionary rules is stressed in this proposal, but there is no explicit guidance for the evolutionary process. 

Theorem 1 (Corner Point Theorem). Assume the following (1) constant heat capacities and no phase change, (2) supply and target temperature uncertainties only (no uncertainties in flow rates or heat transfer coefficients), (3) constant stream split fractions (Saboo et al., 1987b), and... 

Heat exchanger network resilience analysis can become nonlinear and nonconvex in the cases of phase change and temperature-dependent heat capacities, varying stream split fractions, or uncertain flow rates or heat transfer coefficients. This section presents resilience tests developed by Saboo et al. (1987a,b) for (1) minimum unit HENs with piecewise constant heat capacities (but no stream splits or flow rate uncertainties), (2) minimum unit HENs with stream splits (but constant heat capacities and no flow rate uncertainties), and (3) minimum unit HENs with flow rate and temperature uncertainties (but constant heat capacities and no stream splits). 

The remaining ATm constraints (r4 - rf > 10, Tj - r3 10) can be disregarded since for the values of the heat capacity flow rates chosen in this example, Arm will always occur on the hot ends of the exchangers for any value of the stream split fractions. 

Floudas and Grossmann (1987b) have shown that for HENs with any number of units, with or without stream splits or bypasses, and with uncertain supply temperatures and flow rates but with constant heat capacities, the active constraint strategy decomposes the resilience test (or flexibility index) problem into NLPs which have a single local optimum. Thus the resilience test (or flexibility index) also has a single local optimum solution. 

Different algorithms are required if the HEN resilience problem is nonlinear. Special algorithms were presented for testing the resilience of minimum unit HENs with piecewise constant heat capacities, stream splits, or simultaneous flow rate and temperature uncertainties. A more general algorithm, the active constraint strategy, was also presented which can test the resilience or calculate the flexibility index of a HEN with minimum or more units, stream splits and/or bypasses, and temperature and/or flow rate uncertainties, but with constant heat capacities. 

Develop techniques to test the resilience of class 2 HENs with stream splits and/or bypasses, temperature and/or flow rate uncertainties, and temperature-dependent heat capacities and phase change. It may be possible to extend the active constraint strategy to class 2 problems. This would allow resilience testing of class 2 problems with stream splits and/or bypasses and temperature and/or flow rate uncertainties. However, the uncertainty range would still have to be divided into pinch regions (as in Saboo, 1984). 

Saboo, A. K., Morari, M., and Colberg, R. D., Resilience analysis of heat exchanger networks—Part II Stream splits and flowrate variations. Comp. Chem. Eng., 11, 457 (1987b). 

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