Stream split constraints

These stream split constraints replace the two ATm constraints used in their derivation. 

In general, assuming that any exchanger may be connected to either a cold stream split or a hot stream split (but not to both a cold stream split and a hot stream split), the stream split constraints have the form... 

For the feasibility test, stream split constraints (Bl) are linear In particular, each g,y is constant since the feasibility test is for specified supply temperatures. Also note that the energy balance and energy recovery constraints are not included in this feasibility test they are used to determine constants aijh cijh and vk, in the stream split, load, and A7 constraints. 

Example 9. Resilience of the minimum-unit stream-splitting HEN shown in Fig. 12 is to be tested in the uncertainty range 415 K < rf 515 K. The stream split constraints were derived in Example 8 from the ATm constraints on the hot ends of the exchangers. 

The resilience test incorrectly identifies the network as not being resilient. The values of vk max and gy max for this uncertainty range are listed in Table VI. Since all the vk max are nonpositive, the load constraints are satisfied throughout the uncertainty range. To test the stream split constraints, LP (29) is applied ... 

When these expressions for r4, Ts, and T6 are substituted into the A7 m, load, and stream split constraints, the following NLP can be written for feasibility measure i/ ... 

Note that in Table VIII, u3i violates the stream split constraints (0 s u31 < 1) at some of the solutions of NLP (34 ) for example, m3, < 0 at the solution of NLP (3418). This means that potential set of active constraints MA(18) = (/5, /7) is not active at the solution of the resilience test problem, since u31 violates the nonnegativity constraint [which was not included in NLP (3418)]. However, u31 does satisfy the constraint m3i = 1 (f7 0), since this constraint was included in NLP (3418). 

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. 

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. 

The nonlinear resilience tests developed by Saboo et al. (1987a,b) are each for a rather specific case. A more general resilience analysis technique based on the active constraint strategy of Grossmann and Floudas (1985,1987) is also presented. The active constraint strategy can be used to test the resilience of a HEN with minimum or more units, with or without stream splits or bypasses, and with temperature and/or flow rate uncertainties (Floudas and Grossmann, 1987b). 

Example 8 (from Saboo et al., 1987b). Resilience of the HEN in Fig. 12 is tested in Example 9. In this example, the constraints on the stream split fractions are formulated. 

A necessary condition for the HEN to be resilient is that v max be nonpositive for every k (Saboo et al., 1987b). This condition is necessary since constraint functions v are linear in Tf and independent of stream split fractions u. If any of the vk max is positive, then one of the ATm or load constraints in problem (28) is violated (at the critical corner point for v max) and no choice of stream split fractions u will make the network feasible (at that critical corner point). If all the v max are nonpositive, then all of the A Tm and load constraints in problem (28) are satisfied at every corner point, and thus (by linearity in Ts) throughout the entire uncertainty range. 

The minimum unit HEN with stream splits is resilient if < 0. This test is sufficient, but not necessary, for HEN resilience. It is necessary only if the same critical corner point maximizes all of the and vk constraint functions simultaneously. 

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. 

Example 11. The active constraint strategy is to be used to test the resilience of the same stream splitting HEN as in Example 9 (Fig. 12) in the uncertainty range 0 = TII415 < Tf < 515 K. ... 

Note that the energy recovery constraint reduces to 0 = 0 since there is no heater in this network. The energy balances for both exchangers and the stream split can be solved simultaneously for intermediate stream tempera-... 

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). 

Knowledge of the critical uncertainty point and constraint allows the designer to strategically place exchangers and stream splits where they will most improve resilience. This feature tends to minimize the number of synthesis-analysis iterations required to achieve a resilient HEN. 

Finally, an interesting feature of the MINLP model is that it is possible to add constraints to avoid generating structures with stream splits. This is accomplished simply by requiring that not more than one match be selected for every stream at each stage that is. 

A five-component stream splits into a distillate product with composition and a bottoms product with composition A,. Which of the following sets of specifications uniquely define the separation, without taking into account vapor-liquid equilibrium or energy balance constraints ... 

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