Resilience tests

The most straightforward way to measure the effect of low temperatures on recovery is by means of a compression set or tension set test. Tests in compression are favoured and a method has been standardised internationally. The procedure is essentially the same as set measurements at normal or elevated temperatures and has been discussed in Chapter 10, Section 3.1. As the recovery of the rubber becomes more sluggish with reduction of temperature the dynamic loss tangent becomes larger and the resilience lower (see Chapter 9), and these parameters are sensitive measures of the effects of low temperatures. Procedures have not been standardized, but rebound resilience tests are inherently simple and quite commonly carried out as a function of temperature. It is found that resilience becomes a minimum when the rubber is in its most leathery state and rises again as the rubber becomes hard and brittle. 

The solution 8C of max-min problem (8) defines a critical point for feasible operation—it is the point where uncertainty range 0 is closest to feasible region R if (d) 0 (Fig. 3a), or it is the point where maximum constraint violations occur if (d) > 0 (Fig. 3b). In qualitative terms, the critical points in the resilience test are the worst case conditions for feasible operation. 

It can be shown that the resilience test for this HEN is linear (see Section III,B,1). Therefore the HEN is resilient if and only if it is feasible at every vertex of the specified uncertainty range. 

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 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). 

To allow algebraic equations to be used to locate ATm, assume that the heat capacities can be approximated by piecewise constant functions of temperature, with discontinuities at temperature breakpoints TBRj. Then for each exchanger, Arm can occur only at either end or at a breakpoint location inside the exchanger. However, a remaining difficulty is that since the intermediate stream temperatures are not known before the resilience test, the breakpoint locations are also not known a priori. 

Resilience analysis for HENS can become nonlinear and nonconvex if varying stream split fractions are allowed. In this section nonlinear feasibility and resilience tests are presented for networks with stream splits, with the assumption that the network has a minimum number of units. This assumption often is not restrictive since many stream split networks do have a minimum number of units. 

The resilience test correctly identifies that the HEN is not resilient in the specified uncertainty range. In order to apply the resilience test, the values °f vk.max and gjj max shown in Table V are calculated. The values of v max are all nonpositive thus the load constraints are satisfied throughout the... 

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 ... 

Solution of this LP yields x = 0.0305, thus implying that the network is not resilient in the new uncertainty range. The resilience test is conservative because LP (29) looks for values of stream split fractions u which, if held... 

Saboo (1984) has generalized resilience test (32) to class 2 problems. However, his method is still limited to minimum unit HENs with no stream splits. 

The active constraint strategy has been developed for both the resilience (flexibility) test and the flexibility index (Grossmann and Floudas, 1985, 1987). However, only the active constraint strategy for the resilience test will be discussed here. Recall that the resilience test is based upon a resilience measure x(d) ... 

A HEN is resilient in a specified uncertainty range 0 if and only if X(d) 0. If (d) > 0, then at least one of the feasibility constraints fm is violated somewhere in the uncertainty range. Geometrically, the resilience test determines whether uncertainty range 0 lies entirely within feasible region R. 

The basic idea of the active constraint strategy is to use the Kuhn-Tucker conditions to identify the potential sets of active constraints at the solution of NLP (4) for feasibility measure ip. Then resilience test problem (6) [or flexibility index problem (11)] is decomposed into a series of NLPs with a different set of constraints (a different potential set of active constraints) used in each NLP. 

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. 

Now the active constraint strategy for performing the resilience test can be summarized (Grossmann and Floudas, 1987) as follows. 

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). 

Resilience Test with Active Constraint Strategy for Example 11... 

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). 

The operability test (resilience test in Section III,A,2, but without the energy recovery constraint) is applied at the stage of matches to determine whether the predicted stream matches can lead to an operable HEN. The general form of this operability test is... 

The obvious ways to measure the temperature at which the ability to recover from a deformation is lost are by the loss tangent from dynamic tests or by compression or tension set measurements. Dynamic analyzers arc an excellent way of characterizing low-temperature characteristics, stiffness as well as viscous loss, but they are a relatively modern invention and expensive. It is, how ever. a little. surprising that rebound resilience tests have not been commonly used. Set is quite often used, with compression set being favored over tension set. A particular form of recovery test developed and standardized for measurement of low-temperature behavior is the so-called temperature retraction test. This consists of stretching a dumbbell test piece, placing it in a bath at -70 C. and allowing it to retract as the temperature is raised- in a sense a variation on tension set. 

The resrilting foam is conditioned at 60% R.H. at 20 C for one week. Mechanical tests, i.e. compressive testing and resilience testing, are performed according to the method described by Tatarka and Cutuiingham. The foam density was calculated by sand replacernent volumetric measurement. The cell structure was analyzed by Scanning Electron Microscopy. 

Each box contained six test pieces for tensile stress/strain measurements, two strain test pieces, one test piece for low temperature measurements, two large compression set pieces which also serve for hardness measurements, four small compression set discs for swelling measurements, one Lupke disc for both electrical resistivity and resilience tests and the special compression annulus test piece for long term set. All test pieces except the long term compression set annulus were exposed in the unstrained state. A completed box without the lid is shown in Figure 3. 

The fractional return, to an impacting hody, of the energy with which it strikes a resilient test specimen. ASTM D 1054 details a pendulum-rehound test, while D 2632 and D 3574 describe drop-weight-rebound tests, all employing this principle and all in section 09.01. 

In three-phase three-limb transformer, DC fluxes compensate each other in the main limbs and connecting yokes. Thereby, this transformer type has the highest resiliency. Test results showed that even in case of impact of GIC equal to 50 nominal magnetization current ( 50/l), the transformer current... 

Resilience testing of failed samples showed that the failures are oriented parallel to the direction of extrusion, being propagated in a normal direction fragile, quasifragile, and ductile [10]. 

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