Multiple hysteresis loops

CH3)3NCH2C00 CaCl2 2H2O Family (CH j NCHiCOO- CaCh -lH O (LB Number 66A-I). This crystal is ferroelectric below 46 K. It exhibits at least ten phase transitions (eight can be recognized in Fig. 4.5-87), and the crystal stmcture is commensurately or incommensurately modulated between 46 and 164 K, with a feature known as a devil s staircase. Multiple hysteresis loops are observed in the modulated phases, as shown in Fig. 4.5-88. 

In between these tangencies, the curves R and L have three intersections, so the system has multiple stationary states (Fig. 7.3(b)). We see the characteristic S-shaped curve, with a hysteresis loop, similar to that observed with cubic autocatalysis in the absence of catalyst decay ( 4.2). 

It cuts. the axis at 0ad = 4 as 1/tn tends to zero (adiabatic limit). We have already seen that this is the condition for transition from multiple stationary states (hysteresis loop) to unique solutions for adiabatic reactors, so the line is the continuation of this condition to non-adiabatic systems. Above this line the stationary-state locus has a hysteresis loop this loop opens out as the line is crossed and does not exist below it. Thus, as heat loss becomes more significant (l/iN increases), the requirement on the exothermicity of the reaction for the hysteresis loop to exist increases. 

Even with the largest values of y, corresponding to systems with very little temperature sensitivity, some limited multiplicity is possible. Condition (7.95) again gives the range within which the cooling temperature must lie and within which the system can display the inverse hysteresis loop for some combinations of 0ad and tn. 

Fig. 9.4. (a) The dependence of the stationary-state concentration of reactant A at the centre of the reaction zone, a (0), on the dimensionless diffusion coefficient D for systems with various reservoir concentrations of the autocatalyst B curve a, / = 0, so one solution is the no reaction states a0i>8 = 0, whilst two other branches exist for low D curves b and c show the effect of increasing / , unfolding the hysteresis loop curve d corresponds to / = 0.1185 for which multiplicity has been lost, (b) The region of multiple stationary-state profiles forms a cusp in the / -D parameter plane the boundary a corresponds to the infinite slab geometry, with b and c appropriate to the infinite cylinder and sphere respectively. 

The unfolding of the hysteresis loop gives rise to a cusp in the D-f ex parameter plane, as shown in Fig. 9.4(b). Also shown there are the cusps for infinite cylinder and spherical geometries. For the latter, multiple stationary states cease for / ei = 0.1129 and 0.1078 respectively, values still smaller than the 5 for the CSTR. 

Typical values for the parameters D and x u might be D = 0.05 and ku = 0.01. We now examine how the stationary-state concentration profiles ass(p) and /Jss(p) depend on the dimensionless concentration of the precursor reactant, p0. Figure 9.10 shows the stationary-state concentrations at the centre of the reaction zone ass(0) and / ss(0) as functions of p0. These loci each draw out a hysteresis loop, with a range of corresponding multiplicity of solutions. 

We now consider the dependence of the stationary-state solution on the parameter d. To represent a given stationary-state solution we can take the dimensionless temperature excess at the middle of the slab, 0ss(p = 0) or 60,ss-With the above boundary conditions, two different qualitative forms for the stationary-state locus 0O,SS — <5 are possible. If y and a are sufficiently small (generally both significantly less than i), multiplicity is a feature of the system, with ignition on increasing <5 and extinction at low <5. For larger values of a or y, corresponding to weakly exothermic processes or those with low temperature sensitivity, the hysteresis loop becomes unfolded to provide... 

Fig. 12.5. (a) The region of multiple stationary-state behaviour for the Takoudis-Schmidt-Aris model of surface reaction, with = 10 3 and k2 = 2 x 10-3 (b), (c), and (d) show how the stationary-state reaction rate varies with the gas-phase pressure of reactant R for different values of p, giving isola, mushroom, and single hysteresis loop respectively. (Adapted and reprinted with permission from McKarnin, M. A. et al. (1988). Proc. R. Soc., A415, 363-87.)... 

Figure 7.10 shows the hysteresis loops of Yrd vs. Yfa for various values of Fq. It is clear that the multiplicity range is in general quite large and increases as the gas oil flow rate Fqm increases. 

For a system with a fixed y, the steady-state multiplicity may be revealed by the curve of superficial velocity of solids Up versus Ap, as illustrated in Fig. 8.17, where a hysteresis loop is formed between two branch (upper and lower) curves [Chen et al., 1984], The upper branch corresponds to Regimes 2, 3, or 4, whereas the lower one corresponds to Regime 1. At Ap less than (Ap)i, the system is operated on the upper branch of the curves, while it is operated on the lower branch of the curves at Ap larger than (Ap)2- Between these two... 

Figure 4.30 Hysteresis loop for a CSTR with multiple steady states variation of inlet temperature T. ...

Isotherms associated with N2 adsorption and desorption are directly correlated with the porosity of an LDH. An adsorption isotherm that rises rapidly at low pressures is a result of the adsorption of multiple layers of N2, indicating microporosity, whereas a gentle rise in the adsorption isotherm at low pressure represents the formation of a monolayer or mesoporosity. A sharp increase in the adsorption isotherm at relatively high pressures illustrates that condensation is occurring, which is indicative of mesoporosity (320). When the adsorption isotherm is coupled with a desorption isotherm, the result is a hysteresis loop. The structure of this loop can impart information related to the geometry of the pores (regular or irregular) and the distribution of pore dimensions (101,387). 

A local model of a solder joint can be found in Hg. 59.17. The local model is subjected to multiple thermal cycles and a final damage parameter is extracted. A typical thermal cycling loading profile to which the model is subjected is similar to that shown in Fig. 59.3. About two to four cycles are run to achieve stability of the hysteresis loop. 

There are other indirect methods. Consider a one-variable system (or an effectively one variable). Let the system have multiple stationary states and in Fig. 11.1, taken from [1], we show a schematic diagram of the hysteresis loop in such systems. 

It has been shown in the earlier paper ( ) that isothermal CO oxidation over Pt-alumina catalysts with appreciable intrapellet diffusion resistances can exhibit a wide range of steady state multiplicities in the conversion-temperature, conversion-inlet CO concentration, and conversion-mass flow rate domains. Fig. 2 shows the steady state CO conversion as a function of reactor inlet temperature for a fixed set of concentrations ( 0.3 vol. % CO, 2 vol. % Op). The details of the experimental conditions of Fig. 2 and the subsequent figures are given in Table II. Hegedus et al. ( ) pointed out that the hysteresis envelope shown in Fig. 2 corresponds to the highest and lowest stable steady state conversions, and also demonstrated the existence of several intermediate stable steady states within the envelope of the hysteresis loop. They also have shown that each of these multiple conversion levels can be achieved by a properly chosen sequence of steady state operations. 

In the present study we investigate the possibility of enhancing the reactor conversion in the regime of multiple conversions by means of deliberate perturbations of the initial steady state. For all the results reported here, the reactor was initially set to operate at the lowest conversion steady state, and the reactor inlet CO concentration was perturbed by temporarily reducing the CO concentration to a lower value, while maintaining the total flow rate, the reactor pressure, and oxygen concentration essentially unperturbed. Before each pulse, the reactor was cooled down to a temperature well below the hysteresis loop of Fig. 2 and then slowly heated up to a desired temperature to establish the lowest conversion state. 

In this experiment, the inlet CO concentration was reduced from 0.316 vol. % to 0.108 vol. % for the duration of 3 sec. It can be readily seen that the steady state conversion is enhanced upon pulsing in the multiple conversion regime, while the reactor quickly returns to the original conversion level at temperatures outside the hysteresis loop (e.g., = 145°C or 299°C). 

In order to examine in more detail the conversion enhancement in the multiple conversion regime, we investigated the effects of pulse amplitude and duration at a temperature which falls in the middle of the hysteresis loop in the conversion-temperature plane (see Fig. 2). Fig. 4 shows the observed time variations of CO conversion for various pulse amplitudes at = 212°C. In each case of Fig. 4, a pulse of 3 sec duration was injected at time t = 0 to the reactor which had been previously stabilized at the lowest conversion steady state. As expected, the degree of conversion enhancement was found to increase with increasing pulse amplitudes. Also, it appears that the reactor already attained the highest conversion level (about 40%) with the pulse amplitude of 0.321-0.076-0.321, and thus the even larger pulse amplitude of 0.317-0-0.317 did not provide any further conversion enhancement. 

The stiffness of soil at cyclic loading conditions is even more specific and very often a hysteresis loop can be found during cyclic testing (e.g. cyclic triaxial testing). Multiple loading cycles can cause degradation of the soil strength. This has... 

Typically, multiplicity of steady states is accompanied by hysteresis, a term derived from the Greek word meaning lagging behind. A characteristic mark of hysteresis is that the output— here the reaction rate—forms a loop, which can be oriented clockwise or counterclockwise. Hysteresis causes the reaction rate to jump up or fall down, signifying, respectively, ignition and extinction of the reaction. Fig. 7.5 shows some examples of kinetic dependences involving hysteresis. 

The degree of stability that can be achieved in this temperature control loop depends on the rate at which the heat can be removed from the reactor. In other words, the reactor can be stabilized if the reaction temperature changes fairly slowly when compared with the rate at which the jacket temperature changes. The idea of the control loop shown in Figure 7.26 is that, once the feedstock and catalyst are added, hot water in the jacket is used to initiate the reaction. As the reaction temperature increases, the controller output decreases, closing the hot-water valve (which is air to open) and opening the cold-water valve (which is air to close). A valve positioner, V/P, is used to minimize valve hysteresis. The pump and multiple water inputs to the jacket are used to minimize dead time and to change the jacket temperature as quickly as possible, i.e. minimize the time constant of the jacket. 

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