Steady uniqueness

The latter may be fiirther subdivided into transient experiments, in which the current and potential vary with time in a non-repetitive fashion steady-state experiments, in which a unique interrelation between current and potential is generated, a relation that does not involve time or frequency and in which the steady-state current achieved is independent of the method adopted and periodic experiments, in which current and potential vary periodically with time at some imposed frequency. 

Normally when a small change is made in the condition of a reactor, only a comparatively small change in the response occurs. Such a system is uniquely stable. In some cases, a small positive perturbation can result in an abrupt change to one steady state, and a small negative perturbation to a different steady condition. Such multiplicities occur most commonly in variable temperature CSTRs. Also, there are cases where a process occurring in a porous catalyst may have more than one effectiveness at the same Thiele number and thermal balance. Some isothermal systems likewise can have multiplicities, for instance, CSTRs with rate equations that have a maximum, as in Example (d) following. 

The jump conditions must be satisfied by a steady compression wave, but cannot be used by themselves to predict the behavior of a specific material under shock loading. For that, another equation is needed to independently relate pressure (more generally, the normal stress) to the density (or strain). This equation is a property of the material itself, and every material has its own unique description. When the material behind the shock wave is a uniform, equilibrium state, the equation that is used is the material s thermodynamic equation of state. A more general expression, which can include time-dependent and nonequilibrium behavior, is called the constitutive equation. 

These equations hold if an Ignition Curve test consists of measuring conversion (X) as the unique function of temperature (T). This is done by a series of short, steady-state experiments at various temperature levels. Since this is done in a tubular, isothermal reactor at very low concentration of pollutant, the first order kinetic applies. In this case, results should be listed as pairs of corresponding X and T values. (The first order approximation was not needed in the previous ethylene oxide example, because reaction rates were measured directly as the total function of temperature, whereas all other concentrations changed with the temperature.) The example is from Appendix A, in Berty (1997). In the Ignition Curve measurement a graph is made to plot the temperature needed for the conversion achieved. 

The chemical process industries (CPI), petroleum and allied industries apply physical as well as chemical methods to the conversion of raw feedstock materials into salable products. Because of the diversity of products, process conditions and requirements, equipment design is often unique, or case specific. The prime requirement of any piece of equipment is that it performs the function for which it was designed under the intended process operating conditions, and do so in a continuous and reliable manner. Equipment must have mechanical reliability, which is characterized by strength, rigidness, steadiness, durability and tightness. Any one or combination of these characteristics may be needed for a particular piece of equipment. 

As a consequence, good, safe, steam-sampling points are required, and automatic, real-time continuous analyzer systems for monitoring of steam and condensate quality are very useful. These requirements usually are not a problem in larger power and process HP boiler plants. Here, each facility tends to have a unique combination of operating conditions and waterside chemistry circumstances that necessitate the provision of a steady stream of reliable operational data, and this can be obtained realistically only from continuous, real-time analysis. 

It should be noted that the steady-state solution of Equation (12) is not necessarily unique. This can easily be seen in the case of the four-reservoir system shown in Fig. 4-7. In the steady state all material will end up in the two accumulating reservoirs at the bottom. However, the distribution between these two reservoirs will... 

Fig. 4-7 Example of a coupled reservoir system where the steady-state distribution of mass is not uniquely determined by the parameters describing the fluxes within the system but also by the initial conditions (see text).

The steady-state design equations (i.e., Equations (14.1)-(14.3) with the accumulation terms zero) can be solved to find one or more steady states. However, the solution provides no direct information about stability. On the other hand, if a transient solution reaches a steady state, then that steady state is stable and physically achievable from the initial composition used in the calculations. If the same steady state is found for all possible initial compositions, then that steady state is unique and globally stable. This is the usual case for isothermal reactions in a CSTR. Example 14.2 and Problem 14.6 show that isothermal systems can have multiple steady states or may never achieve a steady state, but the chemistry of these examples is contrived. Multiple steady states are more common in nonisothermal reactors, although at least one steady state is usually stable. Systems with stable steady states may oscillate or be chaotic for some initial conditions. Example 14.9 gives an experimentally verified example. 

The turbine and generator components of a nuclear power plant have exact counterparts in power plants fueled by fossil fuels. The uniqueness of the nuclear power plant lies in its core. The core is a nuclear reactor where fission takes place under conditions that keep the reactor operating just below the critical level. The core contains three parts fuel rods, moderators, and control rods. These components act on the flow of neutrons within the core, as shown in Figure 22-13. The fate of neutrons must be controlled carefully. Fission must be sustained at a steady rate that produces sufficient energy to mn a generator, but the rate must not be allowed to increase and destroy the reactor. 

The temperature rise due to this exothermic reaction then approaches the adiabatic temperature rise. The final steady state is always characterized by conditions T = T, and c = 0. A batch reactor, in which a zero order reaction is carried out, always has a unique and stable mode of operation. This is also true for any batch and semibatch reactor with any order or combination of reactions. 

The main inconvenient of this methodology is that the results cannot be considered stricto sensu as obtained in operando conditions, because the system was perturbed from the steady state to reveal hidden species. It could be even hypothesized that such compounds are uniquely due to the particular test conditions and not to the real reaction pathway. A method to discard such kind of criticism is to maintain the chemical steady state of the reaction, while introducing a perturbation via a sudden exchange of one... 

In the treatment of steady-state pipeline network problems so far we have tacitly assumed that there is a unique solution for each problem. For certain types of networks the existence of a unique solution can indeed be rigorously established. The existence and uniqueness theorems for formulation C were proved by Duffin (DIO) and later extended by Warga (Wl). In Warga s derivation the governing relation for each network element assumes the form,... 

A second equation is needed to determine the surface tension as a function of axial position. We adopt the quasistatic assumption that a is a unique equilibrium function of the surface excess concentration, T, even during dynamic events (17). A surface species continuity balance dictates how T varies along the interface. Upon neglect of surface diffusion and for h <1, the steady state form of this balance is... 

In order to apply the concepts of modern control theory to this problem it is necessary to linearize Equations 1-9 about some steady state. This steady state is found by setting the time derivatives to zero and solving the resulting system of non-linear algebraic equations, given a set of inputs Q, I., and Min In the vicinity of the chosen steady state, the solution thus obtained is unique. No attempts have been made to determine possible state multiplicities at other operating conditions. Table II lists inputs, state variables, and outputs at steady state. This particular steady state was actually observed by fialsetia (8). 

We would be remiss in our obligations if we did not point out that the regions of multiple solutions are seldom encountered in industrial practice, because of the large values of / and y required to enter this regime. The conditions under which a unique steady state will occur have been described in a number of publications, and the interested student should consult the literature for additional details. It should also be stressed that it is possible to obtain effectiveness factors greatly exceeding unity at relatively low values of the Thiele modulus. An analysis that presumed isothermal operation would indicate that the effectiveness factor would be close to unity at the low moduli involved. Consequently, failure to allow for temperature gradients within the catalyst pellet could lead to major errors. 

System (1) is of the same type as other systems which have recently appeared in the literature on allelopathic competitions (Hsu and Waltman, 1998, Braselton and Waltman, 2001, Fergola et al., 2004). Usually the mathematical analysis of such systems, first requires one to check if their solutions satisfy the properties of global existence in the future, positivity, boundedness and uniqueness. Subsequently, it often happens that, due to the difficulties of integration of these systems, we look for special biologically meaningful solutions such as steady-states or periodic... 

A reaction which follows power-law kinetics generally leads to a single, unique steady state, provided that there are no temperature effects upon the system. However, for certain reactions, such as gas-phase reactions involving competition for surface active sites on a catalyst, or for some enzyme reactions, the design equations may indicate several potential steady-state operating conditions. A reaction for which the rate law includes concentrations in both the numerator and denominator may lead to multiple steady states. The following example (Lynch, 1986) illustrates the multiple steady states... 

Pharmacists also perform an important role as educators of patients (Fleischer, 1999). This is a safety role, because many drugs are safe if used as intended but unsafe if used differently from the way they are intended. Pharmacists routinely inform patients about the need to ingest medications at certain times of the day to ensure steady blood levels. They caution about drowsiness or other manageable side effects. Psycho-social issues are also important in patient education by pharmacists, because every patient develops a unique medication use behavior, and sometimes these behaviors can interfere with the effectiveness of drug therapy (Bloom, 1996). 

This follows by a steady state energy balance of the surface heated by qe, outside the flame-heated region S. It appears that a critical temperature exists for flame spread in both wind-aided and opposed flow modes for thin and thick materials. Tstmn has not been shown to be a unique material property, but it appears to be constant for a given spread mode at least. Transient and chemical effects appear to be the cause of this flame spread limit exhibited by 7 smln. For example, at a slow enough speed, vp, the time for the pyrolysis may be slower than the effective burning time ... 

It is suspected that this nonlinear rate form, which has a maximum value, may cause certain regions of unstable operation with multiple steady states. How should the operation be conducted to ensure unique steady conditions ... 

The program THERMFF solves the same dynamic process model equations as THERM, where it was shown that all the parameters, including the inlet temperature and concentration will influence the steady state. In the case of multiple steady states the values of the steady state parameters cannot be set, because they are not unique. This example should, therefore, be mn under parameter conditions that will guarantee a single steady state for all expected values of the CA0 and T0. These can be selected with the aid of the programs THERMPLOT and THERM. 

Figure 18. A simple bistable pathway [96], Left panel The metabolite A is synthesized with a constant rate vi and consumed with a rate vcon V2(A) + V3(A), with the substrate A inhibiting the rate V3 at high concentrations (allosteric regulation). Right panel The rates of vsyn vi const. and vcon V2 (A) + V3(A) as a function of the concentration A. See text for explicit equations. The steady state is defined by the intersection of synthesizing and consuming reactions. For low and high influx v, corresponding to the dashed lines, a unique steady state A0 exists. For intermediate influx (solid line), the pathway gives rise to three possible solutions of A0. The rate equations are specified in Eq. 67, with parameters 0.2, 3 2.0, Kj 1.0, and n 4 (in arbitrary units).

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