Steady state defined

Figures 3 and 4 were obtained using the steady state defined by the variables and parameters of Tables I and II. Table III shows the poles and time constants computed from the A matrix for this steady state.

Computed as explained in the text using initial stationary concentration (Ct 0) given by profile 1 in Figure 12 and new steady-state given by profile 2. Time to steady-state shown for different values of eddy diffusion coefficient (K) and advective velocity (U). Time to steady-state defined as the time when the oxygen concentration has attained 95% of the concentration difference between the new and initial steady-state profiles C — Ct 0 = 0.95(Ct x — Ct= 0). 

Consider the steady state of the well-stirred system, [X]s, [Y]s, and small perturbations that move the system away from the steady state defined by x = [X] - [X]s and y = [Y] - [Y]s. These are substituted into Eqs. [56], and the resulting expressions are linearized by dropping nonlinear terms. As described earlier, this is formally carried out by writing Taylor series expansions for /"([X], [Y]) and g([X], [Y]) around the steady state concentrations [X]s, [Y]s and retaining only the linear terms. This procedure yields equations for the evolution of the perturbation in the linear regime of the steady state... 

The chain reactions treated in Section 3.4 consist of straight chains. There is an initiation step taking place at a rate r,-. The propagation steps do not produce nor do they destroy active centers. This is so because in each step of the closed sequence, the active centers change only in kind, not in number. Therefore, there exists a steady state defined by the condition that the rate of initiation is equal to the rate of termination. 

However, if the concentration of species A is much larger than that of species B, we shall have at the beginning the steady state dehned by the reactions (8.8a) and (8.8b), whereas later, when the concentration of reactant B will be close to zero, the steady state defined by the reactions (8.8a) and (8.8c). In that case, the net stoichiometric equation will also change from A -F B Pi to A P2. The pseudo-steady-state concentration of intermediate X change from the initial value of about fci[A]/fc2[B] to the end value of about ki[K jkj,. 

In these equations, we use the fact that Op, bpp,. .. and Oq, f qq,. .. are essentially derivatives of two inverse functions (0 and P). (dQ/dP)s and (9 2/9P )s are the first and second derivative of the adsorption isotherm written in the dimensional form [56] at steady state defined by P and Q. It can be shown that the low-frequency asymptotic values of the third- and higher-order functions are proportional to the third- and higher-order derivatives of the adsorption isotherm... 

Proposition 13 Suppose that ar i = (Xg i for all i > 1, and that the retailer s forecasting system is in steady state. Define the state vector Zt = ( HrX[ i X[ ). Then, the following system represents the co-evolution of the retailer s order quantities and the supplier s observations ... 

Note the pressure-flow solver is relatively fragile and it is important to build up each case carefully from steady state, defining an appropriate pressure-flow profile and specifications for each sub-task.]... 

Concepts of the psendo-steady and quasi-steady states are apparently veiy close to one another. However the difference is fundamental. Whereas the pseudo-steady states define kinetic modes of reaction, the quasi-steady states are simple mathematical approximations. Moreover the difference is palpable in the sense that the first concept can be directly checked by the experiment independently of the kinetic law, the second is checked only by the conformity between the kinetic law speed, concentrations calculated and the experimental corresponding law. But maity other factors must intervene in addition so that these two laws coincide, in particular a correct choice of the mechanism of the reaction. 

The scan rate, u = EIAt, plays a very important role in sweep voltannnetry as it defines the time scale of the experiment and is typically in the range 5 mV s to 100 V s for nonnal macroelectrodes, although sweep rates of 10 V s are possible with microelectrodes (see later). The short time scales in which the experiments are carried out are the cause for the prevalence of non-steady-state diflfiision and the peak-shaped response. Wlien the scan rate is slow enough to maintain steady-state diflfiision, the concentration profiles with time are linear within the Nemst diflfiision layer which is fixed by natural convection, and the current-potential response reaches a plateau steady-state current. On reducing the time scale, the diflfiision layer caimot relax to its equilibrium state, the diffusion layer is thiimer and hence the currents in the non-steady-state will be higher. 

In this sequence the Cl also acts as a catalyst and two molecules are destroyed. It is estimated that before the Cl is finally removed from the atmosphere in 1—2 yr by precipitation, each Cl atom will have destroyed approximately 100,000 molecules (60). The estimated O -depletion potential of some common CFCs, hydrofluorocarbons, HFCs, and hydrochlorofluorocarbons, HCFCs, are presented in Table 10. The O -depletion potential is defined as the ratio of the emission rate of a compound required to produce a steady-state depletion of 1% to the amount of CFC-11 required to produce the 1% depletion. The halons, bromochlorofluorocarbons or bromofluorocarbons that are widely used in fire extinguishers, are also ozone-depleting compounds. Although halon emissions, and thus the atmospheric concentrations, are much lower than the most common CFCs, halons are of concern because they are from three to ten times more destmctive to O, than the CFCs. 

The analysis of steady-state and transient reactor behavior requires the calculation of reaction rates of neutrons with various materials. If the number density of neutrons at a point is n and their characteristic speed is v, a flux effective area of a nucleus as a cross section O, and a target atom number density N, a macroscopic cross section E = Na can be defined, and the reaction rate per unit volume is R = 0S. This relation may be appHed to the processes of neutron scattering, absorption, and fission in balance equations lea ding to predictions of or to the determination of flux distribution. The consumption of nuclear fuels is governed by time-dependent differential equations analogous to those of Bateman for radioactive decay chains. The rate of change in number of atoms N owing to absorption is as follows ... 

Fig. 15. Example of steady-state feedforward controls, where + indicates the summation of signals. Terms are defined in text.

Thus when an electric field is appHed to a soHd material the mobile charge carriers are accelerated to an average drift velocity v, which, under steady-state conditions, is proportional to the field strength. The proportionality factor is defined as the mobility, = v/E. An absolute mobility defined as the velocity pet unit driving force acting on the particle, is given as ... 

If the crystallizer is now assumed to operate with a cleat feed (n = 0), at steady state (dn jdt = 0), and if the crystal growth rate G is invariant and a mean residence time T is defined as then the population balance can be written as... 

The continuous current rating of a bus system can be defined by the current at which a steady-state thermal condition can be reached. It is a balance between the enclosure and the conductor s heat gain and heat loss. If this temperature is more than the permissible steady-state thermal limit it must be reduced to the desired level by increasing the size of the conductor or the enclosure or both, or by adopting forced cooling. Otherwise the rating of the bus system will have to be reduced accordingly. 

For a first-order plant, proportional eontrol will always produee steady-state errors. This is diseussed in more detail in Chapter 6 under system type elassifieation where equations (6.63)-(6.65) define a set of error eoeffieients. Inereasing the open-loop... 

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