Steady-state behavior determinants

In section 11.4 Che steady state material balance equations were cast in dimensionless form, therary itancifying a set of independent dimensionless groups which determine ice steady state behavior of the pellet. The same procedure can be applied to the dynamical equations and we will illustrate it by considering the case t f the reaction A - nB at the limit of bulk diffusion control and high permeability, as described by equations (12.29)-(12.31). 

The mathematical model most widely used for steady-state behavior of a reactor is diffusion theory, a simplification of transport theory which in turn is an adaptation of Boltzmann s kinetic theory of gases. By solving a differential equation, the flux distribution in space and time is found or the conditions on materials and geometry that give a steady-state system are determined. 

To illustrate the usefulness of the Information Index in determining the best time interval, let us consider the grid point (l, 0.20, 35). From Figure 12.7 we deduce that the best time interval is 25 to 75 h. In Table 12.4 the standard deviation of each parameter is shown for 7 different time intervals. From cases 1 to 4 it is seen that that measurements taken before 25 h do not contribute significantly in the reduction of the uncertainty in the parameter estimates. From case 4 to 7 it is seen that it is preferable to obtain data points within [25, 75] rather than after the steady state has been reached and the Information Indices have leveled off. Measurements taken after 75 h provide information only about the steady state behavior of the system. 

Steady-state behavior and lifetime dynamics can be expected to be different because molecular rotors normally exhibit multiexponential decay dynamics, and the quantum yield that determines steady-state intensity reflects the average decay. Vogel and Rettig [73] found decay dynamics of triphenylamine molecular rotors that fitted a double-exponential model and explained the two different decay times by contributions from Stokes diffusion and free volume diffusion where the orientational relaxation rate kOI is determined by two Arrhenius-type terms ... 

Example 14-7 can also be solved using the E-Z Solve software (file exl4-7.msp). In this simulation, the problem is solved using design equation 2.3-3, which includes the transient (accumulation) term in a CSTR. Thus, it is possible to explore the effect of cAo on transient behavior, and on the ultimate steady-state solution. To examine the stability of each steady-state, solution of the differential equation may be attempted using each of the three steady-state conditions determined above. Normally, if the unsteady-state design equation is used, only stable steady-states can be identified, and unstable... 

Scott, T. C., C. G. Hill, and C. H. Amundson, "Determination of the Steady-state Behavior of Immobilized beta-Galactosidase Utilizing an Integral Reactor Scheme," Biotechnol. Bioeng. Symp. 15 (1985) 431 - 445. 

In order to determine the accuracy of the solution proposed in Eq. (3.101) for the case of a microdisc electrode, in Fig. 3.13 numerical results are compared with this equation and also with the Oldham Eq. (3.95). Fully reversible, c[Jisc ss = 1000 /4, quasi-reversible, cj[lsc ss = nj4, and fully irreversible, cj[lsc ss = 0.001 /4, heterogeneous kinetics were considered under steady-state behavior. It is seen that, for fully reversible kinetics, both equations give almost identical results which are in good agreement with the simulated values. As the kinetics becomes less reversible, however, the results given by the two equations diverge from each other, with the simulated result lying between them. The maximum error in the Oldham equation is 0.5 %, and for Eq. (3.101), the maximum error is 3.6 %. 

Transient vs. Steady-State Behavior in Permeability Determinations. The foregoing derivations raise some intriguing speculations about the measurement and determination of permeabilities for the respective components in a mixture. Thus, if a true or complete steady-state condition exists during the experiment, whereby all of the feed stream passes through the membrane, then the ratio V/F = and the ratio L/F = 0. That is, it can be said that no reject phase whatsoever is produced. 

All but one of the remaining population-balance parameters are determined from steady-state behavior of foam flow. Fortunately, this exercise drastically limits the choice of parameter values. Thus, for our strong foamer solution, we chose a value of Sw = 0.26, which is slightly above connate saturation (20, 61, 78), and Xlmsx = 0.9, based on the experimental tracer studies of trapped gas saturations (38, 39). 

Multiple steady-state behavior is a classic chemical engineering phenomenon in the analysis of nonisothermal continuous-stirred tank reactors. Inlet temperatures and flow rates of the reactive and cooling fluids represent key design parameters that determine the number of operating points allowed when coupled heat and mass transfer are addressed, and the chemical reaction is exothermic. One steady-state operating point is most common in CSTRs, and two steady states occur most infrequently. Three stationary states are also possible, and their analysis is most interesting because two of them are stable whereas the other operating point is unstable. 

We can immediately see that a necessary condition for coexistence is that 0)2 t The values of o) for each species are determined by the same equation as in the single population case, unaffected by the presence of the other species. We can, therefore, move directly to a graphical description of the steady-state behavior for our competition model. Figure 11 shows isoclines of w in the plane of (k,X) values. If we specify values Xi and ki for population 1, this yields a value for wi. We can then immediately discover the permissible steady-states for any species 2 with parameter values X2 and <2 (see Figure 12). If <2 < only species 1 can survive, unless X2 xs such that 0)2 > Wj — which would allow coexistence. If Kj > Kj, only species 2 can survive, unless X2 is such that 0)2 < o)i, which again allows coexistence. 

Even in the absence of the indicated software, one can employ material balances on the first CSTBR in the cascade and the composition of the feed stream (jq, S o> Po) together with a knowledge of the Monod constants (Pmax,i and T i) at the conditions of the first reactor, to ascertain the steady state composition of the effluent from this reactor. In turn, one can utiUze this composition as that of the stream fed to the second CSTBR, together with the values of the Monod constants (Pmax,2 K ), in the steady state material balances on the second reactor to determine the composition of the effluent from the second reactor. One can then employ this approach to analyze the third and sub-seqnent CSTBRs, marching forward with knowledge of the composition of the effluent from the previous CSTBR as an essential element of the analysis of the steady-state behavior of the next CSTBR in the cascade. 

Since short and long times are relative terms, it is useful to determine the times over which transient and steady state behaviors will predominate and how this time regime is affected by the electrode radius. 

Tye [122] explained that separator tortuosity is a key property determining transient response of a separator and steady-state electrical measurements do not reflect the influence of tortuosity. He recommended that the distribution of tortuosity in separators be considered some pores may have less tortuous paths than others. He showed mathematically that separators with identical average tortuosity and porosities could be distinguished by their non-steady-state behavior if they have different distributions of tortuosity. 

As no complete data sets with both reactor and the kinetic data are known in open literature validating the model is difficult. Therefore, the steady state behavior of the model used has been determined and compared to the behavior as reported in literature [12]. 

Although the model contains many assumptions the results can be regarded promising. The steady-state behavior of the model is in agreement with literature reports. The reported cycle time is also in line with earlier reported work in literature. The assumption made about the equilibrium constant of the rate determining step does not influence the results dramatically. Experimental verification of the presented approach will be the future challenge. 

Figure 1 shows a comparison of the evolution of the catalytic activity over the dififerent caihon catalysts in the anaerobic dehydrogenation of EB to ST with time on stream at 5S0°C. An induction period was observed of about 1.5 hours, during which all carbon materials have shown different behavior. The highest activity after having reached steady state was determined for carbon nanotubes as the catalyst with a ST yield of 28%. 

With the assumption of chemisorption equilibrium invoked here, the surface reaction becomes the rate determining step, and thus in Eq. (5) can be reasonably approximated by the empirical rate expression [Eq. (12)] which was found to be adequate in describing the steady state behavior of CO oxidation under our operating conditions ( ). That is. 

It is of interest to discuss the predictions of a diffusion-reaction model which accounts for no surface accumulation phenomena [that is, the model with Nj = 0 in Eq. (5)]. We found that such a model was inadequate in describing the transient behavior of our system the simulation based on the operating conditions of Fig. 6 predicted a conversion enhancement to the highest conversion level even with the very short pulse duration of 0.25 sec. In view of the small characteristic response time (about 0.1 sec) for intrapellet diffusion, surface phenomena seem to play a key role in determining the transient response of supported catalysts (12, 14, 15, 28). This, of course, does not invalidate the use of the simpler diffusion-reaction model for the description of steady state behavior, as shown in our previous paper (9). 

Voltage fluctuation caused by the charging current of cables is an important factor for the design of a wind farm. The voltage fluctuation can be simulated by steady-state analysis. In the analysis, the cable can be approximately expressed by a lumped parameter equivalent circuit. Since the steady-state behavior of a three-phase circuit is determined by its positive sequence component, the wind farm can be expressed by a single-phase circuit. 

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 ... 

Dyna.mic Viscometer. A dynamic viscometer is a special type of rotational viscometer used for characterising viscoelastic fluids. It measures elastic as weU as viscous behavior by determining the response to both steady-state and oscillatory shear. The geometry may be cone—plate, parallel plates, or concentric cylinders parallel plates have several advantages, as noted above. 

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