Transient approach to steady state

Time Dependence—The Transient Approach to Steady-State and Saturation Kinetics... 

The approach to steady-state can be extremely slow (see Figure 9), and would lead to considerable error in the determination of the characteristic parameters of the system if an inappropriate transient value is taken as the final... 

The process response is presented in Figure 4.6. Observe that all the state variables exhibit a fast transient, followed by a slow approach to steady state, which is indicative of the two-time-scale behavior of the system, and is consistent with our observation that processes with impurities and purge are modeled by systems of ODEs that are in a nonstandard singularly perturbed form. 

The fractional approach to steady state following an Increase in the throughput from 30, 50, and 80% to full load is shown in Figure 7. There is a single time constant of about three hours. The transient time for startup is shorter than for turndown because a sudden increase of the flux increases the flame velocity. 

For the same physical conditions, the same initial concentration-depth profile (Figure 12, curve 1), and the same new concentration at the lower boundary z — 0 (5.0 ml/liter), transient concentration-depth profiles have been computed for a semi-infinite water layer extending upwards from z = 0 (40). Although the computations were done for a semi-infinite layer model, the oxygen concentrations were considered only within the 3-km-thick layer between z = 0 and z = 3. In the semiinfinite layer model, the oxygen concentration at z — 1 km attains the 95% value of the steady-state concentration after approximately 1000 years. After 500 years, the concentration is only 88% of the steady-state value. The difference between the estimates of time to steady-state from the two models is understandably accounted for by a faster approach to steady-state in a water layer the two boundaries of which are maintained at constant concentrations. 

Figure 14.5 shows a comparison between experimental results and the model. The startup transient has an initial overshoot followed by an apparent approach to steady state. Oscillations begin after a phenomenally long delay, t > 10 , and the system... 

Concentration profiles predicted by Equations 3-57 and 3-58 are shown in Figure 3.9. In panel (a), the approach to steady state is shown for the situation where D = 1 x 10 cm /s and = 1 x 10 /s. After approximately sufficient time, the transient equations are identical to the steady-state solution. In panel (b), steady-state solutions are shown for the same D, with k varying between 1 X 10 and 1 x 10 /s. Clearly, the rate of elimination of a solute from the tissue space can have a profound influence on the ability of the solute to penetrate from a localized source. 

Since the Fourier Equation (3) is a linear and homogeneous differentia] equation, the transient that describes the initial temperature approach to steady state %r) and any subsequent transients Tsfr) can be evaluated independentJy and then combined with the steady-slate solution of Eq. (7) for Ti(r). An additional simplification will be that all subsequent calculations are made only for the center of the cylindrical sample, i.e for r = 0, the position of the thermocouple. Thus, one can write... 

So far, examples to illustrate experimental methods for following the time course of the approach to steady states and of their kinetic interpretation have been restricted to enzymes which do not have a natural chromophore attached to the protein although reference has been made to the classic studies of Chance with peroxidase (see p. 142). Qearly the application of these techniques to the study of enzymes with built in chromophores, such as the prosthetic groups riboflavine, pyridoxal phosphate or haem, contributed considerably to the elucidation of reaction mechanisms. However, the progress in the identification of the number and character of intermediates depended more on the improvements of spectral resolution of stopped-flow equipment than on any kinetic principles additional to those enunciated above. This is illustrated, for instance, by the progress made between the first transient kinetic study of the flavoprotein xanthine oxidase by Gutfreund Sturtevant (1959) and the much more detailed spectral analysis of intermediates by Olson et al. (1974) and Porras, Olson Palmer (1981). 

FIGURE 4.1 Transient approach to a stable steady state in a CSTR. 

Using the MBL formulation, we performed additional transient hydrogen transport calculations with L — 5.10, 9.96, 16.04, 21.36. 31.28. 41.63, 50.38 mm, stress intensity factor K, =34.12 MPaVm. T Icsa =-0.316, and zero hydrogen concentration C, prescribed on the outer boundary. For these domain sizes, we found the values of the effective time to steady state r to be 240. 608. 1105. 1538. 2297, 2976. and 3450 sec, respectively. Although the MBL approach does not predict the effective time to steady state accurately in comparison to the full-field solution, it can be used to provide a rough approximation. The non-dimensional effective times to steady-state r = Dl jb and the... 

It is often the case that after a sufficiently long time, a transient problem approaches a steady-state solution. When this is the case, it can be useful to calculate the steady solution independently. In this way it can be readily observed if the transient solution has the correct asymptotic behavior at long time. 

We found earlier that transient (nonsteady-state) sedimentation required density gradients to stabilize against convection. Isopycnic sedimentation relies on density gradients not only for anticonvective purposes but also as the secondary gradient needed to establish steady-state conditions. The difference in the two cases is found in the magnitude of the density gradient and in the degree to which components are allowed to approach their steady-state condition. The equipment is similar the zonal rotor developed by Anderson is used for isopycnic as well as transient zonal separations [45]. 

Unlike macroelectrodes which operate under transient, semi-infinite linear diffusion conditions at all times, UMEs can operate in three diffusion regimes as shown in the Figure for an inlaid disk UME following a potential step to a diffusion-limited potential (i.e., the Cottrell experiment). At short times, where the diffusion-layer thickness is small compared to the diameter of the inlaid disc (left), the current follows the - Cottrell equation and semi-infinite linear diffusion applies. At long times, where the diffusion-layer thickness is large compared to the diameter of the inlaid disk (right), hemispherical diffusion dominates and the current approaches a steady-state value. 

Chains, Stokes law with, 68-70 Chan, T., 120 Chapman, S. J., 168-169 Characteristic charge, 187 Characteristic length and diffusion, 155 Charging mechanisms, 179 collisions with ions, 185 contact electrification, 182-183 corona discharge, 195-198 diffusion, 185-189,195 electric, 179-183 equilibrium with, 200-201 steady-state theory of, 201-207 transient approach to, 207-208 field charging, 185,189-195 flame ionization, 184-185 and force, 179-180 frictional, 184... 

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