Time to steady state

Analysis of nonstationary convective mass transfer, under well-defined hydrodynamic conditions, may be helpful to understand the way in which the limiting current is established at electrodes of appreciable dimensions. As referred to in the previous section, the transition time to steady-state... 

Note that previously known time characteristics, such as moments of FPT, decay time of metastable state, or relaxation time to steady state, follow from moments of transition time if the concrete potential is assumed a potential with an absorbing boundary, a potential describing a metastable state or a potential within which a nonzero steady-state distribution may exist, respectively. Besides, such a general representation of moments dn(c,xo,d) (5.1) gives us an opportunity to apply the approach proposed by Malakhov [34,35] for obtaining the mean transition time and easily extend it to obtain any moments of transition time in arbitrary potentials, so i9n(c, xo, d) may be expressed by quadratures as it is known for moments of FPT. 

AED i/2 Time to Steady State (days) Unchanged (%) VD (L/kg) Metabolite Protein Binding (%)... 

Vary the system model parameters and see how these affect the time to steady state... 

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... 

Figure 9. MBL formulation results plotted against normalized domain size L lb. under zero concentration boundary condition on the remote boundary while the crack faces are in equilibrium with 15 MPa hydrogen gas (a) non-dimensionalized effective time to steady state = >t lb (b) peak values of the normalized hydrogen concentration in NILS at / =/ (effective time to steady state).

The simulations show that steady-state diffusion conditions throughout the wall thickness are attained in 2 hr for the case when hydrogen outgases through the OD surface and 10 hr when the OD surface is impermeable. We defined an effective time to steady state / as the time at which the hydrogen concentration at NILS at the hydrostatic-stress peak location reaches 98% of the final steady state value. We found the corresponding / values equal to 6.4 min for an outgassing pipeline and 34.25 min for an impermeable pipeline. 

Lastly, we studied the effect of 7-stress on the effective time to steady state and the corresponding magnitude of the peak hydrogen concentration. We found that a negative T -stress (which is the case for axial pipeline cracks) reduces both the effective time to steady state and the peak hydrogen concentration relative to the case in which the T -stress effect is omitted in a boundary layer formulation under small scale yielding conditions. This reduction is due to the associated decrease of the hydrostatic stress ahead of the crack tip. It should be noted that the presented effective non-dimensional time to steady state r is independent of the hydrogen diffusion coefficient D 9. Therefore, the actual time to steady state is inversely proportional to the diffusion coefficient (r l/ ). 

Table 10.5 Time to Steady State During the Heating Cycle...

Probe Time to Steady State, minutes Steady-State Temperature, °C... 

Screw Speed, rpm Time to Steady State, min T17, °C Discharge Temperature, °C Rate, kg/h... 

Substance Mechanism Half-life Time to steady-state Clinical comments... 

Fentanyl Plaster formulation determines time to steady state Used as plaster according to special instructions... 

In patients with heart failure, lidocaine s volume of distribution and total body clearance may both be decreased. Thus, both loading and maintenance doses should be decreased. Since these effects counterbalance each other, the half-life may not be increased as much as predicted from clearance changes alone. In patients with liver disease, plasma clearance is markedly reduced and the volume of distribution is often increased the elimination half-life in such cases may be increased threefold or more. In liver disease, the maintenance dose should be decreased, but usual loading doses can be given. Elimination half-life determines the time to steady state. Thus, although steady-state concentrations may be achieved in 8-10 hours in normal patients and patients with heart failure, 24-36 hours may be required in those with liver disease. Drugs that decrease liver blood flow (eg, propranolol, cimetidine) reduce lidocaine clearance and so increase the risk of toxicity unless infusion rates are decreased. With infusions lasting more than 24 hours, clearance falls and plasma concentrations rise. Renal disease has no major effect on lidocaine disposition. 

As an interesting fact, we can learn from Eq. 12-21 that the time to steady-state (or time to equilibrium) depends on the sum of the forward and reverse reaction rate constants. Thus, even if one rate constant is very small, time to equilibrium can be small, provided that the other rate constant is large. By using Eq. 4 in Box 12.1 (95% of equilibrium reached) we obtain ... 

Note that the denominator of these equations consists of the sum of all relevant (transport and reaction) rate constants, k +, ot = k + Lk]. Thus, we can immediately assess the relative influence of these processes on Cj or M by looking at the relative size of all /t-values. This is also important when we are interested in time-to-steady-state (Eq. 7 in Box 12.2) ... 

What does time to steady-state mean Why is there no unique definition of time to steady-state for a linear first-order differential equation ... 

It can be shown that all the coefficients kxi of Eq. 1 are positive or zero, if the model equations result from a mass balance scheme. Then k, > 0, where, is the smaller of the two eigenvalues. By analogy to Eq. 4 of Box 12.1, the time to steady-state can be defined by ... 

In Illustrative Example 21.5 we discussed the behavior of tetrachloroethene (PCE) in a stratified lake. As mentioned before, our conclusions suffer from the assumption that the concentrations of PCE in the lake reach a steady-state. Since in the moderate climate zones (most of Europe and North America) a lake usually oscillates between a state of stratification in the summer and of mixing in the winter, we must now address the question whether the system has enough time to reach a steady-state in either condition (mixed or stratified lake). To find an answer we need a tool like the recipe for one-dimensional models (Eq. 4, Box 12.1) to estimate the time to steady-state for multidimensional systems. 

We have to remember that one does not usually calculate the steady-state values by hand by solving Eq. 21-46 explicitly. However, our general discussion helps to decide whether such steady-states exist and how long it takes to reach them. Since it is the exponential function with the smallest ka value which decreases most slowly we conclude that, in analogy to Eq. 4 of Box 12.1, the overall time to steady-state of the system is determined by... 

Inhomogeneous systems. If Eq. 21-46 is an inhomogeneous system, that is, if at least one Ja is different from zero, then usually all eigenvalues are different from zero and negative, at least if the equations are built from mass balance considerations. Again, the eigenvalue with the smallest absolute size determines time to steady-state for the overall system, but some of the variables may reach steady-state earlier. In Illustrative Example 21.6 we continue the discussion on the behavior of tetrachloroethene (PCE) in a stratified lake (see also Illustrative Example 21.5). Problem 21.8 deals with a three-box model for which time to steady-state is different for each box. 

There is still another point to be discussed, which may limit the calculations presented in Tables 23.4 and 23.5. In 1986, when the concentrations were measured, the lake may not have been at steady-state. In fact, the PCB input, which mainly occurred through the atmosphere, dropped by about a factor of 5 between 1965 and 1980. However, the response time of Lake Superior (time to steady-state, calculated according to Eq. 4 of Box 12.1 from the inverse sum of all removal rates listed in Table 23.4) for both congeners would be less than 3 years. This is quite short, especially if we use the model developed for an exponentially changing input (Chapter 21.2, Eq. 21-17) with a specific rate of change a = - 0.1 yr 1 (that is, the rate which... 

In Part 2 of the PCB story, we introduced the exchange between the water column and the surface sediments in exactly the same way as we describe air/water exchange. That is, we used an exchange velocity, vsedex, or the corresponding exchange rate, ksedex (Table 23.6). Since at this stage the sediment concentration was treated as an external parameter (like the concentration in the air, Ca), this model refinement is not meant to produce new concentrations. Rather we wanted to find out how much the sediment-water interaction would contribute to the total elimination rate of the PCBs from the lake and how it would affect the time to steady-state of the system. As shown in Table 23.6, the contribution of sedex to the total rate is about 20% for both congeners. Furthermore, it turned out that diffusion between the lake and the sediment pore water was much more important than sediment resuspension and reequilibration, at least for the specific assumptions made to describe the physics and sorption equilibria at the sediment surface. 

The accumulation of HOCs in phytoplankton plays an important role in food-web bioaccumulation. Both the increased times to steady state and the effect of dilution decrease accumulation in phytoplankton. This decrease results in a lower phytoplankton body burden and a decreased exposure in higher organisms. As a result, an equilibrium-based model will tend to overestimate concentrations in phytoplankton, and this overestimate will be evident throughout the food web. 

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