Quasi-steady state solution

A] = b/a (equation (A3.4.145)) is stationary and not [A ] itself This suggests d[A ]/dt < d[A]/dt as a more appropriate fomuilation of quasi-stationarity. Furthemiore, the general stationary state solution (equation (A3.4.144)) for the Lindemaim mechanism contams cases that are not usually retained in the Bodenstein quasi-steady-state solution. 

As in Fig. 11.13, the loop can be represented by an array of point sources each of length R0. Using again the spherical-sink approximation of Fig. 11.126 and recalling that d Rl Ro, the quasi-steady-state solution of the diffusion equation in spherical coordinates for a point source at the origin shows that the vacancy diffusion field around each point source must be of the form... 

As solidification continues and solute is continuously rejected into the remaining liquid, the concentration in the bulk liquid increases slowly and the quasi-steady-state solute distribution in the boundary layer evolves. This, in turn, produces... 

Because the solute diffusivity in the solid is far smaller than in the liquid, any diffusion in the solid will be neglected. In most cases of interest, the transient period required to produce a quasi-steady-state solute distribution at the interface is relatively small.1 At a relatively short time after the establishment of the quasi-steady-state concentration spike, the flux relative to an origin at the interface moving at velocity v is... 

Statement 3. If the concentrations are fixed, iVa = const, Nb = const, the set of kinetic equations (8.2.17), (8.2.22) and (8.2.23) as functions of the control parameter demonstrates two kinds of motions for k k the stationary (quasi-steady-state) solution holds, whereas for k < k a regular (quasi-regular) oscillations in the correlation functions like standing waves... 

Remark 2.4. In a general nonlinear system, there may be several distinct solutions X2i G 1,..., k. In such a case, one focuses on a particular solution and the corresponding representation for the slow subsystem (2.13) in an appropriate neighborhood. The choice of a particular quasi-steady-state solution depends on the initial condition x x . The solution of the fast system will... 

Thus the multiperiod optimisation problem is formulated as a sequence of two independent dynamic optimisation problems (PI and P2), with the total time minimised by a proper choice of the off cut variables in an outer problem (PO) and the quasi-steady state conditions appearing as a constraint in P2. The formulation is very similar to those presented by Mujtaba and Macchietto (1993) discussed in Chapter 5. For each iteration of PO, a complete solution of PI and P2 is required. Thus, even for an intermediate sub-optimal off cut recycle, a feasible quasi-steady state solution is calculated. The gradients of the objective function with respect to each decision variable (Rl or xRl) in problem PO were evaluated by a finite difference scheme (described in previous chapters) which again requires a complete solution of problem PI and P2 for each gradient evaluation (Mujtaba, 1989). 

Statement 2. A set (8.3.22) to (8.3.24) with fixed concentrations = P/K and N = VIP has two kinds of motions dependent on the value of parameter n. As k kq, the stationary (quasi-steady-state) solution occurs, whereas for /c < kq the correlation functions demonstrate the regular (quasiregular) oscillations of the standing wave type. The marginal magnitude is Ko = o(p,/3)-... 

In the absence of convection the behavior can often be analyzed using a quasi-steady-state solution to the diffusion equations because the time required for diffusion to produce equilibration in the drop, which is of order d lD with a the drop radius (typically 25 to 50 pm) and D the diffusivity, is normally much less than the time of the experiment (several minutes). This quasisteady-state approach predicts that drop composition is uniform but varies with time and that the time required for intermediate phase formation to begin for given drop and solution compositions is proportional to the square of the initial drop radius. Results obtained using the oil drop technique that are consistent with these predictions are discussed below. 

Figure 5-1 shows the behavior of both the quasi steady-state solution, Eq. [27], for t 100 d, and the unsteady-state solution, Eq. [21], for t = 1, 10, 20 and 100 d. Equations [A24] and [A25] have been incorporated into Eq. [21] to develop the concentration curves. The values of the parameters are presented on Fig. 5-1, except co = 1/24 hr1 and R = 1. For t 100 d, the quasi steady-state solution, Eq. [27], gives the same results as are given by Eq. [21], the unsteady-state solution. Obviously, 100 d is a long enough time for the unsteady-state effects to disappear from the 500 cm length of column in Fig. 5-1. Dispersion acts to reduce the height of the peaks and fill in the valleys of the concentration vs>distance curve. Although the sinusoidal fluctuations in concentration amplitude are) dampened by dispersion, they are still visible after the solute travels 500 cmN- ...

A quasi steady-state solution for the tracer distribution in a soilpolutnn has been developed for the inlet boundary concentration being a constant plus a Sinusoidal component. Then an unsteady state solution for tracer distribution a soil column was developed for the same inlet boundary condition as above. The unsteady-state tracer concentration distribution applies to the section of a soil column that still remembers the initial condition. The two solutions may be applicable to those planning experiments to measure parameters such as the dispersion coefficient from tracer tests. A sinusoidal loading of tracer at the inlet boundary may enable one to obtain repeated data traces at the column outlet as part of an extended experiment. Continued collection of tracer concentration vs. time data at the column outlet over a number of periods would enable one to collect data from repeated experiments, for each period of the sine wave would represent another experiment. This should enable one to obtain more replicates of data to improve statistical estimates of the dispersion coefficient than could be obtained by experimental methods that use a slug loading or a step change of concentration at the column inleL"... 

The groundwater composition was relatively constant during the one-year collection period. The modelling approach, therefore, is focused on finding (quasi) steady-state solutions of the differential equations. Further details, concerning the filed site and measurements are given by Massmann et al. (2001). 

A quasi-steady state solution of Eqs. (25)- (27) yielded the following square root of time dependence of the normalized gel layer thickness, 8 ... 

The quasi-steady-state solution of the problem corresponds to solving the problem with vanishing time-derivatives. Hence the problem is solved only for the instantaneous field values and for the fluxes in the system. Thus the problem is described with the following governing equations ... 

The third term on the rhs of Eq. 202 is zero due to axi-symmetry. We further restrict ourselves to a quasi steady-state solution, i.e. we assume that at any given time /, the flow can be approximated as being steady. This would mean, for instance, that the impulsive loading involved in the start-up of the squeeze would not be covered by the solution. The quasi steady-state assumption allows us to discard the term on the Ihs of Eq. 202. The simplified z-momentum equation is thus... 

Two approaches can be used for calculating interparticle and particle surface collision heat transfer (Amritkar et al., 2014). The first approach is based on the quasi-steady state solution of the coUisional heat transfer between two spheres (Vargas and McCarthy, 2002). The other approach is based on the analytical solution of the one-dimensional unsteady heat conduction between two semi-infinite objects. This approach was proposed by Sun and Chen (1988) based on the analysis of the elastic deformation of the spheres in contact. 

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