Quasi-steady state sequences

Equality of speeds law for homogeneous linear quasi-steady state sequences with invariant volume... 

Existence and uniqueness of the particular solution of (5.1) for an initial value y° can be shown under very mild assumptions. For example, it is sufficient to assume that the function f is differentiable and its derivatives are bounded. Except for a few simple equations, however, the general solution cannot be obtained by analytical methods and we must seek numerical alternatives. Starting with the known point (tD,y°), all numerical methods generate a sequence (tj y1), (t2,y2),. .., (t. y1), approximating the points of the particular solution through (tQ,y°). The choice of the method is large and we shall be content to outline a few popular types. One of them will deal with stiff differential equations that are very difficult to solve by classical methods. Related topics we discuss are sensitivity analysis and quasi steady state approximation. 

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). 

Using the anticipated method of startup the sequence of events was broken down into distinct quasi steady state steps as follows ... 

Task (a) is addressed by multiple acquisition and co-addition of solid echoes according to the sequence of Ostroff and Waugh . Fast repetition of pulses in the OW4 sequence leads to an effective slowing down of the apparent transverse relaxation (spin locking in a quasi steady state) such that the gain is over-proportional. Combined with an active Q-switch for dead time reduction, a number of echoes (up to 20) could be detected for averaging. 

The principles of quasi-steady-state and quasi-equilibrium hypotheses are illustrated in Figures 2.1 and 2.2. If we consider the reversible reaction sequence A R S and assume that R is a rapidly reacting intermediate, its concentration remains at a low, practically constant level during the reaction Figure 2.1 shows the concentrations of A, R, and S in a batch reactor (Chapter 3) as a function of the reaction time. The net generation rate of R is practically zero (Rr = 0), except during a short initial period of time. On the other hand, if one of the reaction steps is very rapid compared with the others, for instance. 

Clearly the analysis of such sequences is in its early stages, although the key characteristics of long times and the potential for arresting the accident at the calandria shell boundary are well recognized. Integrated system models need to be developed to cover the transient behaviour from initiating event to quasi-steady state, supported by small scale experiments [12] aimed at phenomena unique to PHWRs such as channel collapse and core debris retention. 

Linear sequences in quasi-steady state mode... 

We will now focus our attention on linear sequences in quasi-steady state mode, which are of great practical importance in homogeneous kinetics. 

In quasi-steady state mode, for a linear sequence of a homogeneous system we can prove a theorem symmetrical to the very general one obtained for pseudo-stea state modes. 

From the quasi-steady-state for the main reaction (more details on two-step sequence are given in Chapter 4), the following relationship between the coverage of the intermediate and the fraction of vacant sites holds... 

An obvious map to consider is that which takes the state (x(t), y(t) into the state (x(t + r), y(t + t)), where r is the period of the forcing function. If we define xn = x(n t) and y = y(nr), the sequence of points for n = 0,1,2,... functions in this so-called stroboscopic phase plane vis-a-vis periodic solutions much as the trajectories function in the ordinary phase plane vis-a-vis the steady states (Fig. 29). Thus if (x , y ) = (x +1, y +j) and this is not true for any submultiple of r, then we have a solution of period t. A sequence of points that converges on a fixed point shows that the periodic solution represented by the fixed point is stable and conversely. Thus the stability of the periodic responses corresponds to that of the stroboscopic map. A quasi-periodic solution gives a sequence of points that drift around a closed curve known as an invariant circle. The points of the sequence are often joined by a smooth curve to give them more substance, but it must always be remembered that we are dealing with point maps. 

The SRTS sequence consists of a preparatory pulse and an arbitrary long train of the phase-coherent RF pulses of the same flip angle applied with a constant short-repetition time. As was noted above, the "short time" in this case should be interpreted as the pulse spacing T within the sequence that meets the condition T T2 Hd. The state that is established in the spin system after the time, T2, is traditionally defined as the "steady-state free precession" (SSFP), ° and includes two other states (or sub-states) quasi-stationary, that exists at times T2effective relaxation time) and stationary, that is established after the time " 3Tie after the start of the sequence.The SSFP is a very particular state which requires a specific mechanism for its description. This mechanism was devised in articles on the basis of the effective field concept and canonical transformations. Later approaches on the basis of the average-Hamiltonian theory were developed. ... 

Thus, the mechanism of catalytic processes near and far from the equilibrium of the reaction can differ. In general, linear models are valid only within a narrow range of (boundary) conditions near equilibrium. The rate constants, as functions of the concentration of the reactants and temperature, found near the equilibrium may be unsuitable for the description of the reaction far from equilibrium. The coverage of adsorbed species substantially affects the properties of a catalytic surface. The multiplicity of steady states, their stability, the ordering of adsorbed species, and catalyst surface reconstruction under the influence of adsorbed species also depend on the surface coverage. Non-linear phenomena at the atomic-molecular level strongly affect the rate and selectivity of a heterogeneous catalytic reaction. For the two-step sequence (eq.7.87) when step 1 is considered to be reversible and step 2 is in quasi-equilibria, it can be demonstrated for ideal surfaces that... 

Instead of the steady state of (quasi) homogeneous nucleation, during which nuclei appear at constant rate, both the total number of critical nuclei and the duration of the nucleation process are confined by the maximum number of preexisting sites Mo. If there is a uniform probability with time of converting these sites into critical nuclei S — Mno one obtains the following first-order nucleation law as an approximate solution of the reaction sequence (17)... 

Even with a small number of steps, the rate expression of a sequence is rather complicated in spite of the steady-state approximation. A simplification is in order whenever possible. Frequently, for a restricted range of experimental conditions, it can be assumed with considerable success that a given step in the sequence is rate-determining. If this occurs, all the other steps in the sequence will be in quasi-equilibrium and the kinetic problem is reducec to a consideration of the kinetics of a single step and the thermodynamic equilibrium of all the other steps. Sequences that are not amenable to the genera treatment of the preceding chapter can still be treated readily. 

A relatively long sequence of steps, frequently encountered in practice, evidently requires quite a number of rate equations. In many cases one of the steps is intrinsically much slower than the others. A steady state is established in which the rates of the other steps adapt to the rate of this step — it is the rate determining step. For steady state conditions only one rate equation will suffice to describe the process. All the other steps will be in quasi equilibrium. The rate determining step may change with the operating conditions so that care has to be taken when using this concept. The change will be revealed by shifts in the product distribution. 

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