Equilibrium manifold

Tang, Q. and S. B. Pope (2002). Implementation of combustion chemistry by in situ adaptive tabulation of rate-controlled constrained equilibrium manifolds. In Proceedings of the Combustion Institute, vol. 29, pp. 1411-1417. Pittsburgh, PA The Combustion Institute. 

X2 will quickly reach its quasi-steady-state value ay = 0, and a 2 0 constitutes an equilibrium manifold of this system. 

Within the equilibrium manifold, ay will slowly evolve towards the equilibrium point (0,0) of the entire system. 

Figure 2.4 presents the trajectories of the two concentrations in the phase plane, revealing the presence of the equilibrium manifold the phase trajectories starting from any initial condition (aq o,ay o) [0,1] x [0,1] approach the horizontal line a 2 = 0, followed by convergence towards the equilibrium point (0,0). 

Clearly, in the second case, the rate of change of concentration of the two reactants is identical and the equilibrium manifold is not present in the phase portrait, confirming the absence of a two-time-scale behavior. 

This result also lends itself to an intuitive interpretation temperature equilibration is a fast phenomenon and T = T2 - a line in the (T), T2) coordinate system -is the equilibrium manifold of the fast dynamics. 

Figure 2.7 reveals the presence of the equilibrium manifold phase trajectories approach the Ti=T2 line and converge along this line to the equilibrium point T1=T2=Te = 273 K. 

This chapter has reviewed existing results in addressing the analysis and control of multiple-time-scale systems, modeled by singularly perturbed systems of ODEs. Several important concepts were introduced, amongst which the classification of perturbations to ODE systems into regular and singular, with the latter subdivided into standard and nonstandard forms. In each case, we discussed the derivation of reduced-order representations for the fast dynamics (in a newly defined stretched time scale, or boundary layer) and the corresponding equilibrium manifold, and for the slow dynamics. Illustrative examples were provided in each case. 

From physical considerations, at most C equations (with C being the number of chemical components) are required in order to completely capture the above overall, process-level material balance. Thus, we can expect the dimension of the system of equations describing the slow dynamics of the process to be at most C, and the equilibrium manifold (3.12) of the fast dynamics to be at most C-dimensional. 

Remark 3.1. In contrast to the theory presented thus far (Section 2.3), the algebraic constraints of (3.12) incorporate a set of (unknown) manipulated inputs, u1. The equilibrium manifold described by (3.12) is thus referred to as control-dependent. 

Equation (5.12) effectively corresponds to the dynamics of the individual process units that are part of the recycle loop. The description of the fast dynamics (5.12) involves only the large flow rates u1 of the recycle-loop streams, and does not involve the small feed/product flow rates us or the purge flow rate up. As shown in Chapter 3, it is easy to verify that the large flow rates u1 of the internal streams do not affect the total holdup of any of the components 1,..., C — 1 (which is influenced only by the small flow rates us), or the total holdup of I (which is influenced exclusively by the inflow Fjo, the transfer rate Af in the separator, and the purge stream up). By way of consequence, the differential equations in (5.12) are not independent. Equivalently, the quasi-steady-state condition 0 = G (x)u corresponding to the dynamical system (5.12) does not specify a set of isolated equilibrium points, but, rather, a low-dimensional equilibrium manifold. 

Consequently, the steady-state condition associated with the fast dynamics specifies a six-dimensional equilibrium manifold in which a slower dynamics evolves. 

The choice of manipulated input for the supervisory controller (the setpoint of a controller that belongs to the primary control structure used to stabilize the fast dynamics) is dictated by the low number (more precisely, one - the liquid product flow rate) of stream flow rates available as manipulated inputs in the intermediate time scale. The implementation of the resulting cascaded control structure is more elaborate from a technical point of view, since the equilibrium manifold of the fast dynamics (5.41) becomes control-dependent. We used the method proposed in Contou-Carrere cl, al. (2004) as discussed earlier in the book to overcome this difficulty. 

From the preceding discussion regarding the equilibrium manifold of the fast component of the system dynamics having a maximum dimension of one, we can infer that, in effect, G Q1 C IR1. 

The munber of theoretical and reactive stages is determined from the distillation line and from the intersection of the distillation line and chemical equilibrium manifold (GEM) and represents the boimdary of the forward and backward reactions) (Giessler et al., 1999). Since there are multiple pairs of X and product composition that satisfy the mass balance, the method sets one of the product composition as reference point and solves for the other two (for a 3-component system) by using material balance expressions. Thus, two of the components compositions and X lie on the same line of mass balance (LMB) in the diagram and allow the estimation of the ratio D/B at a certain reboil ratio only by exploring the ratio of the line segments (figure 3.1f>). 

From Eq. (1.6.39) it is obvious that the quantity Ajfj is positive at all points outside the equilibrium manifold. This means that in chemical reaction systems with kinetics given by Eq. (1.6.34), each reaction individually leads to positive entropy production, a result beyond the requirements of Postulate 1.5.1. 

The equilibrium state u in a given manifold r Uo) is the intersection of r(Uo) with the kinetic equilibrium manifold Q, defined by Eq. (1.5.4). In terms of the variables, the equilibrium state is the solution of y( )==0. Obviously, the equilibrium state in distributed systems depends on the transport properties. In some cases of empirical kinetics, such as in Example 2.S2, there may exist more than one equilibrium states. 

In the case of a reversible reaction, fj has no fixed sign, need not be monotonic, and sustained oscillations cannot be ruled out except by introducing entropy considerations. Suppose that there exists a positive definite form (t(u), vanishing only on the equilibrium manifold and satisfying a conservation equation of the type... 

It is obvious that the kinetic and equilibrium manifolds coincide. By introducing the expressions for // in Eq. (2.8.51) there is obtained... 

Example 2.8.2. Two Irreversible Exothermic Reactions. This will be a continuation of the analysis on the physical system treated in Examples 2.4.2, and 2.7.2. In contrast with Example 2.8.1, the rate expressions (2.7.12) and (2.7.13) are not consistent with thermodynamics. Indeed, the kinetic equilibrium manifold /i=/2 = 0 consisting of two disjoint pieces... 

Kinetic equilibrium point 15 Kinetic equilibrium manifold 19 Kinetics 2 Knudsen diffusion 44... 

Lee et al. have used a relay metathesis trigger to further enable their studies of metallatropic [l,3]-shifts. For example, both of the relay-activated diene-diynes 98 underwent CM with (Z)-l,4-diacetoxy-2-butene (99) to give ene-diyne 100 (Scheme 9.24) [30]. This process was designed to involve insertion of ruthenium onto the relay subunit in 98 to give 101, followed by rapid migration of [Ru] to the isomeric species 102 and 103 and drainage of the most reactive of these, 103, from the equilibrium manifold by final CM with 99. 

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