Concentration space dimensionality

Deterministic dynamics of biochemical reaction systems can be visualized as the trajectory of (ci(t), c2(t), , c v(0) in a space of concentrations, where d(t) is the concentration of ith species changing with time. This mental picture of path traced out in the N-dimensional concentration space by deterministic systems may prove a useful reference when we deal with stochastic chemical dynamics. In stochastic systems, one no longer thinks in terms of definite concentrations at time t rather, one deals with the probability of the concentrations being xu x2, , Wy at time t ... 

A -h B 0 reaction (provided equal reactant concentrations). Peculiarity of the diffusion-controlled regime of reaction is the existence of the marginal space dimensionality do-... 

Glasser et al. (1987) and Hildebrandt et al. (1990) demonstrated this two-dimensional approach on a number of small reactor network problems, with better results than previously reported. Moreover, Omtveit and Lien (1993) were able to consider higher-dimensional problems as well through projections in concentration space that allow a complete two-dimensional represention. These projections were accomplished through the principle of reaction invariants (Fjeld et al., 1974) and the imposition of system specific constraints. 

The segregated-flow model described by (P2) forms a basis to generate an AR. We now develop conditions for the closure of this space with respect to the operations of mixing and reaction by means of a PFR, a CSTR, or a recycle PFR (RR). Consider the region depicted by the constraints of (P2). Our aim is to develop conditions that can be checked easily for the reaction system in question so that, if these conditions are satisfied, we need to solve only (P2) for the reactor targeting problem. We will analyze these conditions based on PFR trajectories projected into two dimensions. Here, a PFR, which is an n-dimen-sional trajectory in concentration space and parametric in time, is generated by the solution of the initial value differential equation system in (PI). Figure 3 illustrates a PFR trajectory and its projections in three-dimensional space, where the solid line represents the actual PFR trajectory and the dotted lines represent the projected trajectories. 

S.2 Higher Dimensions Take as another example a set of points given in three-dimensional concentration space. This might occur if a third component, in addition to NaCl and KOH, were introduced into the system. There are now more ways in which to move in space via mixing. 

Table 2.8 shows a set of 15 points in six-dimensional concentration space. Using a convex hull program, determine which points would form part of the convex hull. 

These concepts were demonstrated with simple thought experiments in two-dimensional concentration space, although, these ideas are valid for higher dimensions as well. When dealing with higher dimensions, visualizing the full set of data is not always intuitive, and so we must be comfortable with relying on the convex hull to determine unique points in space. 

We can integrate the system of ODEs in the usual manner, to produce a set of points that represent a solution function C(t), which may be plotted in concentration space. This is given in Figure 4.10, where a hypothetical two-dimensional solution trajectory for C(t) in c -Cg space is shown. 

If a rate vector exists on the AR boundary that points out of the region, then it would be possible to expand the region by initiating a PFR from that point. Figure 4.33 shows three points on a hypothetical AR boundary in two-dimensional concentration space. 

Assume that the feed is pure in A with a feed concentration of CAf= l.Omol/L. Since the kinetics is a function of only components A and B, it is possible to construct the AR in two-dimensional concentration space (c - Cg). Hence, we aim to construct the AR in c - Cg space using a feed vector of Q= [c f, CBf] = [1.0, 0.0] mol/L for three distinct scenarios using dimensionless rate constants aj and Si2-... 

I Reaction and Feed Specification All of the examples described in this chapter possess equivalent versions involving residence time as well. If it is desired to retain the same components as before, and also include residence time in the state vector, then the resulting AR must be of one dimension higher than that originally posed— two-dimensional problems in concentration space are thus three-dimensional problems if residence time is considered as well. With this in mind, consider now the single autocatalytic reaction involving components A and B... 

Observe that columns 1 and 2 of A are linearly dependent (column 2 is the negative of column 1). Calculation of the rank of A gives 2, indicating that only two independent reactions are preset in this system. The columns in A thus describe a two-dimensional subspace in R" concentration space in this instance. 

Notice that if we substitute matrix X discussed in this example with the stoichiometric coefficient matrix A, then the columns in X (Xj and X2) represent two reactions participating in n-dimensional concentration space, R". Hence, to compute N, we simply determine the stoichiometric coefficient matrix A (as in Section 6.2.1.3), and then compute the null space of A. From linear algebra, we can show that if A has size nxd (n components participating in d reactions), the size of matrix N will benx(n-d). 

We can hence calculate the extreme points of the region defined by Equation 8.2b in extent space by vertex enumeration. The set of extents forming the extreme points of the stoichiometric subspace is found to be [1.0, 0.0], [-0.25, 0.75]T, [0.0, 1.0]T, [0.0, 0.0]. From this set, equivalent extreme points in concentration space may be computed using Equation 8.1. The stoichiometric subspace resides as a two-dimensional subspace in IR . This subspace may be projected onto different component spaces for visualization. A number of example component pairs are shown in Figure 8.4(a). 

For example, a triangle, representing 8 in two-dimensional concentration space, might be viewed as a triangular prism in C-t space as in Figure 8.6. A similar argument applies to temperature, when temperature can be assumed to obey a linear mixing law. 

The system involves three independent reactions with four components. It follows that the AR is a three-dimensional subspace in Later on, it will be useM to provide a comparison of the AR generated in this chapter in mass fraction space, to that produced in Chapter 7 originally in concentration space. For this reason, the AR shall be generated in z -Zb-Zjj space. Components C may be found by mass balance. The mass fraction and rate vectors are then defined as z = [z, Zg, Zjj] and r(z) = [r (z), rg(z), rjj(z)] . It is assumed that the feed available is pure in component A. The feed molar flow rate vector is hence given as = [1, 0, 0]. Since the feed is pure in A, it follows that the mass fraction feed vector be given as Zj = [1,0,0]. ... 

Introduction All of the examples up to this point have approached construction of the AR for various systems under different conditions. In the initial chapters of the book, we investigated lower dimensional systems under constant density, isothermal operation in concentration space. We have slowly relaxed many of these assumptions throughout the course of the book. In this final example, we wish to show how the constmction of the AR for a more realistic system might be addressed. 

Assume that there is a chemical kinetic system described by Eqs. (1) and (2), We decompose the concentration space into 2 volumes, which are arranged in space so that they are homeomorphic (topologically equivalent) to the 2 orthants in the neighborhood of the origin in an N-dimensional coordinate space. There are boundaries between the contiguous... 

One very important mathematical result facilitates the analysis of two-dimensional (i.e., two concentration variables) systems. The Poincare Bendixson theorem (Andronov et al., 1966 Strogatz, 1994) states that if a two-dimensional system is confined to a finite region of concentration space (e.g., because of stoichiometry and mass conservation), then it must ultimately reach a steady state or oscillate periodically. The system cannot wander through the concentration space indefinitely the only possible asymptotic solution, other than a steady state, is oscillations. This result is extremely powerful, but it holds only for two-dimensional systems. Thus, if we can show that a two-dimensional system has no stable steady states and that all concentrations are bounded—that is, the system cannot explode—then we have proved that the system has a stable periodic solution, whether or not we can find that solution explicitly. 

Concentration space of an n-component mixture C is a space in which every point corresponds to a mixture of definite composition. Dimensionality of concentration space corresponds to the number of concentrations of components that can be fixed independently. 

The (n - 1) concentration for an n-component mixture can be fixed independently because concentration of the nth component can be found from Eq. (1.2). That is why the dimensionality of the concentration space of binary mixture Ci is one, of ternary mixture C3 - two, of four-component mixture C4 - tree, etc. 

Equation (1.5) represents the so-called lever rule points xip, Xio, and xib are located on one straight line, and the lengths of the segments [xif, xid and [xib, Xip] are inversely proportional to the flow rates D and B (Fig. 1.1b). Mixture with a component number n > 5 cannot be represented clearly. However, we wiU apply the terms simplex of dimensionality (n -1) for a concentration space of n-component mixture C , hyperfaces C i of this simplex for (n - l)-component constituents of this mixture, etc. 

Trajectory bundles of bottom and intermediate sections in the mode of minimum reflux should join with each other in the concentration space of dimensionality (n - 1). Therefore, joining is feasible at some value of the parameter (L/T) if the summary dimensionality of these bundles is equal to (n - 2). 

The phase singularity is dimensionally trivial in the simplest examples. In a limit-cycle reaction implicating only two chemical concentrations, in the absence of diffusion, it is a reaction steady-state, a state of zero amplitude of the oscillations going on all around it (at azimuthally staggered phase, thus averaging out at the center). If there are A > 2 reactants, the singularity in the A-dimensional concentration space has codimension 2 it is a set of... 

It is convenient to analyse tliese rate equations from a dynamical systems point of view similar to tliat used in classical mechanics where one follows tire trajectories of particles in phase space. For tire chemical rate law (C3.6.2) tire phase space , conventionally denoted by F, is -dimensional and tire chemical concentrations, CpC2,- are taken as ortliogonal coordinates of F, ratlier tlian tire particle positions and velocities used as tire coordinates in mechanics. In analogy to classical mechanical systems, as tire concentrations evolve in time tliey will trace out a trajectory in F. Since tire velocity functions in tire system of ODEs (C3.6.2) do not depend explicitly on time, a given initial condition in F will always produce tire same trajectory. The vector R of velocity functions in (C3.6.2) defines a phase-space (or trajectory) flow and in it is often convenient to tliink of tliese ODEs as describing tire motion of a fluid in F with velocity field/ (c p). 

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