Stationary algebraic equation

For a CSTR the stationary-state relationship is given by the solution of an algebraic equation for the reaction-diffusion system we still have a (non-linear) differential equation, albeit ordinary rather than partial as in eqn (9.14). The stationary-state profile can be determined by standard numerical methods once the two parameters D and / have been specified. Figure 9.3 shows two typical profiles for two different values of )(0.1157 and 0.0633) with / = 0.04. In the upper profile, the stationary-state reactant concentration is close to unity across the whole reaction zone, reflecting only low extents of reaction. The profile has a minimum exactly at the centre of the reaction zone p = 0 and is symmetric about this central line. This symmetry with the central minimum is a feature of all the profiles computed for the class A geometries with these symmetric boundary conditions. With the lower diffusion coefficient, D = 0.0633, much greater extents of conversion—in excess of 50 per cent—are possible in the stationary state. 

If the aim is to determine the stability limit of the trickling flow (that is, the limit of existence of a truly stationary flow), one may neglect the effective viscosity terms in Eqn. (5.2-9) and Eqn. (5.2-10). Furthermore, in trickling flow, all derivatives with respect to time or height (z) are equal to zero a stability analysis of the solution of the system of equations against small variations of t or z yields an algebraic equation which permits to calculation of the limit of liquid mass-flow-rate for a given gas flow rate. 

The behavior of a mixture is determined by a system of ordinary differential equations, while the required state, either equilibrium or stationary, is determined by a time-independent system of algebraic equations. Therefore, at first glance one would not expect any qualitative difference between the equilibrium and stationary states. Ya.B. shows that in the equilibrium case, even for an ideal system, a variational principle exists which guarantees uniqueness. Such a principle cannot be formulated for the case of an open system with influx of matter and/or energy. 

The numbers 0, 1, 2 and 3 of central parts of graph correspond to the working condition of compressor stations therefore to find the probabilities of appropriate conditions is important. In this case concerning the stationary probabilities, the system of the algebraic equations is received from the system of the differential equations (1) ... 

Unlike standard Markovian rate constants these constants depend on the excitation lifetime and light intensity. Using them, we can obtain a set of algebraic equations for the stationary concentrations of reactants and reaction products passing to the limit t —> 00 in Eqs. (3.505) ... 

Any trajectory can end when p - I at a stationary point (SP), in which all the right-hand parts of equations (5.2) equal zero. In the case of the terminal model (2.8) all such SPs are those solutions of the non-linear set of the algebraic equations (4.13) which have a physical meaning. Inside m-simplex one can find no more than one SP, the location of which is determined by the solution of the linear equations (4.14). In addition to such an inner azeotrope of the m-simplex, azeotropes can also exist on its boundaries which are n-simplexes (2 S n m - 1). For each of these boundary azeotropes (m — n) components of vector X are equal to zero, so it is found to be an inner azeotrope in the system of the rest n monomers. Moreover, the equations (4.13) always have m solutions x( = 8is (where 8js is the Cronecker Delta-symbol which is equal to 1 when i = s and to 0 when i =(= s) corresponding to each of the homopolymers of the monomers Ms (s = 1,. ..,m). Such solutions together with all azeotropes both inside m-simplex and on its boundaries form a complete set of SPs of the dynamic system (5.2). 

Now that we have a procedure for finding Tm for given E and a we can determine E time-dependently from Eq. (3.4) or stationary for given M from Eq. (3.6). The latter is a local algebraic equation at each radius r, solved analytically if the opacities are power-law formulae, or numerically otherwise. 

Despite the evident complexities of handling additional initiation or propagation reactions, the algebra of these steps is always first-order in radical concentration and the stationary-state equations, although cumbersome, can always be solved explicitly. However, as soon as we introduce more than one termination reaction involving bimolecular participation of two radicals, the equations become nonlinear and explicit solutions arc not always possible. Suppose for example that we were to include in the simple scheme Eq. (XIII.10.5), the additional termination reactions... 

Any highly reactive intermediate that is and remains at trace level attains a quasi-stationary state in which its net chemical rate is negligibly small compared separately with its formation and decay rates. This is the basis of the Bodenstein approximation, which allows the rate equation of the intermediate to be replaced by an algebraic equation for the concentration of the intermediate, an equation which can then be used to eliminate that concentration from the set of equations. The approximation can be applied in succession for each trace-level intermediate. It is the most powerful tool for reduction of complexity. It is the basis of general formulas to be introduced in Chapter 6 and widely used in subsequent chapters. 

It turns out, in fact, that the large k limit sought leads to an approximation in which the kinetic energy is suppressed and thus the leading contribution to the ground-state energy is determined by the minimum of the effective potential V(u, v). The stationary points of K = V(u, v) are evidently roots of the algebraic equations dV/du = 0 = 6K/0t . Explicitly, one then finds that [25]... 

The application of the z-transform and of the coherence theory to the study of displacement chromatography were initially presented by Helfferich [35] and later described in detail by Helfferich and Klein [9]. These methods were used by Frenz and Horvath [14]. The coherence theory assumes local equilibrium between the mobile and the stationary phase gleets the influence of the mass transfer resistances and of axial dispersion (i.e., it uses the ideal model) and assumes also that the separation factors for all successive pairs of components of the system are constant. With these assumptions and using a nonlinear transform of the variables, the so-called li-transform, it is possible to derive a simple set of algebraic equations through which the displacement process can be described. In these critical publications, Helfferich [9,35] and Frenz and Horvath [14] used a convention that is opposite to ours regarding the definition of the elution order of the feed components. In this section as in the corresponding subsection of Chapter 4, we will assume with them that the most retained solute (i.e., the displacer) is component 1 and that component n is the least retained feed component, so that... 

We may seek stationary values of the parameters jr,- by setting the right-hand sides of these equations to zero. This results in a set of algebraic equations and if there arc finite positive real solutions, these are the stationary solutions. They coirespimd to the balance between aerosol formation and depletion by How through the CSTR. 

By setting the time-derivative of each metaboUte concentration to zero under pseudo-stationary assumption, the set of differential equations for mass balance equations is converted into a system of algebraic equations (see the E-coli.txt file in the folder Chapter 13 on the attached compact disk). Each nonlinear equation contains several rate expressions and terms. The glucose impulse term, fpuise, in the mass balance equation... 

Figure 2.5a and b show points at which the composition profiles originate from, terminate at, and tend toward. These points, interchangeabiy referred to as stationary points, pinch points or nodes, represent compositional steady states in the beaker where the liquid composition within the beaker is no longer changing with time, and may be determined by solving the system of nonlinear algebraic equations (see Equation 2.8) such that... 

This approximation resolves the computational difficulty encountered in the direct exact formulation that requires repeated computations of the solution of linear simultaneous algebraic equations and determinants of the matrices with huge dimensions. The efficiency in the approximated expansion is gained by the appreciation that the conditioning information can be truncated within one period of the system only. For linear systems, the expressions for the reduced-order likelihood function p(yi, yj, - - -, yNp W, C) and the conditional PDFs p(.yn 0, yn-Np, yn-Np+1, , y -i, C) are available since they are Gaussian and the correlation functions are known in closed forms regardless of the stationarity of the response. For stationary response, the method is very efficient in the sense that evaluation of all the conditional PDFs p(ynW, yn-Np,yn-Np+i,, y -i, C) requires the inverse and determinant of two relatively small matrices only. 

For example, assumingyl = 1 x s,, EJR = 5,000 K,/= 100Km kmol T = 60 s, initial concentration Cao — 1 kmolm and initial temperature Tq = 270 K, we can find three stationary states. Their quantitative properties are determined by the solutions of an algebraic equation set, to which the differential equation set is transformed when both dCA t)/dt2LnAdr i)/dt equal zero ... 

One would obtain the pseudo-stationary state concentrations for each of the intermediate chemical species from the thermodynamics of irreversible processes if it were supposed that the relaxation time for the production or destruction of each of these species is very small compared to the half-time for the overall chemical reaction of the fuel to go to the product molecules. In the pseudo-stationary states approximation, the net rate of formation, Kp of each of the intermediate chemical species by chemical reactions is set equal to zero. This provides exactly the right number of simultaneous algebraic equations to express the concentration of each of the chemical intermediates in terms of powers of the concentrations of the fuel and product molecules. For example, in the hypothetical chain system given by Eqs. (130), (131), and (132), the pseudo-stationary mole fraction of B (which we shall designate as x ) is the solution of the equation ... 

The system of integral equations [Eq. (66)] is eventually discretized and solved with numerical linear algebra procedures. At each energy, the system (66) must be solved for each of the open channels. A complete set of linearly independent degenerate real (i.e., stationary) continuum solutions if"E is thus obtained. The stationary scattering states xjr E are not orthogonal it can be shown that their superposition is given by... 

In addition to openness and feedback, a third condition is often found in conjunction with oscillation, though it has not yet been proven to be necessary the existence of multiple stationary states. Most chemists are aware of the steady state condition, wherein the rate of change of the concentration of an intermediate can be equated to zero. Thus, the concentration of this intermediate can be obtained by solving an algebraic, rather than a differential equation. The familiar solution to this equation is a single concentration for a given set of initial conditions. [The reader may be familiar with a mechanism k k... 

The mathematical techniques most commonly used in chemical kinetics since their formulation by Bodenstein in the 1920s have been the quasi-stationary state approximation (QSSA) and related approximations, such as the long chain approximation. Formally, the QSSA consists of considering that the algebraic rate of formation of any very reactive intermediate, such as a free radical, is equal to zero. For example, the characteristic equations of an isothermal, constant volume, batch reactor are written (see Sect. 3.2) as... 

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