Multiplicity of solution

The mechanism of these transitions is nontrivial and has been discussed in detail elsewhere Q, 12) it involves the development of a homoclinic tangencv and subsequently of a homoclinic tangle between the stable and unstable manifolds of the saddle-type periodic solution S. This tangle is accompanied by nontrivial dynamics (chaotic transients, large multiplicity of solutions etc.). It is impossible to locate and analyze these phenomena without computing the unstable, saddle-tvpe periodic frequency locked solution as well as its stable and unstable manifolds. It is precisely the interactions of such manifolds that are termed global bifurcations and cause in this case the loss of the quasiperiodic solution. 

To prevent internal flow rates having zero shadow costs at the solution and therefore to avoid a multiplicity of solutions, a penalty of 0.05 times the price is incurred for each ton transported between a source and a destination. 

The multiplicity of solutions at the continuum level can be viewed as arising from a constitutive deficiency in the theory, reflecting the need to specify additional pieces of constitutive information through some kind of phenomenological modeling (see, for instance, Truskinovsky, 1987 Abeyaratne and Knowles, 1991). Here we take a different point of view and interpret the nonuniqueness as an indicator of essential interaction between macro and micro scales. 

In the models discussed here we have considered primarily as bifurcation parameters the affinity of reaction as measured by the parameter B in model (1) or the length l as in Section VI. The results have illustrated that when B or l increase, the multiplicity of solutions increases. This is not astonishing, as a variation in length is a simple way through which the interactions of the reaction cell with its environment can be increased or decreased. 

In parallel, the simplicity of handling the locally electro-neutral case provides a convenient ground for studying the far less tractable one-dimensional version of the nonreduced system (4.1.1)-(4.1.2), asymptotically for small e. For an example of such a study we refer to [14] where the difficult question of multiplicity of steady solutions of the nonreduced system was approached through studying the multiplicity of solutions in the LEN approximation for a four layer (quadrupolar) arrangement. The theory of a bistable electronic device (thyristor) which resulted from this study will be presented in 4.3. 

The b.v.p. (4.3.1)-(4.3.3) is solved explicitly, following the scheme of the previous section, and it is shown that for a certain range of the parameters Aj, TVj, V multiplicity of solutions of (4.3.1)-(4.3.3) occurs. The appropriate multiple solutions may be described as follows. There exists a lower solution branch, which is explicitly constructed in the asymptotic limit co — 0. This lower solution appears right at equilibrium (V = 0) and whenever multiplicity occurs. The electric current I,... 

Typical values for the parameters D and x u might be D = 0.05 and ku = 0.01. We now examine how the stationary-state concentration profiles ass(p) and /Jss(p) depend on the dimensionless concentration of the precursor reactant, p0. Figure 9.10 shows the stationary-state concentrations at the centre of the reaction zone ass(0) and / ss(0) as functions of p0. These loci each draw out a hysteresis loop, with a range of corresponding multiplicity of solutions. 

Communications on the theory of diffusion and reaction-V Findings and conjectures concerning the multiplicity of solutions. (with I. Copelowitz). Chem. Eng. Sci. 25, 906-909 (1970). 

Although most engineering systems have well defined unique optima, multiplicity of solutions resulting from the use of variational methods in optimal control problems may appear as shown by Luus and Cormack (1972). The uniqueness can be checked using different initial guesses, for example, for the correction factor of the Hamiltonian. 

McGreavy and Thornton [134] included the film surrounding the particle into their analysis of the multiplicity of solutions. With gradients over the film, multiplicity is possible even when the particle is isothermal, of course. As mentioned in Chapter 3, Luss [135] came to the following criterion for uniqueness for a first-order reaction ... 

The existence of two branches points out the multiplicity of solutions, which become even more complicated along the ZNS, ZNS boundaries, also due to the discontinuities which evolve from laminar-turbulent flow regime transitions in either of the two phases (and the associated change in the shear stresses). Note, that for low gas holdup (H/D > 0.5), the destabilizing effect of may be dampened even when the gas flow is turbulent due to the strong effect of the upper wall on the turbulent structures. Therefore, the ZNS line in Figure 8 is extended (beyond gas phase L-T transitional boundary) up to the H/D = 0.5 line. 

Marek M, Hlavacek V. Modelling of chemical reactors. VI. Heat and mass transfer in a porous catalyst particle on the multiplicity of solutions for the case of an exothermic zeroth-order reaction. Collection of Czechoslovak Chemical Communications 1968 33 506-517. 

The loss of stability of a nonequilibrium state can be analyzed using the general theory of stability for solutions of a nonlinear differential equation. Here we encounter the basic relationship between the loss of stability, multiplicity of solutions and symmetry. We also encounter the phenomenon of bifurcation or branching of new solutions of a differential equation from a particular solution. We shall first illustrate these general features for a simple nonlinear differential equation and then show how they are used to describe far-from-equilibrium systems. 

Note the multiplicity of solutions related to symmetry. When a solution does not possess the symmetries of the differential equation, i.e. when a —a, it is said to be a solution with a broken symmetry or a solution that has broken the symmetry. In this case the solution a = 0 is invariant when a is replaced by —a. 

Here X is the nonequilibrium constraint. If the system under consideration is a homogeneous chemical system, then Zk is specified by the rates of chemical reactions. For an inhomogeneous system, Zk may contain partial derivatives to account for diffusion and other transport processes. It is remarkable that, whatever the complexity of Zk, the loss of stability of a solution of (19.2.4) at a particular value of A, and bifurcation of new solutions at this point are similar to those of (19.2.1). As in the case of (19.2.1), the symmetries of (19.2.4) are related to the multiplicity of solutions. For example, in an isotropic system, the equations should be invariant under the inversion r —r. In this case, if Xk[r,t) is a solution then Xk -r,t) will also be a solution if Xk(r,t) Xk —r,t) then there are two distinct solutions which are mirror images of each other. 

These are called the bifurcation equations. In fact, though (19.2.1) is an equation in its own right, it is also a bifurcation equation for systems that break a two-fold symmetry. The multiplicity of solutions to (19.2.6) corresponds to the multiplicity of solutions to the original equation (19.2.4). 

In this manner, instability, bifurcation, multiplicity of solutions and symmetry are all interrelated. We shall now give a few detailed examples of instability of the thermodynamic branch leading to dissipative structures. 

The locally stable solutions of the TV-charge surface Coulomb problem are constrained solely by spherical boundary conditions and the 0(4) symmetry of the Coulomb interaction. The exponential growth of the multiplicity of solutions—M(N) e Eq. (3.2a)—shows that these restrictions are compatible with a great variety of geometric structures. Only in the simplest systems is there an overlap with the criteria of strict regularity... 

Fig. 11.4. Multiplicity of solutions for an exothermal reaction performed in an adiabatic CSTR. The straight line represents the heat balance and the S-shaped line the mass balance.

A further specific aspect of the reactor heat balance is the multiplicity of solutions to the system of equations. This situation may arise with a CSTR in which an exothermal reaction is performed. The mass balance [Eq. (12)] is coupled with the heat balance [Eq. (13)], which gives a system of equations [Eq. (14)] that is represented graphically in Figure 11.4. 

The point above can be made plausible for the ground state when one considers certain properties of the first eigenfunction. It is not only valid for the ground-state, but also for a threefold infinite multiplicity of solutions. I do not wish to base this on special considerations, since it is the immediate result of our method of solution. 

Note that multiples of equation (4.23) appear in equations (4.24) and (4.25), because the points corresponding to them, namely, (1, —2), (2, —4), (3, —6) all lie on a common ray within the solution domain. This will also occur with other solutions, being a natural consequence of the fact that equations (4.13) give all solutions of (4.10), including multiples of solutions. 

The multiplicity of solutions at finite time values may be analyzed more explicitly by applying the following treatment of Amundson and Raymond [17] to noncatalytic gas-solid reactions in a slab. Equation (4.3.100) may be written for an infinite slab as follows ... 

Schaefer et al. ( ) theoretically examined the stability of a coimtercurrent shaft furnace by treating a simple step change in heat generation rate and reported the multiplicity of solution depends on model parameters and boundary conditions. Mori and Muchi (83) treated the case of a first order reaction occurring in a catalytic moving bed and examined the reactor stability. 

The multiplicity of solutions can be reduced by considering the isotopic invariance property (within the Born - Oppenheimer approximation), of the dipole moment derivatives with respect to Cartesian projection of bonds. 

It is necessary to explore the multiplicity of solutions of the model and the validity of extrapolation for predicting the maximum boiling point, although this last fact could not be a critical issue. 

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