Solution middle-branch

For the adiabatic condition in which RHL is suppressed, the flame response exhibits the conventional upper and middle branches of the characteristic ignition-extinction curve, with the upper branch representing the physically realistic solutions. It can be noted that the effective Le of this lean methane/air mixture is sub-unity. It can be seen from Figure 6.3.1 that, with increasing stretch rate, first increases owing to the nonequidiffusion effects (S > 0), and then decreases as the extinction state is approached, owing to incomplete reaction. Furthermore, is also expected to degenerate to the adiabatic flame temperature, when v = 0. 

These two latter solution branches are connected via a turning point in the VI plane at V = Va. The current Iu associated with the upper solution branch grows unboundedly as V — oo, while the current Im associated with the middle branch decreases with increasing V (negative differential... 

Stability of the described solution branches (the middle branch with negative differential resistance is expected to be unstable). 

Fig. 6.9. Domains of attraction for the two stable branches of stationary-state solution systems which have initial extents of conversion lying within the shaded region evolve to the lower branch those in the unshaded region approach the highest extent of reaction. In the region of multiple stationary states, the middle branch acts as a separatrix.

It may also seem sensible, if there are multiple solutions, to ask which of the states is the most stable In fact, however, this is not a valid question, partly because we have only been asking about very small disturbances. Each of the two stable states has a domain of attraction . If we start with a particular initial concentration of A the system will move to one or other. Some initial conditions go to the low extent of reaction state (generally those for which 1 — a is low initially), the remainder go to the upper stationary state. The shading in Fig. 6.9 shows which initial states go to which final stationary state. It is clear from the figure that the middle branch of (unstable) solutions plays the role of a boundary between the two stable states, and so is sometimes known as a separatrix (in one-dimensional systems only, though). 

Thus the upper root of eqn (8.10), which gives the middle branch of stationary-state solutions and requires the minus sign above, has a negative value for trelax. It then follows that the eigenvalue A for this branch is positive, so perturbations grow. This is an unstable state. 

Fig. 8.3. The approach to, or departure from, stationary-state solutions following small perturbations for simple cubic autocatalysis again showing the instability of the middle branch. The turning points (ignition and extinction) have one-sided stability as perturbations in one direction decay back to the saddle-node point, but those of the opposite sign depart for the other...

For the present scheme, when there is a unique stationary state, we find Apr < 0 and local stability. Under circumstances with multiple solutions, the highest and lowest states always have Apr < 0 and hence are stable the middle branch of solutions has Apr > 0 and hence is a branch of unstable saddle points. 

The Hopf bifurcation analysis proceeds as described previously, the required condition being that the trace of the Jacobian matrix corresponding to eqns (12.45) and (12.46) should become equal to zero for some stationary-state concentration given by the lower root from (12.51). (The solution with the upper root corresponds to the middle branch of stationary states for... 

In Figure 5.14 note that for this set of Peclet numbers the middle uniformly dotted branch steady-state instabilities seem not to be as severe as before. There is only one gap in the family of solutions where double stiffness is an issue. To corroborate, note that the middle branch of the plot takes only 135 individual BVPs, each indicated by a pair of green circles on a computer screen and a pair of gray circles in Figure 5.14. This represents about two thirds of our earlier effort for Pejj = 4 and PeM = 8 for obtaining these graphs. Each BVP now takes about 3 seconds on average. 

These indicate the limit of our successful numerical BVP integrations near the bifurcation points. In between the x and o marks on the middle branch, the curve is drawn using interpolation of our successful BVP solutions data, while in between two adjacent x or two adjacent o marks, the curve is drawn by extrapolating nearby computed function data. This is done automatically by MATLAB s plot commands. 

Consider again the steady-state temperature versus the residence-time curve shown in Figure 6.9 We have labeled seven points on this cun e A, B and C on the lower branch D on the middle branch and E, F and G on the upper branch. The numerical values at these points are listed in Table 6.2. If we first substitute the solution of the mass balance, Equation 6,.36, for CA into the energy balance, Equation 6.37, we obtain... 

Fig. 40.2 Possible scenarios for particle formation via Emulsion Combustion Method. Right branch, each micro-solution droplet produces one particle. Left branch, shrinkage of the emulsion droplet is faster than that of micro-solution droplets, and therefore, some of the micro-solution droplets merge to form larger particles. Middle branch. microexplosion of the emulsion droplet and the formation of smaller emulsion droplets. (Reprinted from [2] with permission. Copyright 2009 of Bentham Science Publishers)...

If an emulsion droplet undergoes microexplosion, while the suspended microsolution droplets are not formed yet, then several smaller emulsion droplets will form. These secondary emulsion droplets probably contain smaller micro-solution droplets than the original ones. As a result of microexplosion, compared to the case where microexplosion is absent, smaller particles will form (see Fig. 40.2, middle branch). 

Figure 2.70. Typical examples of chromatograms. The solution is branched polyethylene in tetrahydrofuran. Top light-scattering intensity 1 (LS 90°) and refractive index difference A (Rl). Middle molecular weight M. Bottom radius of gyration R, plotted as a function of the retention volume Vr. (From Ref. 9.)...

When there are multiple solutions, the middle branch is always unstable. 

If C 1 the dynamics of the system is very much determined by the shapes and positions of the nullclines of Equation (32), i.e. the curves j//(x) (x-nullcline) and yg x) (y-nullcline) that are solutions to the Equations /(x, y) = 0 and g(x, y) = 0, respectively. Often, the x-nullcine is N shaped (Figure 10). It consists of three branches the left (AB), middle (BC) and right (CD) branch. The steady state of system (32) is the intersection of the x-nullcline and the y-nullcline. Normally, when the intersection occurs at the middle branch (BC) df x, y) jdx > 0 - for this reason this branch is called an unstable branch -and for sufficiently small e this steady state is unstable. If this steady state is unique the system exhibits limit cycle oscillations. 

In Fig. 4.3.3b we present a V — I curve with a turning point and a negative differential resistance region with current saturation, computed for the same values of parameters Ni, A, cq as in Fig. 4.3.3a. This V—I curve corresponds to the upper and the middle solution branches. The range of parameters in which the high current solutions exist is again evaluated below, via an asymptotic treatment for /— oo. 

Samples of branched polyethylene were investigated by the combination of crystal-lizability fractionation and fractionation by molar mass. The first step was the precipitation of the polymer sample onto the glass-beads in a column (150 x 8 mm) used for the fractionation by crystallinity. The precipitation was performed by cooling a solution of 10 g/1 polyethylene in o-dichlorobenzene from 140 to 40 °C within 60 min. Subsequently, the polymer was extracted from the column by the same solvent at stepwise increased temperature. In an example given, the first fraction was eluted at 40 °C and additional 17 fractions at temperatures each raised by 2-10 K. The finer steps were employed in the middle of the fractionation procedure the last one reached even from 110 to 140 °C. About 10 min equilibration time proved to be adequate. The fractions were analyzed subsequently by SEC on a polystyrene-gel column. The whole process was automated 128). 

Si NMR Spectroscopy of Colloidal Silica. 29Si NMR spectroscopy has been used extensively to study aqueous solutions of silicates and has provided detailed information on the types of polymeric species in solution (3-5). However, despite the power of the technique it has not been exploited previously to investigate colloidal silica. The usefulness of the technique arises from the successive shift of the 29Si resonance to high field on replacement of Si-OH bonds in silicate species with siloxy bonds, Si-O-Si. Thus monomeric species are observed furthest downfield (Q°) and other resonances are observed at intervals of —10 ppm for end units (Q1), middle units (Q2), branching units (Q3), and tetrafunctional units (Q4). (Q refers to four possible coordination bonds the superscript refers to the actual number.) From the relative intensities of the different... 

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