Disturbed region, stability

This can be interpreted as the condition that the temperature within the disturbed region will not diverge exponentially from that in the normal region of the bed. G, R, L and d are fixed parameters for a given reactor configuration and flow rate, hence Equation (8) can be interpreted as a stability criterion which simply states that the normal-bed temperature rise must not exceed some limiting value, otherwise hot spots will develop. 

The condition for stability of a disturbed region is that a solution exist for the system of equations ... 

As discussed in Chapters 3 and 4 of this book, when such conditions prevail, a large disturbance may shift the reactor outside the region of stability of the optimum steady state and towards the stability region of another, but undesirable steady states. When the disturbance is removed, the reactor does not generally restore itself to its original steady state and an irreversible drop of productivity will generally occur. 

It has been shown that the transition from the two peak periodic oscillation to the chaotic behavior occurs with a loss of stability of the periodic oscillation an unstable two peak oscillation is embedded In the chaotic region. During this transition the Liapunov characteristic exponent changes sign from negative to positive. Furthermore, calculations indicate that a small amplitude regular disturbance does not have a significant effect on the character of the oscillations. 

Plug Flow With a significant amount of axial dispersion, Equation (3), describing the normal bed temperature profile, must be modified to account for this dispersion. The effect of this modification is that the ultimate vertical asymptote in temperature is moved forward in the extended bed. Dispersion enhances the tendency of a reactor to run away. However, with the type of dispersion that occurs in a trickle bed, by variations in velocity from point to point, the profile retains its vertical asymptote. The solution of Equation (3) plus dispersion is almost identical with Equation (5), but with a different value of SD. Since SD drops out in the ultimate stability criterion, axial dispersion cannot be of any particular significance in the development of local hot spots. It affects the global stability of the normal part of the reactor, but it has little influence on the way disturbances grow, relative to the normal regions. 

If these conditions hold, then instability first appears as an oscillation when the Rayleigh numbers satisfy (12-282). It should be remembered that the conditions (12-283) pertain only to the question of whether oImag is zero or nonzero at the point of neutral stability. It says nothing about whether the growth of disturbances will be oscillatory or not in the unstable regions where oReai > 0. 

The thermal and mechanical stability criteria (8.1.23) and (8.1.31) apply both to pure fluids and to mixtures however, for homogeneous mixtures, those criteria are not sufficient to identify stable systems because, in addition to energy and volume fluctuations, mixtures have concentration fluctuations. These fluctuations occur in localized regions of a system when material spontaneously aggregates and redisperses. If such fluctuations are not to disturb a system s stability, then the mixture must satisfy a set of conditions known as the material or diffusional stability criteria. These criteria are derived in a manner similar to that given in 8.1.2 for (8.1.23) and (8.1.31), so we only sketch the procedure here. 

The setup for internal reflection spectroscopy is shown in Kg. 3. The electrode consists of a germanium reflection element, covered on one side with PPy in contact with the electrolyte solution. The IR beam is totally reflected at the interface between the reflection element and the PPy layer. The penetration depth of the IR beam into the optical thinner medium (polymer and electrolyte) at each total reflection is in the order of some m. Spectral information on this small region is available without disturbances by the bulk electrolyte. Since the counter electrode and the reference electrode can be mounted in an advantageous arrangement, no electrochemical hmitations, connected with a thin electrolyte layei occur using this method. Due to the limited stability of Ge in aqueous electrochemical systems, the water content of the electrolyte solution has to be very low. 

---