Oscillation Processes Modeling

In some reactions, one can see the periodic changes of the reagent concentrations over time. Correspondingly, the rate of the reaction has an oscUlating character. 

3 Numerical Solution of the Direct Problem in Chemical Kinetics 

Such reactions are called oscillating or periodic. Nowadays several dozens of homogeneous and heterogeneous oscillating reactions have been explored. Investigations of the kinetic models for these complex processes have allowed formulating a series of general conditions, which are required for the stable oscillations of the reaction rates and intermediate concentrations  

These conditions are required but not sufficient for the oscillation to occur in the system. An important role is played also by the ratio between the rate constants of certain steps and starting reagent concentrations. An investigation of the oscillating reactions is still an important chemical kinetics problem because it is crucial in understanding catalysis, periodic process laws for living systems, and chemical technology. 

Sometimes chemical problems can be answered using the knowledge from other sciences that are not related to chemistry at first sight. For example, some information about a complex reactions flow can be gained from the mathematical models of the interspecific competition. A classical example is the predator-prey model, which describes the population trends for predators and prey in living conditions 

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Produced process models can be used for the design of measuring devices based on electromagnetic oscillation effect in the first case and based on charged particle lodging area definition in second. The equations decribing the motions in thermoelectric field have the following form ... 

In later work, Roelofs and co-workers discovered further details of the reaction by investigating the sub-systems, and they suggested a 21 step chemical model (RWJ model) to explain the observed non-linear kinetic patterns (163). According to the experimental observations, the oscillation process can be divided into two distinct alternating stages, the stoichiometries of which can be approximated as follows ... 

The blend of the two active ingredients (B1 and B2) is slugged and then the slugs are oscillated. Slugger model and tooling are listed in the batch instructions. The thickness of the slug is specified, but no information is recorded on the slugging operation, as control of this procedure is left to the experience of the press operator. The batch record permits the use of only one screen size. Since all of the batches have been made in the same manner, this important process step will not be included as one to be studied. 

These processes can be explained by using a harmonic oscillator-like model (Fig. 13(b)). The energy transfer of an electron excites the molecule to a higher energy level and subsequent energy transfer by other electrons causes further excitation in a sequential process. The molecule dissociates when it exceeds the dissociation barrier. 

Control of industrial polymerization reactors is a challenging task because, in general, control engineers lack rigorous polymerization process knowledge, process model, and rapid online or inline sensors to measure polymer properties. Exothermic polymerization processes often exhibit strongly nonlinear dynamic behaviors (e.g., multiple steady states, autonomous oscillations, limit cycles, parametric sensitivity, and thermal runaway), particularly when continuous stirred tank... 

The model thus shows how thresholds in the phosphorylation-dephosphorylation cascade controlling cdc2 kinase play a primary role in the mechanism of mitotic oscillations. The model further shows how these thresholds are necessarily associated with time delays whose role in the onset of periodic behaviour is no less important. The first delay indeed originates from the slow accumulation of cyclin up to the threshold value C beyond which the fraction of active cdc2 kinase abruptly increases up to a value close to unity. The second delay comes from the time required for M to reach the threshold M beyond which the cyclin protease is switched on. Moreover, the transitions in M and X do not occur instantaneously once C and M reach the threshold values predicted by the steady-state curves the time lag in each of the two modification processes contributes to the delay that separates the rise in C from the increase in Af, and the latter increase from the rise in X. The fact that the cyclin protease is not directly inactivated when the level of cyclin drops below C prolongs the phase of cyclin degradation, with the consequence that M and X will both become inactivated to a further degree as C drops well below C. ... 

In chemical reactors an enormous variety of possible regimes, both steady state and non-steady state (transient) can be observed. Steady-state reaction rates can be characterized by maxima and hystereses. Non-steady-state kinetic dependences may exhibit many phenomena of complex behavior, such as fast and slow domains, ignition and extinction, oscillations and chaotic behavior. These phenomena can be even more complex when taking into account transport processes in three-dimensional media. In this case, waves and different spatial structures can be generated. For explaining these features and applying this knowledge to industrial or biochemical processes, models of complex chemical processes have to be simplified. 

Panda investigated the performance of IMC in fluid-bed drying of sand particles, mustard seeds, and wheat grains [19], The structure of the IMC system for the fluid-bed dryer is depicted in the block diagram shown in Figure 49.6. In this study, IMC uses a process-model transfer function (Gm) parallel to the actual plant transfer function (Gp). A filter is used in the control system to ensure robustness in performance. The exit-air temperature is used for set-point tracking by the IMC. If the system is performed without any oscillations, the overshoots will be tolerable, there will be no offset, and the control scheme will be effective and respond rapidly as described by Panda [19]. 

To simulate the viscoelastic flow, the Oldroyd-B model has been implemented in the VOF-code. Stabilization approaches, such as the Positive Definiteness Preserving Scheme and the Log-Conformation Representation approach have been adapted and implemented in the code to stabilize the simulations at high Weissenberg numbers. The collision of viscoelastic droplets behaves as an oscillation process. The amplitude of the oscillation increases and the oscillation frequency decreases when the Deborah number becomes larger. The phenomenon can be explained with the dilute solution theory with Hookean dumbbell models. An increase of the fluid relaxation time yields a decrease of the stiffness of the spring in the dumbbell and restrains the deformation of the droplets. In addition, with larger the viscosity ratio the collision process is more similar to the Newtonian one since the fluid has less portion of polymers. 

Suppose flow rate W2 is varied sinusoidally about a constant value, while the other inlet conditions are kept constant at their nominal values that is, wi(t) = x[(t) = 0. Because wiit) is sinusoidal, the output composition deviation x t) eventually becomes sinusoidal according to Eq. 5-26. However, there is a phase shift in the output relative to the input, as shown in Fig. 14.1, owing to the material holdup of the. tank. If the flow rate W2 oscillates very slowly relative to the residence time t(co 1/t), the phase shift is very small, aiyroaching 0°, whereas the normalized amplitude ratio(A/KA) is very nearly unity. For the case of a low-frequency input, the output is in phase with the input, tracking the sinusoidal input as if the process model were G s) = K. 

Applepolscher has designed a Smith predictor with proportional control for a control loop that regulates blood glucose concentration with insulin flow. Based on simulation results for a FOPTD model, he tuned the controller so that it will not oscillate. However, when the controller was implemented, severe oscillations occurred. He has verified through numerous step tests that the process model is linear. What explanations can be offered for this anomalous behavior ... 

Goussis, D.A., Najm, H.N. Model reduction and physical understanding of slowly oscillating processes the circadian cycle. SIAM Multiscale Model. Simul. 5, 1297-1332 (2006)... 

Foxboro s Model 823 transmitter uses a taut wire stretched between a measuring diaphragm and a restraining element. The differential process pressure across the measuring diaphragm increases the tension on the wire, thus changing the wire s natural frequency when it is excited by an electromagnet. This vibration (1800—3000 H2) is picked up inductively in an oscillator circuit which feeds a frequency-to-current converter to get a 4—20 m A d-c output. 

Another conventional simplification is replacing the whole vibration spectrum by a single harmonic vibration with an effective frequency co. In doing so one has to leave the reversibility problem out of consideration. It is again the model of an active oscillator mentioned in section 2.2 and, in fact, it is friction in the active mode that renders the transition irreversible. Such an approach leads to the well known Kubo-Toyozawa problem [Kubo and Toyozava 1955], in which the Franck-Condon factor FC depends on two parameters, the order of multiphonon process N and the coupling parameter S... 

In an effort to determine the processes responsible for this type of behavior, Akiba and Tanno (A3), Sehgal and Strand (S2), and Beckstead (B6) have studied the coupling between the dynamics of the combustion process and the dynamic ballistics of the combustion chamber as described by Eq. (7). Each of these investigators has postulated admittedly simplified but slightly different combustion models to couple with the transient ballistic equations. Each has examined the combined equations for regions of instability. The results of these studies suggest a correlation between the L of the motor (the ratio of combustion-chamber volume to nozzle throat area) and the frequency of the oscillations. 

I/O data-based prediction model can be obtained in one step from collected past input and output data. However, thiCTe stiU exists a problem to be resolved. This prediction model does not require any stochastic observer to calculate the predicted output over one prediction horiajn. This feature can provide simplicity for control designer but in the pr ence of significant process or measurement noise, it can bring about too noise sensitive controller, i.e., file control input is also suppose to oscillate due to the noise of measursd output... 

However, all rate data for this reaction are not explained simply by this rate expression. At pressures above 10 Torr the rate exhibits multiple steady states, long transients, and rate oscillations ]). Clearly other processes are Involved than those Implied by the simple one state, constant parameter LH model. 

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