scholarly journals Poisoning-induced exchange of steady states in a catalytic chemical reactor

1980 ◽  
Vol 22 (2) ◽  
pp. 237-253 ◽  
Author(s):  
D. D. Do ◽  
R. H. Weiland
Keyword(s):  
Steady State ◽  
Asymptotic Expansions ◽  
Phase Plane ◽  
Exchange Process ◽  
Chemical Reactor ◽  
Matched Asymptotic Expansions ◽  
Catalyst Poisoning ◽  
Independent Variables ◽  
Sudden Failure ◽  
Reactor Operating

AbstractSlow catalyst poisoning can result in the sudden failure of a chemical reactor operating isothermally with substrate-inhibited kinetics. At failure, a satisfactory steady state is exchanged for one of low conversion. The method of matched asymptotic expansions is used to give a detailed description of the exchange process in the phase plane. The structure of the jump is ascertained by separate asymptotic expansions across two adjoining transition regions in which the independent variables contain unknown shifts.

Asymptotic Analysis of a Finite Gas Slider Bearing of Narrow Geometry

1983 ◽  
Vol 105 (3) ◽  
pp. 491-495 ◽  
Author(s):  
J. J. Shepherd ◽  
R. C. DiPrima
Keyword(s):  
Steady State ◽  
Asymptotic Analysis ◽  
Pressure Distribution ◽  
Asymptotic Expansions ◽  
Matched Asymptotic Expansions ◽  
Slider Bearing ◽  
Load Bearing ◽  
Bearing Number ◽  
Isothermal Gas ◽  
State Pressure

The method of matched asymptotic expansions is used to analyze the steady state pressure distribution and load bearing properties of a finite rectangular isothermal gas slider bearing when ε, the ratio of transverse to longitudinal dimensions of the bearing, is small and the bearing number Λ is moderate. General expressions for the pressure and load are obtained. Specific results are given for bearings with shallow crowning. The effects of the bearing number becoming large and the interaction between the two effects ε→0 and Λ→∞ are discussed.

Using the Kalman Filtering for the Fault Detection and Isolation (FDI) in the Nonlinear Dynamic Processes

2008 ◽  
Vol 6 (1) ◽  
Author(s):  
Yahya Chetouani
Keyword(s):  
Fault Detection ◽  
Nonlinear Dynamic ◽  
Nonlinear Dynamic System ◽  
Chemical Reactor ◽  
Fault Detection And Isolation ◽  
Decision Threshold ◽  
Normal Behaviour ◽  
Batch Mode ◽  
The Moment ◽  
Reactor Operating

This paper presents a Fault Detection and Isolation (FDI) method for stochastic nonlinear dynamic systems. First, the developed fault detection method is based on statistical information generated by the extended Kalman filter (EKF) and is intended to reveal any drift from the normal behaviour of the process. A fault of a chemical origin in a perfectly stirred batch chemical reactor, occurring at an unknown instant, is simulated. The purpose is to detect the presence of this abrupt change, and pinpoint the moment it occurred. It is also shown that the convergence of the EKF is accomplished more or less rapidly according to the nature of the noise generated by the measurement sensors. The state estimate is observed and discussed, as well as the time delay in detection according to the decision threshold. Then, this study shows another method of tackling the problem of the physical origin diagnosis of faults by combining the technique based on the standardized innovations and the technique using the multiple extended Kalman filters for a strongly non-stationary nonlinear dynamic system. The usefulness of this combination is the implementation of all the fault dynamics models if the decision threshold on the standardized innovation exceeds a determined threshold. In the other case, one EKF is enough to estimate all the process state. An algorithm is described and applied to a perfectly stirred chemical reactor operating in a semi-batch mode. The chemical reaction used is an exothermic second order one.

Asymptotic Methods for an Infinite Slider Bearing With a Discontinuity in Film Slope

1976 ◽  
Vol 98 (3) ◽  
pp. 446-452 ◽  
Author(s):  
J. A. Schmitt ◽  
R. C. DiPrima
Keyword(s):  
Boundary Layer ◽  
Film Thickness ◽  
Asymptotic Expansions ◽  
Asymptotic Expression ◽  
Asymptotic Methods ◽  
Matched Asymptotic Expansions ◽  
Slider Bearing ◽  
Trailing Edge ◽  
Slider Bearings ◽  
Special Cases

The method of matched asymptotic expansions is used to develop an asymptotic expression for the pressure for large bearing numbers for the case of an infinite slider bearing with a general film thickness that has a discontinuous slope at a point. It is shown that, in addition to the boundary layer of the pressure at the trailing edge, there is also a boundary layer in the derivative of the pressure at the point of discontinuity. The corresponding load formula is also derived. The special cases of the taper-flat and taper-taper slider bearings are discussed.

Root locus-based stability analysis for biological systems

2021 ◽  
pp. 2150023
Author(s):  
Shinq-Jen Wu
Keyword(s):  
Steady State ◽  
Theoretical Analysis ◽  
Dynamic Behavior ◽  
Biological System ◽  
Setting Time ◽  
Mode Analysis ◽  
Root Locus ◽  
Stability Margins ◽  
Independent Variables ◽  
Simulation Results

Background: The first objective for realizing and handling biological systems is to choose a suitable model prototype and then perform structure and parameter identification. Afterwards, a theoretical analysis is needed to understand the characteristics, abilities, and limitations of the underlying systems. Generalized Michaelis–Menten kinetics (MM) and S-systems are two well-known biochemical system theory-based models. Research on steady-state estimation of generalized MM systems is difficult because of their complex structure. Further, theoretical analysis of S-systems is still difficult because of the power-law structure, and even the estimation of steady states can be easily achieved via algebraic equations. Aim: We focus on how to flexibly use control technologies to perform deeper biological system analysis. Methods: For generalized MM systems, the root locus method (proposed by Walter R. Evans) is used to predict the direction and rate (flux) limitations of the reaction and to estimate the steady states and stability margins (relative stability). Mode analysis is additionally introduced to discuss the transient behavior and the setting time. For S-systems, the concept of root locus, mode analysis, and the converse theorem are used to predict the dynamic behavior, to estimate the setting time and to analyze the relative stability of systems. Theoretical results were examined via simulation in a Simulink/MATLAB environment. Results: Four kinds of small functional modules (a system with reversible MM kinetics, a system with a singular or nearly singular system matrix and systems with cascade or branch pathways) are used to describe the proposed strategies clearly. For the reversible MM kinetics system, we successfully predict the direction and the rate (flux) limitations of reactions and obtain the values of steady state and net flux. We observe that theoretically derived results are consistent with simulation results. Good prediction is observed ([Formula: see text]% accuracy). For the system with a (nearly) singular matrix, we demonstrate that the system is neither globally exponentially stable nor globally asymptotically stable but globally semistable. The system possesses an infinite gain margin (GM denoting how much the gain can increase before the system becomes unstable) regardless of how large or how small the values of independent variables are, but the setting time decreases and then increases or always decreases as the values of independent variables increase. For S-systems, we first demonstrate that the stability of S-systems can be determined by linearized systems via root loci, mode analysis, and block diagram-based simulation. The relevant S-systems possess infinite GM for the values of independent variables varying from zero to infinity, and the setting time increases as the values of independent variables increase. Furthermore, the branch pathway maintains oscillation until a steady state is reached, but the oscillation phenomenon does not exist in the cascade pathway because in this system, all of the root loci are located on real lines. The theoretical predictions of dynamic behavior for these two systems are consistent with the simulation results. This study provides a guideline describing how to choose suitable independent variables such that systems possess satisfactory performance for stability margins, setting time and dynamic behavior. Conclusion: The proposed root locus-based analysis can be applied to any kind of differential equation-based biological system. This research initiates a method to examine system dynamic behavior and to discuss operating principles.

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