Sanjeev Ahuja

EFFECTS OF INITIAL DISCONTINUITIES ON NONLINEAR SYSTEMS REPRESENTED BY DIFFERENTIAL EQUATIONS WITH TERMS CONTAINING DIFFERENTIALS OF THE INPUT FUNCTION

This article treats the effects of initial jump discontinuities of the inputs on solutions of systems represented by nonlinear second-order ordinary differential equations with terms containing differentials of the input function. A methodology for analysis of discontinuities is presented. In some cases, an initial discontinuity of the calculated impulse response cannot be accounted for in the input function. The value of that discontinuity is thus required for the correct solution of such cases. However, that value cannot be obtained unless the solution profile is known. This paradoxical effect is investigated for different systems and validated by the comparison of experimental and simulated data of nonlinear models of flow-level tanks. A few systems are not affected by discontinuities upon linearization. All others, however, retain their maiden character of being affected or not affected by discontinuities upon linearization. Some interesting cases of calculations of time required for emptying flow-level tanks using impulse responses are also discussed.

Second-order numerator-dynamics systems: Effects of initial discontinuities

Linear second-order systems with numerator-dynamics are considered and their transient responses are investigated for discontinuous inputs. Characteristic parameters for second-order systems with numerator dynamics are identified. The solution profiles of these systems depend upon the value of one such parameter relative to that of the others on the number line. Effects of initial jump discontinuities of the inputs on the correctness of the solutions are also treated. A methodology for analysis of discontinuities is presented. Initial discontinuities of the calculated response either get accounted for in the input function or the initial condition of the input function for all but the following case. For the impulse response alone, that too among inherent numerator-dynamics systems only, an initial di scontinuity cannot be accounted for in the input. The value of that discontinuity is thus required for correct solutions of these cases. The results are general and extendable to different inputs and higher order systems.

Discontinuity analysis for the treatment of nonlinear lumped-parameter systems for singular inputs

Different approaches suggested for the treatment of nonlinear systems subjected to singular inputs are either approximate or difficult to apply. In this paper, discontinuity analysis is used for this treatment, and is found to represent a good compromise between accuracy and ease of applicability. An example of non-isothermal CSTR representing nonlinear lumped-parameter chemical engineering systems is considered for the treatment. In the treatment, the integration of a model is carried out across the time of jump discontinuity by including singularity in the cause. This allows a priori estimation of initial conditions. The estimated initial conditions are then used for the initialization of the numerical solution of the model, whose singular input variables are set at their pre-initial values. The accuracy of the procedure in providing a reliable estimation of the system behavior is quantified by comparison with the numerical solutions initialized through the application of physical balances to the initial effects of singularity on the system, and with the numerical solutions obtained through pulse approximation for impulse. The procedure has simple and uniform applicability in comparison to the application of physical balances, and is substantially accurate than the commonly used pulse approximation method.

A Kinetic Model of Alkali Catalyzed Phenol-furfural Novalac Resinification

In this paper, experimental studies were carried out on the preparation of phenol-furfural novalac resins in the presence of potassium carbonate catalyst with a wide range of furfural to phenol (F/P) mole ratios in the feed. A reaction scheme, applicable to alkali catalyzed phenol-furfural novalac resins, was postulated. A model was developed by the lumping of the different reactivities of the sites of phenol into their functionalities. The effects of the mole ratio of the reactants and the functionality of the phenol on the rate constant were examined.