Discontinuity analysis for the treatment of lumped parameter chemical engineering systems for singular inputs

Abstract

Abstract Chemical engineering presents unique and interesting cases consisting of chemical reactions, phase changes, and the interacting capacities of material, thermal and mechanical energy. Singular inputs cause initial discontinuities in the physical system and inconsistency in initial conditions. The mathematical problem of finding the dynamic response to these inputs arises in many fields of engineering and science. The literature studies on singularity deal mainly with electrical and mechanical engineering systems and pure mathematical systems that involve symbolically manipulated and transformed complicated models, and advocate either the use of approximate methods, or the demanding framework of generalized functions. The analysis of linear time-invariant systems in the Laplace domain, also, involves inconsistency. Proposing a framework based on the direct, time domain approach of analysis, which aims to estimate reliably accurate initial conditions for the solution of the original un-manipulated models of chemical engineering for the singular inputs, and to reveal and resolve the inconsistencies, are the main objectives of the present study. Limitations of several transformed and approximated models are also indicated. The framework should also resolve the inconsistency in Laplace domain analysis. Analysis of initial discontinuities is carried out for the nonlinear, linearized and linear systems perturbed by the singular inputs. Nonlinear lumped-parameter chemical engineering models, viz., non-isothermal CSTR, single component condenser, gravity-flow tank, interacting tanks, U-tube manometer, closed-loop stirred tank heater with dead time, etc. under singular inputs or initial conditions are considered. Upon linearization these are found to exhibit second-order numerator-dynamics behavior for some of the output variables that are represented by ODEs with terms containing differentials of the input function, and such systems have been the subject of extensive studies.