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. These systems contain singular terms of differentials of the input function even for the step perturbation and, thus, exhibit initial discontinuities in the step and impulse responses. To explore the range of behavior of these systems in general, conditions are worked out for their assorted solution profiles to identify characteristic parameters, which unfold interesting cases of maximum, minimum, inflection, input multiplicity, pole-zero cancellation, reduction to standard systems, etc., as the value of one such parameter changes relative to that of the others on the real axis. A preliminary methodology is initially presented to ascertain and validate the inconsistency in solution profiles. The analysis is carried out by matching the coefficients of input and output terms in the models; the basic approach is to combine physical principles with mathematical analysis and expose the inconsistencies. In some cases, an initial discontinuity exposed through physical principles can’t be accounted for through mathematical analysis. The value of that discontinuity is, thus, required for correct solutions of such cases for an impulse input. This effect is investigated for different systems and validated by the comparison of experimental and simulated solution profiles of nonlinear models of flow-level tanks. The same is also established by the calculations of time required for emptying the flow-level tanks using impulse responses and verifying them with physical principles. Right through all of these, the significance of putting the theoretical analysis consistent with the physical principles is indicated. This preliminary methodology brings out the qualitative effects of initial discontinuities. Following the above analysis and building upon its results, a general framework of methodology to quantitatively describe the transient response of nonlinear first-order ODE systems, is, then, proposed. It works on the basis of using original non-transformed models, and initializing them through the following alternative methods: Direct inclusion of singularities in the cause through initial integration of the models (Initialization A), whose accuracy in making a reliable estimation is quantified by comparison with the method of application of physical balances to the initial effects of singularity (Initialization B); the latter is, thus, closer to the real as ascertained in the last paragraph. For many cases, the initial conditions calculated from initialization A are inconsistent with that calculated from initialization B. The solution profiles thus obtained are also compared with that obtained by using an increasingly sharpened Gauss pulse function for impulse (Initialization C). It is found that whereas the Gaussian pulse approximation can’t reliably model the singularity of the δ-Dirac function, it can be done fairly accurately through the procedure of initialization A. The ready-to-use initialization A can be adequately used for a system operated not significantly far off from its normal operating conditions. The validity of initialization A is substantiated by its conformity to the definitely known cases, and to the well established Laplace transform approach for the linear time-invariant systems. On application of the proposed framework to the linear systems, it is elicited that the L+ Laplace transform approach doesn’t involve inconsistency and doesn’t require physical balances on the effects for the simple cases; whereas, the L− approach is inconsistent/inexact, especially, for many nontrivial chemical engineering cases indicated at the outset, as their models can’t account for the initialization inconsistency under an impulse. Various transformed and approximated models, viz., Laplace transformed models, linearized models; smoothened impulse models, symbolically manipulated models, etc. can’t predict the native consistencies/inconsistencies of the physical systems, and lead to wrong solution profiles and convergence. The numerical solutions of the proposed framework that uses un-manipulated models are found to converge to the right values, predict accurate behaviors, and work over a wider range of inconsistent initial values, in contrast to the solutions of the implicit and symbolically modified indirect models. Thus, the proposed framework is direct and avoids the complicated machinery of generalized functions.