Lumped-parameter models

A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities the boundaiy or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial valueproblem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinaiy differential equations become two-point boundaiy value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. 

The lumped parameter model of Example 13.9 takes no account of hydrodynamics and predicts stable operation in regions where the velocity profile is elongated to the point of instability. It also overestimates conversion in the stable regions. The next example illustrates the computations that are needed... 

What are some of the mathematical tools that we use In classical control, our analysis is based on linear ordinary differential equations with constant coefficients—what is called linear time invariant (LTI). Our models are also called lumped-parameter models, meaning that variations in space or location are not considered. Time is the only independent variable. 

For other discussions of two-phase models and numerical solutions, the reader is referred to the following references thermofluid dynamic theory of two-phase flow (Ishii, 1975) formulation of the one-dimensional, six-equation, two-phase flow models (Le Coq et al., 1978) lumped-parameter modeling of one-dimensional, two-phase flow (Wulff, 1978) two-fluid models for two-phase flow and their numerical solutions (Agee et al., 1978) and numerical methods for solving two-phase flow equations (Latrobe, 1978 Agee, 1978 Patanakar, 1980). 

Wulff, W, 1978, Lump-Parameter Modeling of One-Dimensional Two-Phase Flow, in Transient Two-Phase Flow, Proc. 2nd Specialists Meeting, vol. 1, pp. 191-219, OECD Committee on Safety of Nuclear Installations, Paris. (3)... 

Dynamic simulations are also possible, and these require solving differential equations, sometimes with algebraic constraints. If some parts of the process change extremely quickly when there is a disturbance, that part of the process may be modeled in the steady state for the disturbance at any instant. Such situations are called stiff, and the methods for them are discussed in Numerical Solution of Ordinary Differential Equations as Initial-Value Problems. It must be realized, though, that a dynamic calculation can also be time-consuming and sometimes the allowable units are lumped-parameter models that are simplifications of the equations used for the steady-state analysis. Thus, as always, the assumptions need to be examined critically before accepting the computer results. 

During the flight of droplets in the spray, the forced convective and radiative heat exchanges with the atomization gas lead to a rapid heat extraction from the droplets. A droplet undergoing cooling and phase change may experience three states (a) fully liquid, (b) semisolid, and (c) fully solid. If the Biot number of a droplet in all three states is smaller than 0.1, the lumped parameter model 1561 can be used for the calculation of droplet temperature. Otherwise, the distributed parameter model 1541 should be used. 

In the lumped parameter model, the transient temperature of a single droplet during flight in a high speed atomization gas is calculated using the modified Newton s law of cooling, 1561 considering the frictional heat produced by the violent gas-droplet interactions due... 

The model is a 1-dimensional lumped parameter model with the following assumptions ... 

Lumped models (usually called lumped parameter models , which is wrong terminology since the state variables are lumped, not the input variables or parameters) described by transcendental equations for the steady state and ODEs for the unsteady state. 

Tronconni E, Forzatti P. Adequacy of lumped parameter models for SCR reactor with monolith structure. AIChE J 1992 38 201-210. 

Process-scale models represent the behavior of reaction, separation and mass, heat, and momentum transfer at the process flowsheet level, or for a network of process flowsheets. Whether based on first-principles or empirical relations, the model equations for these systems typically consist of conservation laws (based on mass, heat, and momentum), physical and chemical equilibrium among species and phases, and additional constitutive equations that describe the rates of chemical transformation or transport of mass and energy. These process models are often represented by a collection of individual unit models (the so-called unit operations) that usually correspond to major pieces of process equipment, which, in turn, are captured by device-level models. These unit models are assembled within a process flowsheet that describes the interaction of equipment either for steady state or dynamic behavior. As a result, models can be described by algebraic or differential equations. As illustrated in Figure 3 for a PEFC-base power plant, steady-state process flowsheets are usually described by lumped parameter models described by algebraic equations. Similarly, dynamic process flowsheets are described by lumped parameter models comprising differential-algebraic equations. Models that deal with spatially distributed models are frequently considered at the device... 

At the process level, efficient flowsheet optimization strategies based on lumped parameter models are now widely used in practice (Biegler et al., 1997). At this scale, the PEFC is embedded within a power plant flowsheet model, as shown in Figure 3. The process comprises... 

In polymer processing, the mathematical models are by and large deterministic (as are the processes), generally transport based, either steady (continuous process, except when dynamic models for control purposes are needed) or unsteady (cyclic process), linear generally only to a first approximation, and distributed parameter (although when the process is broken into small, finite elements, locally lumped-parameter models are used). 

The lumped-parameter model approach becomes particularly useful when dealing with the plasticating extrusion process discussed in the next subsection, where, in addition to melt flow, we are faced with the elementary steps of solids handling and melting. 

APPENDIX 13.1 LUMPED PARAMETER MODEL OF A TUBULAR POLYMERIZER... 

This model can be applied not only to a tank-type equipment but also to a tubular type equipment. Mathematically, the use of the SPMV model implies that the mixing process is expressed by a lumped parameter model and not by a distributed parameter model. The RTD function in this model is expressed as... 

Lumped parameter model The parameter is concentrated at a finite point. 

Mathematical modeling of systems for which characteristic variables are time-dependent only and not space-dependent is done by ordinary differential equations (ODEs). The situation is found in a nearly well-mixed batch reactor. There one may find differences in temperature or concentrations from one site to another due to imperfect mixing. When space changes are not important to the model, the process variables can be approximated by means of lumped parameter models (LPMs). When the... 

Consider the dynamic behavior of a process that can be considered perfectly mixed. The lumped parameter model has the following form ... 

Figure 6.12 provides a block-diagram representation of the energy flows in the process. Postponing the derivation of an expression of the heat-exchanger duty Hlec (which is, evidently, central to this process and was originally captured in the partial differential equations) until later in this section, we can use this representation to develop a lumped-parameter model of the process ... 

The simplified lumped parameter model (M2) can also be used to match the data of Figure 6. This suggests the following correlations for the overall mass transfer coefficients in presence of reaction. 

These authors [32, 33] have considered an alternative classification based on the nature of the variables involved in the model. They classify models by grouping them into opposite pairs deterministic vs. probabilistic, linear vs. non-linear, steady vs. non-steady state, lumped vs. distributed parameters models. In a lumped parameters model, variations of some variable (usually a spatial one) are ignored and its value is assumed to be uniform throughout the entire system. On the other hand, distributed parameters models take into account detailed variations of variables throughout the system. In the kinetic description of a chemical system, lumping concerns chemical constituents and has been widely used (see Sects. 2.4 and 2.5). 

However, such a complex system would not be helpful to describe organic-removal wastewater-treatment processes because of its high degree of complexity and, therefore, in an attempt to achieve a useful model, some assumptions could be made in order to simplify the model. Hence the transformation of this distributed-parameter model in a simpler lumped-parameter model is very common in the modeling of wastewater-treatment processes, because it is not very important to obtain detailed information about what happens in every point of the cell but simply to know in a very simple way how the pollution of a influent waste decreases at the outlet of the electrochemical cell. In this context, there are three types of approaches typically used ... 

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