Grouping of models into opposite pairs

In this section, we will examine various types of mathematical models. There are many possible ways of classification. One possibility is to group the models into opposite pairs  

Linear models exhibit tiie important property of superposition nonlinear ones do not. Equations (and thus models) are linear if the dependent variables or tiieir derivatives appear only to the first power otherwise they are nonlinear In practice, the ability to use a hnear model for a process is of great significance. General analytical methods for equation solving are all based on linearity. Only special classes of nonlinear models can be attacked with mathematical methods. For the general case, where a numerical method is required, the amoimt of computation is also much less for linear models, and in addition error estimates and convergence criteria are usually derived under linear assumptions. 

Other synonyms for steady state are time invariant, static, or stationary. These terms refer to a process in which the point values of die dependent variables remain constant over time, as at steady state and at equilibrium. Non-steady-state processes are also called unsteady state, transient, or dynamic, and represent a situation in which the process dependent variables change with respect to time. A typical example of an non-steady-state process is the startup of a distillation column which would eventually reach a pseudosteady-state set of operating conditions. Inherently transient processes include fixed-bed adsorption, batch distillation, and reactors, drying, and filtration/ sedimentation. 

A lumped-parameter representation means that spatial variations are ignored, and the various properties and the state of a system can be considered homogeneous throughout the entire volume. A distributed-parameter representation, in contrast, takes into accoimt detailed variations in behavior from point to point throughout the system. All real systems are, of course, distributed in that there some variations occur throughout them. As the variations are often relatively small, they may be ignored, and the system may then be lumped.  

The answer to the question whether or not lumping is valid for a process model is far from simple. A good rule of thumb is that if the response of the process is instantaneous throughout the process, then the process can be lumped. If the response shows instantaneous differences throughout the process (or vessel), then it should not be lumped. Note that the purpose of the model affects its vaUdity. Had the pinpose been, for example, to study mixing in a stirred tank reactor, a lumped model would be completely unsuitable because it has assumed from the first that the mixing is perfect and the concentration a single variable. 

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