Model Hierarchy and Its Importance in Analysis

Write the conservation law for the volume element Express flux and reaction rate terms using general symbols, which are taken as positive quantities, so that signs are introduced only as terms are inserted according to the rules of the conservation law, Eq. 1.1. 

After rearrangement to the proper differential format, invoke the fundamental lemma of calculus to produce a differential equation. 

Write out all possibilities for boundary values of the dependent variables the choice among these will be made in conjunction with the solution method selected for the defining (differential) equation. 

Search out solution methods, and consider possible approximations for (i) the defining equation, (ii) the boundary conditions, and (iii) an acceptable final solution. 

It is clear that the modeling and solution effort should go hand in hand, tempered of course by available experimental and operational evidence. A model that contains unknown and unmeasurable parameters is of no real value. 

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To further demonstrate the concept of model hierarchy and its importance in analysis, let us consider a problem of heat removal from a bath of hot solvent by immersing steel rods into the bath and allowing the heat to dissipate from the hot solvent bath through the rod and thence to the atmosphere (Fig. 1.4). 

Model Hierarchy and Its Importance in Analysis 19 Solving for T from Eq. 1.52 yields... 

The obvious question arises When is a model of a process good enough This is not a trivial question, and it can only be answered fiiUy when the detailed economics of design and practicality are taken into account. Here, we have simply illustrated the hierarchy of one simple process, and how to find the limits of validity of each more complicated model in the hierarchy. In the final analysis, the user must decide when tractability is more important than precision. 

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