AND STATICS DYNAMICS ME 221
C. J. Kobus
The Concept of Phenomenological Modeling
Whether it is recognized as such or not, every concept and theory, which is accepted by the scientific community for use in predicting the outcome of a physically observable phenomenon, is based upon a model of physical reality. The word "model," as used here, is not meant to imply a geometrically similar, but smaller, object as per the general usage of the word, but rather a theoretical invention or concept that provides a reasonable description of the mechanisms controlling the physical phenomenon. Inevitably, the model is a simplification of the actual or real physical mechanisms. The amount of simplification might be dictated by the degree to which the phenomenon is understood, or, intentional in order to make the model more amenable to mathematical description. The primary idea to be emphasized is that regardless of the degree of elegance that a theory appears to possess, it is still based upon a model of physical reality; thus, theories and models are inseparable. Many concepts and theories are interconnected. That is, the theory of a particular phenomenon may be abased upon a particular model that in itself is constructed from several other more basic or fundamental models. For example, the theory used for predicting the performance of a refrigeration system is based upon a model which in turn is constructed from such basic models as the conservation of mass principle, the conservation of energy principle, the thermodynamic model, the perfect gas model, models associated with a compressor, a condenser, an expansion valve, and an evaporator, just to mention a few. Perhaps additional meaning to some of the ideas associated with phenomenological modeling can be illustrated by another example with which there should be reasonable wide familiarity. Consider a simple D.C. circuit consisting of a battery and a resistor.
R
i
V
D.C. Circuit
If the voltage, V, and the resistance, R, are known, then the steady state current, i, which will flow through the resistor, can be predicted with a model which is commonly referred to as "Ohm's Law." A mathematical description of this law is given by:
i = V/R
The above is an example of what is called a "lumped parameter system or model." Each component within the system is in general described by a single characteristic quantity (lumped parameter). In the lumped parameter model, the physical detail associated with the geometry and the materials used to give the resistor the resistance, R, is immaterial, as long as its numerical value is known, for example by measurement. However, looking at the first step down the fundamental ladder, the value of the resistance, R, depends upon the geometry of the resistor and the material from which it is constructed. There exists a more detailed theory, based upon a more fundamental model (macroscopic model), which is capable predicting of the resistance, R, providing the geometry of the resistor and a macroscopic property called resistivity, , of the resistor material are both known. This model is described mathematically by the following expression: 1 V n
fi =
This expression states that the current flux (current per unit area), fi, is proportional to the potential or voltage gradient, where the proportionality constant is the inverse of the macroscopic material property called resistivity, . Since the resistivity of the material is a measurable property, the above macroscopic model theoretically makes it possible to predict the resistance of a given resistor geometry. Looking at the next step down the fundamental ladder, one might ask whether or not it is theoretically possible to predict the resistivity of a given material. The answer to the question is yes. This is accomplished by considering the microscopic makeup of the material on a molecular or atomic level. Such a theory is said to be based upon a "microscopic model." Thus, from an understanding of the microscopic structure of various materials, it is possible to predict their relative resistivities. The point of this discussion then is to point out that there are at least three distinct models associated with the flow of current through a resistor: the lumped model, the macroscopic model, and
the microscopic model. The lumped parameter and macroscopic models are capable of yielding quantitative results since both of these models can be readily represented by mathematical expressions. On the other hand, the microscopic model is primarily useful in a qualitative capacity, since, except for very special cases where statistical descriptions are possible, the model is too complex to be expressed or eva...