# 3.7: Constitutive Equations

- Page ID
- 82342

Based upon *Introduction of Chemical Engineering Analysis* by T.W.F. Russell and M. M. Denn, John Wiley & Sons, Inc, New York, 1972, p.41- 43.

As we have worked out problems involving the conservation of mass and the accounting of chemical species, we have frequently found problems where the direct application of the conservation and accounting equation was insufficient to develop a complete model, and hence a solution to our problem or required much more time and effort than we could spend. When this occurs we will often require additional relationships between important variables that do not appear to fit in the conservation and accounting framework. These equations are called constitutive relations or equations.

A **constitutive relation** is a mathematical relationship between variables that describes some physical phenomena. There are several common sources for these relationships. They may be based solely on experiments, such as Ohm's Law that relates voltage drop and electric current through a resistor. They may be based upon purely theoretical grounds, such as the predictions of statistical mechanics for the properties of gases. They may be based on a combination of experiments and theory, such the mass flow rate through an orifice as a function of the height of fluid above the orifice. All of these are constitutive relations.

Constitutive relations are by their nature *specific*. They cannot be applied in general and are only valid under a restricted set of conditions. This is in contrast to the fundamental conservation laws and the entropy accounting equation that are always valid under any circumstances. A prime example of this is the ideal gas "law" we will discuss next. Although commonly presented as a "law," it is really a model for the behavior of substances under restricted conditions of pressure and temperature and is not applicable in every situation.

Constitutive relations typically do one of two things: describe microscopic phenomena or represent empirical relationships between variables for a specific phenomena. Fourier's "law" of heat conduction, Ohm's "law", and Fick's "law" of diffusion all describe the flow of an extensive property — thermal energy, charge, and mass, respectively — in terms of a driving force — temperature, voltage, and concentration, respectively. These are all microscopic phenomena. Another example is the ideal gas "law" that relates pressure, density, and temperature of a gas. All three of these *macroscopic* variables are related to the microscopic behavior of the gas. In the other category are things like the Moody diagram that relates the pressure drop in a pipe to several variables describing the flow situation. The orifice equation \(\dot{V}=C_d A_o \sqrt{2 g h}\) is an example of how empirical data and theory can be used to develop a constitutive relation.

Constitutive relations play an extremely important role in modeling physical systems. You should be on the lookout for constitutive relations and how they are used in the modeling process. Although we will use several constitutive relations during the quarter, we may not provide much background on how they were developed. Later you will learn more about their specific limitations and their development. *Remember that constitutive relations by their very nature have limited ranges of applicability*. One of the major tasks of the engineer is to learn what these ranges are and how to determine the appropriate constitutive relation for a given situation.