In Part 1, Chapters 1 and 2 of this text, we explored a very general and universally applicable model based on conservation of energy: the Energy-Interaction Model. The approach of focusing on initial and final values was extended in Chapter 4, Thermodynamics, to include all state variables. We now turn back to energy conservation, but with a significant difference. Up to now in this course, the changes in energy and other variables occurred over time. We focused on an initial time and a final time and looked at how the variables changed between those two times. We begin Chapter 5 by looking at phenomena that occur in a steady-state fashion. In particular, we will study the steady-state flow of fluids and electric charge in electrical circuits. In these phenomena the change in energy occurs over position, but is constant in time. This is what is meant by “steady-state.” We will not be starting from scratch, however, since we will be able to use many of the ideas and constructs we previously developed. Note: When we use the word “circuit,” it simply implies that the flow of fluid or the flow of electric charge is confined to a fairly well defined path.
We begin by focusing on phenomena that involve real fluids with viscosity and electric circuits with resistance. The effect of resistance in both kinds of flow means that energy will be reduced in the fluid system while thermal energy systems will increase. Frequently the flow is described as being dissipative. We call the model/approach we use to make sense of dissipative flow the Steady-State Energy Density Model.
In the second part of the chapter we generalize the underlying ideas about flow to flow phenomena in which changes in energy are not of paramount importance. Rather, the focus is simply on the “fluid” and medium properties and the “driving force” that keeps the flow going. The “thing” that flows can be a real fluid, electric charge, energy, or other things that diffuse – in short, any phenomenon in which the flow of something becomes constant can be understood with this approach/model, which we call the Linear Transport Model.
Frequently in science the same basic relationship is rediscovered by practitioners in different disciplines. Usually a different name is given to the relationship and different symbols are used to express it in each different discipline. These historical differences manage to last through multiple generations of textbooks. This is certainly true of the linear transport equation we develop in this chapter. As an example, some of you will study environmental science or soil and water science. Sooner or later, you will come across Darcy’s law, describing the flow of water in soils. After studying this chapter, Darcy’s law should seem pretty familiar to you, even if some of the symbols that are used are different. Practitioners in other branches of science, technology, medicine and engineering, will use different laws relating to transport, each with its different name and specialized symbols, but all referring to the same basic underlying transport model. Hopefully, you will develop expertise in “reading past the particular symbols” and recognize the fundamental content expressed by the relationship, which, after all, is independent of which letters of the alphabet we choose to use.
The kinds of phenomena we can make sense of using the Steady-State Energy Density Model or the Linear Transport Model do not change in time. That is, the transport phenomena remain constant for the time interval of interest. For example, once a resistor network has been arranged and a battery connected, the current very quickly comes to its steady-state values in the various parts of the circuit and then remains constant. Or, when there is a constant temperature difference from one end of a bar of metal to the other, the rate of heat flow is constant along the bar. These two powerful models, though broadly applicable to many real situations, exclude any situation where the amount of something transported changes with time. There are certainly many physical systems where the rate of change is not constant.
A very common “non-steady state” phenomenon in nature is exponential growth or decay. You might have seen discussions of exponential growth in biology classes, or perhaps in a business or economics class when discussing compound interest. You may also have seen exponential decay in chemistry when exploring radioactivity and nuclear decay. Many of the physical systems that exhibit steady-state flow also exhibit exponential behavior as they evolve to a steady-state condition. The “charging up” of an electrical circuit, the heat flow from a hot cup of coffee or tea, the draining of a container of liquid through a small hole all exhibit exponential behavior. In the last part of this chapter we develop a model that that provides the foundation for understanding this kind of exponential-change phenomena. We include exponential change in this chapter, because all of the physical systems that exhibit steady-state behavior and to which the Steady-State Energy Density Model and the Linear Transport Model apply, also exhibit exponential change under different circumstances.