8: Flow, Transport and Exponential
- Page ID
- 104116
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)This chapter focuses on phenomena involving 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, where the flow is constant over time. 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. Toward the end of the chapter we look at examples where the flow is no longer constant, but displays exponential decay behavior.
- 8.1: Overview of Flow, Transport and Exponential
- This chapter extends energy conservation principles to flow and transport phenomena, focusing on steady-state systems like fluid flow and electric circuits. We develop the Steady-State Energy Density Model and the Linear Transport Model for processes that remain constant over time. In the final part, we explore exponential growth and decay in systems evolving toward equilibrium, linking steady-state and time-dependent behaviors.
- 8.2: Steady-State Energy-Density Model
- Our goal in this present chapter is to understand fluids and the flow of electric charge in electric circuits using the Steady-State Energy Density Model. Historically, different words and symbols have been used for the description of each of these phenomena, making their similarity even more difficult to see. We will generally use conventional notation and vocabulary, but will continuously draw parallels between these physically different flow systems.
- 8.3: Static Fluids
- We examine energy variations in static fluids, focusing on pressure and gravitational potential energy density. Pressure in a fluid, seen as energy density, increases with depth as gravitational potential energy decreases. This balance explains why pressure at equal heights remains constant, forming the basis of Pascal’s Principle. Applications include calculating pressure differences at various depths and using hydraulic systems for force amplification.
- 8.4: Fluid Flow
- In this section we modify the fluid system to allow for flow. The flow is steady-state which means it is constant over time for a given fluid system. To describe flow, kinetic energy-density, in addition to pressure and gravitational potential energy-density, is added to the equation which describes energy-density conservation. We also look dissipative effects of resistance to flow which convert mechanical energy to thermal energy, and a source which adds energy to the fluid flow system, a pump.
- 8.5: Electric Circuits
- In this section we introduce steady-state electric charge flow and make multiple analogies with fluid flow. We start by introducing the idea of a circuit, where a fluid (or charge) returns to its original location or flows in a circular manner. Wire acts as pipes and a batteries like pumps. Resistors introduce dissipative effects.
- 8.6: Resistors in Parallel and Series
- We analyze circuits with resistors arranged in series and parallel configurations. In series, resistors share the same current, and their equivalent resistance is the sum of individual resistances. In parallel, resistors share the same voltage, with total current split among branches; their equivalent resistance is found using reciprocals. These configurations apply to both electric circuits and analogous fluid systems, providing insights into current flow, resistance, and energy distribution.
- 8.7: Circuit Problem Solving
- We outline methods for solving circuit problems, starting with calculating equivalent resistance by combining resistors in series and parallel. By analyzing loops for voltage drops and junctions for current distribution, we solve for total current and voltages across resistors. Examples include predicting effects of circuit changes like shorted or burnt-out components, helping simplify complex circuit analysis.
- 8.8: The Linear Transport Model
- There are many phenomena that involve the motion or transport of some quantity that behave similarly to the way fluids and electric charge flow in those sections of a circuit without sources of energy density. In steady-state, we can model these phenomena the same way we model fluids and electric charge flow. These are such common and general phenomena that it is useful to collect them together under their own specially named model—the linear transport model.
- 8.9: Exponential Change Model
- Things grow (or decay) exponentially when the time rate of change in the quantity is proportional to the amount of “stuff” already present. Thus, bacteria grow exponentially because the more you have, the more they reproduce (until they run out of food, or their waste starts to poison the environment.) The decay of radioactive elements is exponential because as the number of radioactive atoms decreases, there are fewer available to decay, so less decay occurs.
- 8.10: Exponential Fluid Flow
- We examine exponential decay in fluid flow, with examples like a leaking container where fluid flow rate decreases as the fluid level drops. Another example involves two connected containers reaching equilibrium, with fluid levels approaching balance exponentially. These scenarios illustrate non-steady-state flow, where the flow rate depends on pressure differences that decrease over time. Such exponential behavior is a key concept in both fluid and electrical transport systems.
- 8.11: Exponential Charge Flow
- A new electronic element, a capacitor, is introduced. When a capacitor is part of an electronic circuit, exponential decay of current and voltage is observed. Analogies are made between circuits with capacitor and fluid flow scenarios from the previous section.
- 8.12: Wrap-up
- This chapter applied energy conservation to steady-state flow systems, focusing on energy density and gradients in fluid flow and circuits. We expanded to the Linear Transport Model for processes like diffusion, driven by gradients. Lastly, we introduced the Exponential Change Model for systems approaching equilibrium over time. The chapter emphasizes understanding relationships behind symbols, preparing for advanced applications in varied transport phenomena.