Wednesday, February 4, 2015

Statistical Mechanics, Part 6

[This blog post is part of two series. The first series, titled Statistical Mechanics, is a series of lessons that I have developed to illustrate how curriculum and instruction can be designed to encourage vertical learning. We start by drilling down from our understanding of diffusion to develop a dynamic model of diffusion grounded in random particle motion using statistical mechanics. We then build on top of this foundation to construct deeper understandings of fluid flow, heat transfer, and electrical circuits. You can navigate through the lessons in this series using the links below. These lessons are then part of a larger series, Defining Vertical Learning.]

Part 1  |  Part 2  |  Part 3  |  Part 4  |  Part 5  |  Part 6  |  Part 7  |  Summary

Circuits in Series and in Parallel

At the start of this lesson on diffusion, our understanding existed at one level: we had a textbook definition that told us that diffusion is the flow of particles from regions of high concentration to regions of low concentration.

Setting that definition aside for a moment, we then attempted to model diffusion by using what we know is happening at the molecular scale: that the red dye is made up of particles in perpetual and random motion. To simplify things as much as possible, we developed a model where pennies moved randomly left and right, hopping from region to region in one dimension.

When we tested this model, we observed diffusion occurring at the macroscopic scale. We now knew that the process of diffusion could be generated by and was grounded in random particle motion. We also had a tool that would help us visualize diffusion dynamically (over time) and at the molecular scale. We had drilled down to deepen our understanding to two levels.

Flipping pennies and moving them by hand immerses you in the molecular scale. Essentially, you are seeing diffusion from a particle’s point of view. But, in order to analyze diffusion, we needed to pull back and widen our perspective a little bit. We did that by introducing the concepts of statistical mechanics and control surfaces. By observing the boundaries between regions and applying a little probability, we discovered that there was a net flow of particles whenever there was a concentration gradient, or concentration difference, between the two regions.

Net flow caused by a concentration gradient across regions

By building up from a model of diffusion grounded in random particle motion, we had now constructed a third level: using probabilities, gradients, and net flows to analyze diffusion.

At this level, we know that net flow runs downhill and is proportional to the concentration gradient, and we can use those concepts to figure out and explain why things happen, such as why four individual blocks dissolve faster than four blocks stacked together. But there is a difference between analyzing the world with a new tool and seeing the world through a new lens. Our goal in this series of lessons is to construct a fourth level where we are seeing and thinking (with no translation or overhead) in terms of net flows and gradients.

With that goal in mind, we simulated an electrical circuit and introduced the concept of a steady state. When there is a constant potential difference across a system, and one end will always have more pennies than the other, then the system will never reach a state of equilibrium where all regions have the same number of pennies and the net flow between regions is zero. But the system can reach a steady state where the number of pennies in each region no longer changes.

Net flow into region B from region A and the net flow out of region B into region C

In order for that to happen, the net flow in one side of a region has to equal the net flow out the other side of the region. In the example above, it means that the net flow into region B from region A and the net flow out of region B into region C have to be equal, which means that the potential difference between region A and region B and the potential difference between region B and region C also have to be equal.

A 7-region circuit with a 240-penny potential difference

Thinking in terms of net flows and gradients, if the potential difference across six control surfaces is a constant 240 pennies, then the system will only reach a steady state when the potential difference between regions is 40 pennies (240 ÷ 6 = 40). And when the potential difference between regions is 40 pennies, there will be a net flow from region to region of 20 pennies each round. From there, we can derive Ohm’s law:

In our model, every penny moves to a new region each round unless it is blocked by the edge of the simulation. This means that when we are simulating an electrical circuit and our pennies represent electrons, all of our electrons are moving freely. But in reality, the flow of electrons is impeded by the material that the electrons are moving through. Materials with high resistivity, such as insulators, impede the flow of electrons a great deal. Materials with low resistivity, such as conductors, only impede the flow of electrons slightly.

We are going to extend our model to include resistivity. Instead of flipping and moving every penny each round, we are only going to flip and move a fraction of the pennies. The higher a material’s resistivity, the lower the fraction of pennies that will move in a round.

In this material, only 60% of the pennies move each
round. So, on average, 30% of the pennies move to the left,
30% move to the right, and 40% stay where they are.

Here is a simulation of a circuit with two resistors in series:

A circuit with two resistors in series. The potential
difference across the entire circuit is 220 pennies. Resistor
A has a length of 3 and resistor B has a length of 4.

In resistor A, 50% of the electrons move each round. In resistor B, 80% of the electrons move each round. After 84 rounds, the circuit reaches a steady state with a current of 10 electrons per round. To generate a net flow of 10 electrons per round, it takes a potential difference of 40 pennies between regions in resistor A and it takes a potential difference of 25 pennies between regions in resistor B. That is why the potential gradient in resistor A is steeper than in resistor B.

Here is a simulation of a circuit with four resistors, two of them in parallel:

A circuit with four resistors. Resistors B and C are in parallel.
The potential difference across the entire circuit is 1560 pennies.

In resistors A and D, 80% of the electrons move each round; in resistor B, it’s 50% of the electrons; and in resistor C, it’s 60%. The circuit reaches a steady state after 114 rounds with a current of 108 electrons per round. Notice how the net flow of electrons through resistors A and D is 108, but the net flows through resistors B and C are 60 and 48, respectively. That happens because those two resistors are in parallel (60 + 48 = 108).

Applying Ohm’s law (V = 1560 and I = 108), we calculate that the total resistance (R) of the circuit is 220, which matches what we would expect based on the relationship between total resistance and the resistance of resistors in series and parallel that we see in the real world.

There are a lot of application problems that you can do with circuits, but you can also use our model to simulate fluid flow through pipes in series and parallel. Imagine that you are designing an irrigation system to automatically deliver appropriate amounts of water from an elevated water tank through plastic tubing to a collection of different plots in a garden.

Irrigation system for plots in a garden

In this “circuit,” the elevated water tank is the anode: water leaves the water tank under pressure, flows down through the plastic tubing, and empties out into the garden plots. The garden plots are the cathode: when the water reaches the garden plots, it is at atmospheric pressure. This pressure drop is the potential difference across the entire circuit, and it is what generates the current, or net flow of water from the water tank into the garden plots. Each segment of tubing is a resistor. The resistance of any individual resistor is proportional to the plastic tubing’s length and inversely proportional to it’s cross-sectional area. We can use Ohm’s law to predict how much water each garden plot will receive when we design our circuit with various resistors in series and parallel.

There are some issues with using our model to simulate fluid flow down a pressure gradient. We will discuss them next time.

Part 1  |  Part 2  |  Part 3  |  Part 4  |  Part 5  |  Part 6  |  Part 7  |  Summary

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