### Video Transcript

In this video, we’re talking about Sankey diagrams. As we’ll see, this type of diagram shows us visually how an input into a process is divided up amongst the outputs of that process.

To get started talking about these diagrams, imagine that we create a shopping list. And this list has different items we want to buy at the grocery store. And let’s say further that we have a total of 30 dollars that we can spend on these items. And imagine that as we go through our list top to bottom, we keep track of what each item costs.

Let’s say the cereal costs four dollars, the pretzels cost two dollars, the chicken cost eight dollars, the eggs cost one dollar, and the bananas cost one dollar. If we add up the cost of all these items, then that comes out to a total of 16 dollars. Since we started out with 30, that means we have some left over. We could describe this trip to the grocery store as a process that has an input to it, the money we started with, and a series of outputs, the cost of each item we’re buying.

Thinking of it this way, we can depict our input and outputs for this process graphically. One graphical way to understand this process looks like this. Over on the left-hand side of the sketch, we have our input. And then elsewhere around the diagram, we have our outputs, the different food items as well as the money left over. One important point about this diagram is that it’s to scale. In other words, this distance here represents the 30 dollars’ input, our original shopping budget. On that same scale, the distance from here to here for cereal represents four dollars. The distance from here to here represents two dollars for pretzels. This spin right here represents eight dollars for the chicken, and so forth for all our food items. And this also includes the output of our leftover money.

A great advantage of a diagram like this representing the input and the outputs this way is that, at a glance, we can see where our largest and smallest expenditures are. The name for a diagram like this is a Sankey diagram. This kind of diagram is a visual depiction of process inputs and outputs where line width represents relative quantity.

For example, looking back over at the Sankey diagram from our shopping trip, the line width representing eggs, which were a dollar, and bananas, which were also one dollar, is one thirtieth as wide as the line width of our input of 30 dollars. In this sense, a Sankey diagram is to scale.

Now as you may imagine, the first Sankey diagrams weren’t about trips to the grocery store. Instead, they had to do with energy inputs and outputs for a steam engine process. By looking at such a diagram, it was easy to compare the useful energy output against what was lost to heat and other factors. When it comes to these diagrams, it’s important to know that if we add up the line width of all the separate outputs, then that total output must equal the total input. In other words, we’re accounting for 100 percent of it.

Because it’s very important to be able to calculate the relative line width in a Sankey diagram, often we’ll find them with a grid overlaid. Using such a grid, we’re able to quantify the width of the various lines of input and output. For example, when we look at the line representing lost heat in this process, we can see that that line now is one, two, three, grid spaces wide.

We can then measure the width of the useful output line. That’s equal to one, two, three, four grid spaces in width. Since these are the only two outputs for this process, that means that the total input should equal their sum in its line width, that is, seven blocks. So let’s count the number of blocks in the input side. We see that there are one, two, three, four, five, six, seven blocks. This tells us that 100 percent of the energy input is accounted for as output, either useful output, which we can see is equal to four-sevenths of the total energy, or energy lost to heat, which is three-sevenths of the total energy.

This grid is so useful because it lets us quantitatively compare the various outputs and inputs. But in some cases, when the grid isn’t there on the diagram, it’s helpful to have a ruler at hand. That way, we’re still able to get a quantitative understanding of the width of each line, input as well as output. Let’s get a bit of practice now working with Sankey diagrams through an example.

The image shows Sankey diagrams for four different processes. Which process is the most efficient? Which process is the least efficient?

Taking a look at the image, we see these four different processes, labeled a), b), c), and d). For each one, there is some amount of input energy and some amount of useful energy output as well as some wasted energy. Based on our understanding of these diagrams, we wanna figure out which of these four processes is most efficient and which is least efficient.

To answer these questions, we’ll need to recall what efficiency means. Mathematically, efficiency is defined as the output of a system divided by its input. In our case, we’re talking about energy. So we want to solve for the ratio of the useful energy output for each of these four processes to the energy input. We can see that each of the four processes a, b, c, and d has some useful energy output as well as energy input. But to figure out their ratio, their efficiency, we’ll need to understand these Sankey diagrams.

Notice that each of the four diagrams is overlaid by a grid. This grid shows us the relative width, that is, the proportion, of energy output, whether usefully or wasted to the input. Using these diagrams and this grid, the way we figure out this ratio of useful energy output to energy input is by counting blocks or units on this grid for the energy input and useful energy output parts of each of the four diagrams. Once we know those relative values for each of the four different processes, we’ll be able to calculate each one’s efficiency.

That said, let’s get started looking at the first process, shown in diagram a. The first thing we want to do is count how many blocks wide, so to speak, the input energy is. This will give us a sense for the total energy input into this process. So starting at the bottom of this vertical stack, here we have one block, two, three, four, five, six, seven, eight, nine, 10. There are 10 total blocks on the input energy side. So if we calculate the efficiency, we can call it lowercase 𝑒 for process 𝑎. So we’ll give it a subscript 𝑎. Then we know that that will be equal to the useful energy output divided by the input energy, which we just calculated to be 10 blocks or 10 units.

Now that we figured out the input for process a, let’s figure out the useful energy output. And to do that, we’ll once again count blocks. On the useful energy output part of this diagram, we count one and then two units or two blocks on our grid. This means that, scaled to the input energy, the useful energy output is two-tenths of that. And two divided by 10 is point two zero. Or written as a percent, it’s equal to 20 percent. That’s the efficiency of process a.

Now on to the efficiency of process b. We’ll call this 𝑒 sub 𝑏. When we go to count the number of units or number of blocks comprising the input energy for this process, starting at the bottom, we find it’s one, two, three, four, five, six, seven, eight, nine, 10 blocks once again. And now that we look carefully at the processes shown in diagrams c and d, we see that their input energy is a match for the number of blocks of input energy for b and a. All four have 10 units or 10 blocks representing that input.

That’s good to know. It means that, from now on, we only need to measure the useful energy output for each of the processes. We already know the input. So the input for process b as it was for process a is 10 units. And the useful energy output we count to be one block, two blocks, three, four blocks. So the efficiency of process b) is four divided by 10 or 0.40 as a decimal. And written as a percent, that’s 40 percent.

Next, on to calculating the efficiency of process c. We saw that the input for this process is 10 blocks or 10 units. So we’ll write that down. And then we go to count the number of units or blocks of the useful energy output. We count zero, one, two, three, four, five, six grid spaces. So if the energy input for process c is 10, then the useful energy output is six. As a decimal, that’s equal 0.60. And as a percent, it’s 60 percent.

Then last but not least, we calculate the efficiency of process d. Once again, the input energy comprises 10 units or 10 grid spaces. And then the useful energy output is one, two, three, four, five, six, seven, eight spaces. Writing this in our enumerator, we see that we have an efficiency of eight-tenths or 0.80. That’s equal to 80 percent.

Now that we’ve calculated the percent efficiencies for each of the four processes, we can return to our two questions. The first question asks, “Which process is the most efficient?” And looking over our calculated efficiencies, we can see that it’s process d. This has the highest efficiency of 80 percent.

The next question asks, “Which process is the least efficient?” And we can see that that’s process a, at 20 percent. So then, based on our analysis of these Sankey diagrams, we’ve been able to figure out which of the four processes is most as well as least efficient.

As a side note, notice that, for each of these four diagrams, if we had counted up the number of grid spaces represented by the wasted energy output and added that number to the useful energy output grid spaces, then that sum would equal the total energy input. In other words, for each of the four cases, we’ve accounted for 100 percent of the input.

Let’s take a moment now to summarize what we’ve learned about Sankey diagrams in this lesson. At the outset, we learned that Sankey diagrams depict process inputs and outputs, with line width representing the relative quantity of those inputs and outputs. We furthermore saw that often these Sankey diagrams are overlaid with a grid, which lets us measure out the relative width of the outputs and inputs. And if there is no grid, then it’s always possible to use a ruler to make these measurements.

We also saw that, in these diagrams, if we sum or add up the line widths of all the outputs in a process, then that total yields the total input in the process. In other words, the outputs account for 100 percent of what is input. Nothing is lost. Finally, Sankey diagrams allow for calculating efficiencies in a process as well as quickly assessing a process’s relative outputs.