Video: Determining the Efficiency Using Sankey Diagrams

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

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Video Transcript

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.

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