### 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.