All pipes carrying fluids experience losses of pressure caused by friction and turbulence of the flow. It affects seemingly simple things like the plumbing in your house all the way up to the design of massive, way more complex, long-distance pipelines. I’ve talked about many of the challenges engineers face in designing piped systems, including water hammer, air entrainment, and thrust forces. But, I’ve never talked about the factors affecting how much fluid actually flows through a pipe and the pressures at which that occurs. So, today we’re going to have a little fun, test out some different configurations of piping, and see how well the engineering equations can predict the pressure and flow. Even if you’re not going to use the equations, hopefully, you’ll gain some intuition from reading how they work in a real situation. Today, we’re talking about closed conduit hydraulics and pressure drop in pipes.

I love engineering analogies, and in this case, there are a lot of similarities between electrical circuits and fluids in pipes. Just like all conventional conductors have some resistance to the flow of current, all pipes impart some resistance to the flow of the fluid inside, usually in the form of friction and turbulence. In fact, this is a lovely analogy because the resistance of a conductor is both a function of the cross-sectional area and length of the conductor—the bigger and shorter the wire, the lower the resistance. The same is true for pipes, but the reasons are a little different. The fluid velocity in a pipe is a function of the flow rate and the pipe’s area. Given a flowrate, a larger pipe will have a lower velocity, and a small pipe will have a higher velocity. This concept is critical to understanding the hydraulics of pipeline design because friction and turbulence are mostly a result of flow velocity.

I built a demonstration in my video that should help us see this in practice. This is a manifold to test out different configurations of pipes and see their effect on the flow and pressure of the fluid inside. It’s connected to my regular tap on the left. The water passes through a flow meter and valve, past some pressure gauges, through the sample pipe in question, and finally through a showerhead. I picked a showerhead since, for many of us, it’s the most tangible and immediate connection we have to pressure problems in plumbing. It’s probably one of the most important factors in the difference between a good shower, and a bad one. Don’t worry, all this water will be given to my plants which need it right now anyway.

I used these clear pipes because they look cool, but there won’t be much to see inside. All the information we need will show up on the gauges (as long as I bleed all the air from the lines each time). The first one measures the flow rate in gallons per minute, the second one measures the pressure in the pipe in pounds per square inch, and the third gauge measures the difference in pressure before and after the sample (also called the head loss) in inches of water. In other words, this gauge measures how much pressure is lost through friction and turbulence in the sample - this is the one to keep your eye on. In simple terms, it’s saying how far do you have to open the valve to achieve a certain rate of flow. I know the metric folks are giggling at these units. For this video, I’m going to break my rule about providing both systems of measurement because these values are just examples anyway. They are just nice round numbers that are easy to compare with no real application outside the demo. Substitute your own preferred units if you want, because it won’t affect the conclusions.

There are a few methods engineers use to estimate the energy losses in pipes carrying water, but one of the simplest is the Hazen-Williams equation. It can be rearranged in a few ways, but this way is nice because it has the variables we can measure. It says that the head loss (in other words the drop in pressure from one end of a pipe to the other) is a function of the flow rate, and the diameter, length, and roughness of the pipe. Now - that’s a lot of variables, so let’s try an example to show how this works. First, we’ll investigate the effect the length of the pipe has on head loss. I’m starting with a short piece of pipe in the manifold, and I’m testing everything at three flow rates: 0.3, 0.6, and 0.9 gallons per minute (or gpm).

At 0.3 gpm, we see pressure drop across the pipe is practically negligible, just under half an inch. At 0.6 gpm, the head loss is about an inch. And, at 0.9 gpm, the head loss is just over 3 inches. Now I’m changing out the sample for a much longer pipe of the same diameter. In this case, it’s 20 times longer than the previous example. Length has an exponent of 1 in the Hazen-Williams equation, so we know if we double the length, we should get double the head loss. And if we multiply the length times 20, we should see the pressure drop increase by a factor of 20 as well. And sure enough, at a flow rate of 0.3 gpm, we see a pressure drop across the pipe of 7.5 inches, just about 20 times what it was with the short pipe. That’s the max we can do here - opening the valve any further just overwhelms the differential pressure gauge. There is so much friction and turbulence in this long pipe that I would need a different gauge just to measure it.

Length is just one factor that influences the hydraulics of a pipe. This demo can also show how the pipe diameter affects the pressure loss. If I switch in this pipe with the same length as the original sample but which has a smaller diameter, we can see the additional pressure drop that occurs. The smaller pipe has ⅔ the diameter of the original sample, and diameter has an exponent of 4.9 in our equation. That’s because, as I mentioned before, changing the diameter changes the fluid velocity, and friction is all about velocity. We expect the pressure drop to be 1 over (⅔)^4.9 or about 7 times higher than the original pipe. At 0.3 gpm, the pressure drop is 3 inches. That’s about 6 times the original. At 0.6 gpm, the pressure drop is 7.5 inches, about 7 times the original. And at 0.9 gpm, we’re off the scale. All of that is to say, we’re getting close to the correct answers, but there’s something else going on here. To explore this even further, let’s take it to the extreme.

We’ll swap out a pipe with a diameter 5 times larger than the original sample. In this case, we’d expect the head loss to be 1 over 5^4.3, basically a tiny fraction of that measured with the original sample. Let’s see if this is the case. At 0.3 gpm, the pressure drop is basically negligible just like last time. At 0.6 and 0.9 gpm, the pressure drop is essentially the same as the original. Obviously, there’s more to the head loss than just the properties of the pipe itself, and maybe you caught this already. There is something conspicuous about the Hazen-Williams equation. It estimates the friction in a pipe, but it doesn’t include the friction and turbulence that occurs at sudden changes in direction or expansion and contraction of the flow. These are called minor losses, because for long pipes they usually are minor. But in some situations like the plumbing in buildings or my little demonstration here, they can add up quickly.

Every time a fluid makes a sudden turn (like around an elbow) or expands or contracts (like through these quick-release fittings), it experiences extra turbulence, which creates an additional loss of pressure. Think of it like you are walking through a hallway with a turn. You anticipate the turn, so you adjust your path accordingly. Water doesn’t, so it has to crash into the side - and then change directions. And, there is actually a formula for these minor losses. It says that they are a function of the fluid’s velocity squared and this k factor that has been measured in laboratory testing for any number of bends, expansions, and contractions. As just another example of this, here’s a sample pipe with four 90-degree bends. If you were just calculating pressure loss from pipe flow, you would expect it to be insignificant. Short, smooth pipe of an appropriate diameter. The reality is that, at each of the flow rates tested in the original straight pipe sample, this one has about double the head loss, maxing out at nearly 6 inches of pressure drop at 0.9 gpm. Engineers have to include “minor” losses to the calculated frictional losses within the pipe to estimate the total head loss. In my demo here, except for the case of the 20’ pipe, most of the pressure drop between the two measurement points is caused by minor losses through the different fittings in the manifold. It’s why, in this example, the pressure drop is essentially the same as the original. Even though the pipe is much larger in diameter, the expansion and contraction required to transition to this large pipe make up for the difference.

One clarification to this demo I want to make: I’ve been adjusting this valve each time to keep the flow rate consistent between each example so that we make fair comparisons. But that’s not how we take showers or use our taps. Maybe you do it differently, but I just turn the valve as far as it will go. The resulting flow rate is a function of the pressure in the tap and the configuration of piping along the way. More pressure or less friction and turbulence in the pipes and fittings will give you more flow (and vice versa).

So let’s tie all this new knowledge together with an example pipeline. Rather than just knowing the total pressure drop from one end to another, engineers like to draw the pressure continuously along a pipe. This is called the hydraulic grade line, and, conveniently, it represents the height the water would reach if you were to tap a vertical tube into the main pipe. With a hydraulic grade line, it’s really easy to see how pressure is lost through pipe friction. Changing the flow rate or diameter of the pipe changes the slope of the hydraulic grade line. It’s also easy to see how fittings create minor losses in the pipe. This type of diagram is advantageous in many ways. For example, you can overlay the pressure rating of the pipe and see if you’re going above it. You can also see where you might need booster pump stations on long pipelines. Finally, you can visualize how changes to a design like pipe size, flow rate, or length affect the hydraulics along the way.

Friction in pipes? Not necessarily the most fascinating hydraulic phenomenon. But, most of engineering is making compromises, usually between cost and performance. That’s why it’s so useful to understand how changing a design can tip the scales. Formulas like the Hazen-Williams and the minor loss equations are just as useful to engineers designing pipelines that carry huge volumes of fluid all the way down to homeowners fixing the plumbing in their houses. It’s intuitive that reducing the length of a pipe or increasing its diameter or reducing the number of bends and fittings ensures that more of the fluid’s pressure makes it to the end of the line. But engineers can’t rely just on intuition. These equations help us understand how much of an improvement can be expected without having to go out to the garage and test it out like I did. Pipe systems are important to us, so it’s critical that we can design them to carry the right amount of flow without too much drop in pressure from one end to the other.