 When we talk about frictional losses in internal flow, we sort our frictional losses into two categories. Major losses, which are the frictional effects between the fluid and the pipe wall, and what we call minor losses, which are any point where the fluid changes shape or changes size or changes direction. Minor losses can include losses through valves, elbows, bends, nozzles, diffusers, those sorts of things. They typically make up a small proportion of the frictional losses. That's why they're referred to as minor losses. For frictional losses in general, we describe them as head losses, and we can calculate them by considering our conservation of energy equation from chapter 3. We had neglected the frictional losses, the head loss, but we can talk about it now. In general, our friction head, or our head loss, is the Darcy friction factor multiplied by the unitless proportion of length over diameter, multiplied by the velocity squared divided by 2 times gravity. The Darcy friction factor in general is a function of the Reynolds number, a relative roughness, which is the proportion of how bumpy the pipe is divided by diameter, and the shape of the pipe or duct through which the fluid is flowing. We can calculate the friction factor by using 8 times the shear stress at the wall divided by diameter times velocity squared. For laminar flow, we can use our Poiseuille flow velocity profile that we had developed back in our differential relations chapter. And for Newtonian fluids, we can use a shear stress relationship between viscosity and shear rate to represent our shear stress at the wall as 8 times dynamic viscosity times velocity divided by diameter. When we take 8 times that result divided by density times velocity squared, we end up with 64 divided by the Reynolds number. So for laminar flow, the Darcy friction factor is just 64 divided by the Reynolds number. A nice, simple, easy equation. For turbulent flow, our equation looks like this. This is called the Colbrook equation, and it relates the friction factor to relative roughness and the Reynolds number. Unfortunately, it's an implicit relationship. Meaning friction factor appears on both the left and right sides of the equation. In order to solve this, we have to come up with a numerical solution by guessing and checking. If you're using a fancy calculator or MATLAB, it can do the guessing and checking for you to converge on a very accurate number. If you're solving this by hand, you either have to guess and check yourself or use a graphical solution like the Moody chart. The Moody chart relates friction factor to Reynolds number for a variety of relative roughnesses. Relative roughness is a dimensionless proportion, and it represents the average bump height along the walls divided by diameter. That average bump height we represent with an epsilon. Here's a table of some example epsilon values for a variety of substances. If you're buying a pipe, you can either check with the manufacturer to figure out what the epsilon value would be, or you can go through a third-party testing service, or you can measure it yourself. But armed with an epsilon value and a diameter, we can calculate a relative roughness and use that value with our Reynolds number to look up a friction factor on the Moody chart. That is about plus or minus 15 percent accurate across the entire range of the chart.