1.2. Pipe Networks

So until this point in time, what are the things that we know? We need to find to an f, f is the Darcy Weisbach friction factor. Darcy’s friction factor and that is f is equal to phi of function of Reynolds number and epsilon by D. So Reynolds number we can calculate if the flow is given, D is the pipe diameter, so epsilon, we saw that we can find through these tables. Most of the cases, for your significance in practice for the numerical, you will be given the value of epsilon by D. So if you are able to calculate the value of epsilon by D and Reynolds number, then there is a dependence of these two parameters, Re and D on the friction factor f and if we know the friction factor, we can calculate by using these values and find out the head loss, the major head loss.

The friction factor (f) is crucial for understanding head loss in pipes, as it quantifies resistance to flow due to friction. It is affected by two key parameters: the Reynolds number (Re), which indicates if the flow is laminar or turbulent, and the relative roughness (epsilon/D), which represents the roughness of the pipe's internal surface. To compute the friction factor, we can use empirical equations or charts like the Moody chart, which relate f to Re and epsilon/D. Calculating these parameters allows us to determine how much energy is lost due to friction as fluid moves through the pipe.

Think of water flowing through a garden hose. The rougher the inside of the hose (like if it were coated with sandpaper), the more difficult it is for water to flow, similar to what the friction factor does. If you had a smooth hose, like a large, polished tube, water would flow much easier, reflecting a lower friction factor.

So there is something called a Moody chart. So friction factor is a function of Reynolds number and relative roughness for round pipes is a chart like this, okay, where this is Reynolds number on x-axis and f you can find out and epsilon by D is plotted in the right.

The Moody chart is a graphical representation used to find the Darcy friction factor (f) based on the Reynolds number and relative roughness of the pipe. On the chart, the x-axis represents the Reynolds number, while lines represent different values of relative roughness. Depending on the flow conditions, one can locate the point on the chart that corresponds to their specific Reynolds number and relative roughness to determine the friction factor necessary to calculate head loss.

Imagine a cookbook where different recipes tell you how much salt to add based on the type of meat and desired flavor intensity. The Moody chart acts like that cookbook but for engineers, guiding them on how to calculate flow resistance in different pipes.

However, for this course, of course, if you are given this Moody chart, you should be able to find a corresponding Reynolds number and a corresponding epsilon by D line here and then go ahead and find out the respective f, friction factor. But for your convenience, I am going to provide you two formulas, one is a Colebrook formula, which relates this friction factor f to epsilon by D and Reynolds number. Can you see there is one trick in this formula? If you note, you will see f is also in the left hand side and f is also on the right hand side, that makes it implicit in nature, alright. But I still expect you to remember this formula. The solution for this formula can be done through trial and error, alright. Or you can use another formula, which is totally explicit in nature, which is called Haaland equation. So here you see, there is friction factor unknown is only on the left hand side, so if you know epsilon by D and you know Reynolds number, you will be able to find out the value of f.

The Colebrook equation is an implicit equation that requires iterative methods to solve for the friction factor (f) because it appears on both sides of the equation. This means that an initial guess is often made, and the equation is solved multiple times to converge on an accurate value of f. In contrast, the Haaland equation is explicit, allowing for direct calculations of f from known values of epsilon/D and Reynolds number without iteration. Understanding these formulas is essential for engineers as they provide a means to efficiently calculate friction factor values.

Imagine trying to predict the score of a basketball player based on their previous performances. The Colebrook equation is like asking, 'If I know a player usually scores 20 points, how many times should I guess to hit it right?' The Haaland equation, on the other hand, gives a direct prediction based on straightforward observations, allowing you to compute without guessing.

Head loss was a function of friction factor, right? I mean, it was dependent on friction factor and f was a function of Reynolds number and epsilon by D, so we can find f using these two formulas or Moody chart and therefore, we will be easily able to calculate the head loss.

Head loss in a pipe is directly related to the friction factor; as the friction factor increases, so does the head loss. To calculate head loss, engineers can use the formula: head loss (hf) = f * (L * V^2) / (2 * g * D), where L is the length of the pipe, V is the flow velocity, g is the acceleration due to gravity, and D is the diameter of the pipe. By determining the friction factor first using empirical methods or charts, we can then plug it into this equation to find out how much energy is lost due to friction over a specific distance.

Consider riding a bike on a road. If the road is smooth, you maintain speed well (low head loss). If the road turns bumpy, you have to work harder to keep going (high head loss). Similarly, head loss quantifies how much energy is required to maintain flow through a pipe depending on its frictional characteristics.

To demonstrate that, we have a problem question here, that we are going to solve now. So the question is this, a badly corroded concrete pipe of diameter 1.5 m has an equivalent sand roughness of epsilon S 15 mm. So we have already been given epsilon S. At 10 mm thick lining is proposed to reduce the roughness value to epsilon S of 0.2 mm. The question is, for a discharge of 4 meter cube per second, calculate the power saved per kilometer of the pipe.

In this example, we explore how corrosion in a pipe leads to increased roughness and consequently, higher energy losses through head loss. The problem involves calculating power savings achieved by lining the pipe to reduce roughness. By finding the friction factors before and after the lining, the head losses can be determined, leading to an overall energy savings calculation. This illustrates the importance of pipe maintenance and design in hydraulic engineering.

Imagine you own a heavily used water slide at a water park. Over time, the slide becomes rough, causing people to go slower and decreasing enjoyment. If you smooth it out with a new coating, riders will zoom down faster (less friction, less energy loss), just as lining a pipe saves energy by reducing head loss due to friction.

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