2. Problem Demonstration

So do that 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. So this is the oldest method of finding these friction factors, Darcy friction factor. 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.

In fluid mechanics, the Moody chart represents the relationship between the Darcy-Weisbach friction factor, Reynolds number, and relative roughness of the pipe. The Reynolds number is a dimensionless quantity that helps predict flow patterns in different fluid flow situations. The relative roughness is the ratio of the roughness height to the diameter of the pipe. In summary, this chart allows engineers to determine how much resistance (friction) will occur in a pipe, affecting fluid flow.

Imagine trying to slide a book across a table. If the table is smooth (like a pipe with low roughness), the book will slide easily (low friction). However, if the table is rough (like a pipe with high roughness), the book will be harder to slide (high friction). The Moody chart is like a tool that tells you how different surfaces (or pipe materials) will affect the ease of sliding (flow of fluid).

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. So a thick lining would be put, which is 10 mm long and this will bring down the roughness to 0.2 mm.

The problem introduces a scenario where a corroded concrete pipe has a diameter of 1.5 meters and a roughness (epsilon S) of 15 mm, which affects the flow of fluid through it. Plans are made to install a 10 mm thick lining inside the pipe to reduce the roughness to 0.2 mm. This change aims to decrease friction and head loss when fluid flows through the pipe, ultimately making the system more efficient.

Think of a water slide at an amusement park. If the slide is bumpy (like the corroded pipe), water (fluid) will not flow smoothly, resulting in a rough ride (more energy loss). If the operators smoothen the slide by adding some material (similar to the thick lining), riders will have a more enjoyable (efficient) experience as they'll slide down faster (less friction and energy loss).

So like always we are going to start the solution on the white screen. So we see, before the measure of reducing the head loss, before the lining was put, V1 is going to be Q/A, so Q was 4 meter cube per second, area is pi by 4 into 1.5 whole square. So this is going to be 2.264 meter per second, alright.

In fluid dynamics, the flow rate (Q) is defined as the volume of fluid passing through a cross-section of the pipe per unit time. To find the velocity (V1) of the fluid before the lining is added, we can use the formula: velocity = flow rate / cross-sectional area. In this case, the diameter of the pipe is used to calculate the area, which, along with the flow rate, gives us a fluid velocity of approximately 2.264 m/s.

Imagine you have a garden hose. If you have water flowing through it at a steady rate (the flow rate), and you know the size of the hose opening, you can figure out how fast the water is moving. Just like measuring this helps you know how quickly you can water your plants, engineers need to know the velocity of fluids in a pipe to ensure everything works efficiently.

Similarly, let us talk after the lining is put. So after the lining is put, diameter will be reduced, how much? 1.50 was the diameter and lining of this, so the diameter becomes 1.48 meters. Therefore, the velocity will be 4Q by A pi by 4 into 1.48 to the whole square, that means 2.325 meters per second, alright.

Once the lining is installed, the effective diameter of the pipe is reduced to 1.48 m. Consequently, the area changes, which affects the fluid velocity. Using the same principles as before, the new velocity (V2) is calculated based on the new diameter. This slight change in diameter results in a new fluid velocity of approximately 2.325 m/s, which reflects the improved efficiency due to reduced roughness.

Continuing with the garden hose example: If you were to pinch the end of the hose just a little, the water coming out at that point speeds up (increased velocity), while still maintaining the same flow rate. Similarly, when the pipe lining reduces the effective diameter, the fluid flows faster, leading to higher efficiency.

So now we have calculated f1 and f2, f1 is the friction factor before the lining was put and f2 is the friction factor after the lining was put. So to continue further, alright. We are going to calculate head loss hf is equal to fLV square by 2gD.

The friction factor (�12f1 and f2) quantifies the resistance the fluid experiences while flowing through the pipe. These values are obtained based on the Reynolds numbers calculated earlier for the flow before and after adding the lining. Knowing both friction factors allows us to calculate the head loss using the Darcy-Weisbach equation, which relates head loss to flow conditions and characteristics of the pipe.

Imagine sliding down two different water slides—one with bumps and the other smooth. The bumps represent friction that slows you down. The calculations we make for f1 and f2 help us understand how much those bumps affect your speed, just as the friction factor helps engineers predict fluid behavior in pipes.

So now to continue further, we are going to calculate head loss hf is equal to fLV square by 2gD. So before the lining was put, hf1 would be 0.0379 that was the friction factor 1, length is 1000 and velocity was 2.264 whole square divided by 2 into 9.81 into 6.60. So hf1 comes out to be 6.6 meter.

Calculating the head loss (�12hf1 and hf2) involves plugging in the friction factor, the length of the pipe, and the velocity into the Darcy-Weisbach equation. In this case, we find that the head loss before lining is approximately 6.6 meters, which is the total vertical distance that a fluid must lose due to friction as it flows through the pipe.

Think of head loss like the height you lose while sliding down a hill. The steeper the hill (or the more friction there is), the more height you lose. In our case, the calculation tells us how much 'height' or energy is lost simply because of the rough pipe surface.

Now for power, savings in power would be nothing but gamma Qhs, right. So gamma is 9.79 into 4 into 4.15 kilowatt, because we have taken this 9.79 into 10 to the power 9790 into g. So 9790 instead we did 9.79 therefore we are telling in kilowatt and this will come to be 162.5 kilowatt.

To estimate the savings in power due to reduced friction, we calculate how much energy is conserved as a result of the reduced head loss. By applying the power formula (gamma Qhs), where gamma is the weight density of the fluid, Q is the volumetric flow rate, and hs is the saving in head, we find that operating the lined pipe results in approximately 162.5 kilowatts of power savings.

When you reduce friction on a water slide (like smoothing it out), riders come down faster and use less energy to do the same distance. Similarly, our calculations show that by lining the pipe, we save energy, which in terms of power translates to a significant reduction in energy costs.

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