3. Another Problem

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 problem, we are dealing with a corroded concrete pipe that has a large roughness value affecting the flow of water through it. The diameter of the pipe is 1.5 meters, and the roughness can cause significant friction losses that affect how smoothly the fluid can flow. The problem states that a lining will be added to the pipe to reduce this roughness, and it asks us to calculate the power savings associated with this improvement. Power savings occur because reducing the roughness decreases the energy loss (head loss) due to friction in the pipe, allowing less energy (and therefore less power) to be used to maintain the same flow rate.

Think of a water slide. If the slide is smooth (like a newly lined pipe), the water flows quickly with little resistance. But if the slide is rough and full of bumps (like the corroded pipe), the water slows down, and you have to push harder to keep it moving. In this scenario, by smoothing the slide (lining the pipe), we can let the water flow more easily, saving energy and making for a better ride!

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.

To find the velocity of the fluid in the pipe before the lining is applied, we use the formula for flow rate, which is Q = A * V. Here, Q is the volumetric flow rate (4 m³/s), A is the cross-sectional area of the pipe, and V is the velocity of the fluid. The area A of a circular pipe can be calculated as A = π/4 * D², where D is the diameter. Plugging in our values gives us the initial velocity V1 of approximately 2.264 m/s. This is the speed at which water flows through the pipe before the lining is applied.

Imagine a garden hose. When you turn on the water, the speed of the water flowing out can be found by knowing the size of the hose and the amount of water you want going through. The larger the hose (like the wider pipe), the faster the water can flow. Here, we calculated how fast the water flows through our large pipe.

For this particular case, Reynolds number is going to be Re1 as V1 D1 by nu. So V1 we have already found 2.264, diameter we know, it was 1.5 meter and nu is 1 into 10 to the power -6. So Reynolds number comes out to be 3.395 into 10 to the power 6.

The Reynolds number is a dimensionless quantity used to predict flow patterns in fluid dynamics. It is calculated using the formula Re = (V * D) / ν, where V is the flow velocity, D is the diameter, and ν is the kinematic viscosity of the fluid (in this case, water). By substituting the values we've previously calculated, we find that the Reynolds number is 3.395 x 10^6, indicating that the flow in the pipe is turbulent because this number is significantly greater than 4000.

Think of the Reynolds number like a speed limit for your flow. If it's too low, the water flows smoothly without any chaos—like a quiet drive on a country road (laminar flow). But when it's high, the water is like a rush-hour commuter with all kinds of traffic—things start to get a bit chaotic (turbulent flow).

Using the empirical equation of Haaland, okay, then we can find under root 1 by f1 is equal to the things on the right-hand side and on calculating, we can get because this is an explicit formula, alright. So f1 we can get 0.0379.

To find the friction factor f1, we use the Haaland equation, which is an empirical relationship that estimates the Darcy-Weisbach friction factor based on Reynolds number and relative roughness. In this case, we have calculated it to be 0.0379. This friction factor is crucial because it is used in subsequent calculations of head loss due to friction in the pipe.

Imagine riding a bicycle: the surface of the road affects how fast you can go. A smooth road (low friction) lets you glide easily, while a rough road (high friction) slows you down. Here, the friction factor is like the quality of the road for our water flow; the lower the friction factor, the easier the flow.

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, 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...

To calculate head loss, we use the Darcy-Weisbach equation, which relates the head loss in pipes to the friction factor, pipe length, velocity, and diameter. We calculate the head loss before and after the lining using the appropriate friction factors (f1 and f2). This allows us to see how much energy is lost in the system before and after the lining is applied, which is crucial for understanding the impacts of reducing roughness.

Think of a slide again but this time, let's consider the height you slide down. If it's a smooth, well-maintained slide (low head loss), you go down quicker than if there were bumps that slow you down (high head loss). This is similar to how much energy is lost when water flows through a pipe; smoothing it out helps conserve energy.

So savings in head, you see the frictional loss before the lining was 6.6 meter and after the lining is put as 2.450. So saving in head, hs is hf1 – hf2 and this will come out to be 4.150 meter. Now for power, savings in power would be nothing but gamma Qhs.

Once we have the head loss before and after lining, we can find the difference, which represents the head saved due to reduced friction loss. This head savings can be converted into power savings using the formula P = γ * Q * hs, where γ is the specific weight of water, Q is the flow rate, and hs is the head saved. This calculation shows the tangible benefits of reducing friction in the pipe.

If we think of a water tank, lowering the friction and saving head is like using less power from a pump to get the same amount of water to a certain height. You spend less energy on moving the water, just like using less gas to drive the same distance if your car runs smoother.

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