2.2. Power Savings Calculation

A badly corroded concrete pipe of diameter 1.5 m has an equivalent sand roughness of epsilon S = 15 mm. A thick lining of 10 mm is proposed to reduce the roughness to epsilon S = 0.2 mm. The question is to calculate the power saved per kilometer of the pipe for a discharge of 4 m³/s.

This problem involves understanding how the roughness of a pipe affects fluid flow. When a pipe is corroded, it has a greater roughness, which increases the resistance to flow and, therefore, leads to higher energy losses and reduced efficiency. Adding a lining smooths the internal surface of the pipe, thus reducing its effective roughness. The goal is to calculate how much power (energy per unit time) is saved due to this reduction in roughness as we expect lower energy losses in the form of heat.

Imagine trying to slide down a slide covered in sandpaper compared to one that is smooth. The roughness of the sandpaper makes it harder and slower to slide down, just like a rough pipe makes it harder for water to flow through, leading to higher energy costs for pumping.

Before lining: Velocity V1 = Q/A = 4 m³/s / (π/4)(1.5 m)² = 2.264 m/s. Reynolds number Re1 = V1 * D1 / ν. Re1 = 2.264 * 1.5 / (1 x 10^(-6)) = 3.395 x 10^6. Epsilon S1/D1 = 15 mm / 1.5 m = 0.10.

To find the initial conditions before any modifications are made, we calculated the velocity of flow through the pipe using the formula for velocity (Velocity = Flow Rate / Cross-Sectional Area). Once we have the velocity, we can calculate the Reynolds number which helps us understand the flow regime (laminar or turbulent). The friction factor and head loss are dependent on Reynolds number as well as relative roughness (epsilon/D).

Think of this like figuring out how quickly water flows through a hose. We first need to know how much water is flowing and the size of the hose (diameter) to understand how fast it's coming out. The Reynolds number tells us whether the flow is smooth or turbulent, similar to how the rush of water can either be consistent or create splashes depending on how fast it's flowing.

Using the Haaland equation or Moody chart, we calculate the friction factor before lining: f1 = 0.0379 (using Haaland equation).

The friction factor is crucial for determining head loss in the pipe. We can find the friction factor using either the Haaland equation or a Moody chart, both of which relate flow characteristics to friction loss in the pipe. The Haaland equation is particularly useful because it directly gives a formula based on the Reynolds number and relative roughness.

If we compare it to driving a car, the friction factor is like the road surface condition. A smooth road (lower friction) lets your car glide effortlessly, while a rough road (higher friction) makes the ride bumpy and slow. Understanding these factors helps engineers design better, more efficient systems.

After lining, the new diameter D2 = 1.48 m. New velocity V2 = Q/A = 4 m³/s / (π/4)(1.48 m)² = 2.325 m/s. New Reynolds number Re2 = 2.325 * 1.48 / (1 x 10^(-6)) = 3.44 x 10^6. Epsilon S2/D2 = 0.2 mm / 1.48 m = 1.35 x 10^(-4). Calculate friction factor f2 = 0.0132.

After applying the lining, we recalculate the diameter of the pipe, which slightly changes the flow characteristics. The fluid velocity is recalculated with the new area. This change also affects the Reynolds number and relative roughness again, allowing us to calculate the new friction factor using the same methods as before.

Adding a lining to the pipe can be compared to changing the width of a river. A slightly wider river allows more water to flow through without as much resistance, similar to how the lining reduces friction and allows for more efficient fluid flow.

Head loss before lining: hf1 = f1 * L * V1² / (2g * D1) = 6.6 m. Head loss after lining: hf2 = f2 * L * V2² / (2g * D2) = 2.450 m. Savings in head: hs = hf1 - hf2 = 6.6 m - 2.450 m = 4.150 m.

We calculate the head loss before and after the application of lining using the Darcy-Weisbach equation to see how much head loss has been reduced. The difference in head loss directly translates to how much energy is saved in pumping the fluid through the pipe.

Consider head loss like trying to push a cart up a slope. A steep incline (higher head loss) requires more effort than a gentle slope (lower head loss). Reducing the slope (head loss) makes it easier to push the cart (more efficient fluid flow).

Savings in power = γ * Q * hs = 9.79 kN/m³ * 4 m³/s * 4.150 m = 162.5 kW.

Finally, we calculate the power savings which is the energy saved per unit time due to the reduction in head loss. This is computed by multiplying the weight of the fluid (density times gravity) by flow rate and the head saved. The result shows how much less energy is required to pump the water after the lining is applied.

If we think of power savings as the effort saved when pushing a cart, reducing the resistance (head loss) means you expend less energy to get the same amount of work done (moving the fluid). In practical terms, this translates to cost savings on energy bills.

--