3.6. Horizontal Pipeline with Different Diameters

Now the one more question is there is a horizontal pipeline which is 50 meters long and is connected to reservoir at one end and discharges freely into the atmosphere at the other end. So for the first 25 meters, I have shown this diagram here. So the first 25 meters length from the reservoir, the pipe has a diameter of 15 centimeter and it has a square entrance at the reservoir. The remaining 25 meter length of the pipe has a diameter of 30 centimeter, so this is the long, long pipe.

In this example, we have a horizontal pipeline that spans a total length of 50 meters. The pipeline is designed in two sections: the first half (25 meters) has a smaller diameter of 15 centimeters, while the second half (the remaining 25 meters) has a larger diameter of 30 centimeters. One end of this pipeline connects to a reservoir filled with water, and the other end discharges into the atmosphere. The change in diameter affects how water flows through the pipe, especially due to the different velocities and pressures that will be encountered in each section.

Think of this pipeline as a garden hose that narrows at one end. When you turn on the water, it flows easily through the wider end, but once it reaches the narrow part, the speed of the water increases. Similarly, in our example, when water moves from the 15 cm diameter section to the 30 cm diameter section, it experiences different flow velocities and pressures.

Junction of the two pipes is in the form of a sudden expansions. So we have been told sudden expansion that means there is going to be a energy loss, minor loss. It is also told that there is going to be a there is a gate valve in a 15 centimeter pipe. So there is going to be another minor loss here.

When there is a sudden change in the diameter of a pipe, such as moving from a smaller pipe (15 cm) to a larger pipe (30 cm), it creates a minor loss due to turbulence. This is referred to as a minor loss, and it represents the energy loss that occurs when the flow pattern changes abruptly. Additionally, the presence of a gate valve in the smaller pipe can also create a minor loss. Each of these factors needs to be accounted for when calculating the overall efficiency and pressure in the system.

Imagine driving a car on a smooth road, and suddenly hitting a bump that makes the car bounce. This bump is like the sudden change in pipe diameter; it disrupts the smooth flow of water, leading to a loss in energy. Similarly, if you were to stop the car abruptly using brakes, it could represent the loss of pressure caused by the gate valve. Both scenarios illustrate how sudden changes can affect overall flow dynamics.

If the height of the water surface in the tank is 10 meter above the centerline and the velocity, estimate the discharge in the pipe by considering the Darcy Weisbach friction factor of 0.02 for both the pipes.

To estimate the discharge through the pipeline, we apply the principles of fluid dynamics. The height of the water in the reservoir (10 meters) contributes to the potential energy of the water, which drives flow through the pipe. We also consider friction losses described by the Darcy-Weisbach equation, which provides a way to quantify losses due to friction along the length of the pipe. In this case, the friction factor is given as 0.02 for both sections of the pipeline. By combining these elements – the water height and the friction losses - we can calculate the discharge flow rate through the entire system.

Think of this scenario as measuring how fast a fountain shoots water when you fill it with liquid. The taller the water level (10 meters in this case), the stronger the flow will be. However, if there are sections in the fountain's plumbing that are too narrow or have turns (like the friction factors we calculate), it will affect how fast the water can shoot up. Thus, while we calculate the discharge, we consider both the height (pressure) of the water and any resistances along the way.

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