6.11. Calculating Specific Energy in Rectangular Channel

To solve this, so what we know, we write it down, V1 is equal to 0.20 meter and that is the existing depth, area is A1 is By1, B is already given, width of 81.8 meter. So, it is 1.8 into 0.2, that is, 0.36 meter square.

Specific energy in a hydraulic system is defined as the energy per unit weight of water at a specific depth. In this case, we are starting with an existing depth, V1, which is the initial depth of water. The area of flow, A1, is calculated by multiplying the width of the channel (given as 1.8 m) by the depth (0.2 m). Therefore, the area A1 becomes 0.36 m², which is crucial for further calculations regarding energy and flow velocity.

Imagine a river where the water is flowing through a narrow section created by rocks. The width of this section determines how concentrated the water flow is and consequently influences how fast the water moves. In our calculations, we simulate this by using the channel's width and water depth to find how much energy is present at that flow.

Velocity V1 is simply, Q by A1, Q is given 1.80 and area we know is calculated 0.36 and this gives us to be 5 meters per second.

To find the flow velocity V1, we use the equation V1 = Q / A1, where Q is the discharge calculated for the given area. In this scenario, we are given a discharge of 1.80 m³/s. By dividing this discharge by the area of flow (0.36 m²), we get a velocity of 5 m/s. This velocity determines how fast the water is flowing through the channel, which is essential for calculating specific energy and Froude numbers later on.

Think of a garden hose: if you restrict the end with your thumb, the water rushes out faster. Similarly, the area of flow in the channel dictates the velocity. A smaller area means faster water flow, just like a hose where the same amount of water is forced through a smaller opening.

Specific energy; E1 is equal to y1 + V1 square by 2g, so y1 we already know, plus V, V we have already calculated, 2 into 9.81, so this comes out to be 1.47 meter.

Specific energy E is calculated using the formula E = y + V²/2g, where y is the depth of flow, V is the velocity, and g is the acceleration due to gravity (approximately 9.81 m/s²). Using the previously determined depth (0.2 m) and velocity (5 m/s), we substitute these into the equation to find the specific energy, which comes out to be 1.47 m. This specific energy represents the total energy available to the system at that flow condition.

Consider a waterfall. The height of the water that falls is akin to the depth y, while the speed with which it crashes down is similar to the velocity V. Together, they determine the energy of the falling water – how much work it can do, like turning a water wheel for energy.

Now, we have to calculate y2, so let say y2 is depth alternate to y1 and specific energy diagram. I would like to remind you, when we are dealing with specific energy, we found out a cubic equation, for a particular Q, there were existing 3 values of y.

In hydraulic systems, for a certain discharge (Q), there can be multiple depths (y) corresponding to the same specific energy. We denote one of these alternate depths as y2. By analyzing the specific energy curve, we can derive the value of y2, given that we already know y1 and the specific energy E1. It is crucial to remember that in these scenarios, we often find multiple roots, but only the physically meaningful ones are considered.

Imagine designing a dam: there might be multiple levels where water gathers based on how the water flows. Each level represents a different energy state, just like the y values corresponding to the same energy level in our calculations.

So, Froude number at first location is going to be V under root 9.81 into 0.2, that is, 3.57.

The Froude number (Fr) is a dimensionless quantity used to describe flow regimes. It is calculated using the formula Fr = V / √(g*y), where V is the velocity, g is the acceleration due to gravity, and y is the flow depth. By substituting in our calculated values, we determine the Froude number to assess if the flow is subcritical or supercritical. A Fr value of 3.57 indicates supercritical flow, signifying high velocity relative to the gravitational forces acting on the water.

Think of a car going downhill: if it speeds up too quickly (like our supercritical flow with a high Froude number), the stability decreases, just as the flow can become turbulent or unstable if it exceeds a certain threshold in open channels.

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