5.4. Equations of Flow Balance

The Bernoulli’s equation will require that at 2 points we equate y1 + v1² / 2g + Z1 is equal to y2 + v2² / 2g + Z2.

This section introduces Bernoulli's principle, which states that in a flowing fluid, the total mechanical energy is conserved. The left side of the equation represents the energy at point 1 with terms for the height (y1), velocity (v1), and elevation (Z1). The right side corresponds to point 2 with similar terms (y2, v2, Z2). When these points are analyzed, energy lost to friction, viscosity, or other factors must be considered. Here, Z1 is zero, and Z2 is a height of 0.5 feet. Therefore, knowing the flow conditions allows us to relate the states at the two points.

Think of a water slide. At the top, the water has potential energy because it's high up, and as it slides down, it's converted into kinetic energy, making it go faster at the bottom. The slide's height and the speed at various points can be calculated much like Bernoulli's equation.

Now, we also apply the continuity equation and this will give us a second equation. So, V1 y1 = V2 y2.

The continuity equation expresses the principle of conservation of mass: the amount of fluid flowing into a section must equal the amount of fluid flowing out. In this case, it shows how the velocity (V1, V2) and depth (y1, y2) of the fluid relate at two points along the flow. If the channel width is constant, this becomes crucial in solving for unknown variables.

Imagine squeezing a garden hose. If you cover a part of the hose with your thumb, the water flows faster out of the end. The continuity equation explains why: to keep the flow steady, if the area decreases (when covered by your thumb), the velocity must increase to maintain the same flow rate.

Now, if we solve this, we will get 3 solutions, y2 = 1.72 feet, y2 = 0.638 feet, and a negative value.

After applying the continuity equation combined with Bernoulli's equation, we obtain a cubic equation. Solving the cubic provides us with three values for y2, the downstream depth. However, in physics, negative solutions are often impractical, so they can be disregarded. The two valid solutions give insights into possible water surface elevations downstream.

Imagine you're trying to find out how deep a puddle is after a rainstorm. You could measure several points, and one measurement might be negative—meaning that point is above the ground—in which case it doesn't provide useful information.

This question can be answered by use of the specific energy diagram obtained from equation 10 which for the problem is.

The specific energy diagram is a graphical representation of energy relative to the flow depth. It allows engineers to visualize and understand the relationships between energy, depth, and flow conditions. The diagram plots specific energy against flow depth, helping to identify critical flow conditions and overall energy trends in the channel.

Think of it like a financial graph showing income over time. Just as you can see how your savings might grow with interest (energy changes), the specific energy diagram shows how flow conditions change and helps predict future states of the water flow.

Thus, the surface elevation is 2.2. So, because of this discussion y2 + Z2 is not going to be 1.68 feet...

The surface elevation discussed provides detailed insight into the water's state as it transitions from upstream to downstream. It emphasizes the impact of different factors, such as channel shape and energy losses, on determining real water surface heights versus theoretical predictions.

This is much like estimating the height of a tree you saw from a distance. You might estimate lower or higher based on perspective and surroundings. Similarly, engineers use various calculations and diagrams to predict the true height of water elevation in a channel.

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