Fluid Flow on Closed Conduits - 3 Reservoir Problem - Problem #10

The video introduces Problem #10, focusing on determining the discharge in a three-reservoir pipe system. The problem involves a system of pipes connecting three reservoirs, and the objective is to analyze the flow within these closed conduits.

The presenter begins by applying Bernoulli's equation between reservoir 1 and a common junction point. The head loss for pipe section 1 (hf1) is calculated using the Darcy-Weisbach equation. This involves variables such as friction factor, length of the pipe, velocity squared, and diameter of the pipe. The initial calculation results in hf1 being directly proportional to Q1 squared.

Similar to section 1, the head loss for pipe section 2 (hf2) is calculated using the Darcy-Weisbach equation. This involves specific values for the length and diameter of pipe 2, as well as the flow rate Q2, allowing for the determination of hf2 in terms of Q2 squared.

The head loss for pipe section 3 (hf3) is also determined using the Darcy-Weisbach equation. The video demonstrates how to input the specific parameters for pipe 3, including its length, diameter, and the unknown flow rate Q3, to express hf3 in terms of Q3 squared.

The presenter reiterates the importance of the piezometric head at the junction. The HP is also calculated using the elevation of reservoir 2 and the head loss in pipe 2, and similarly with reservoir 3 and head loss in pipe 3. These expressions help in setting up a system of equations to solve for the unknown flow rates.

Since this is an iterative problem, an initial guess for the piezometric head at the junction (HP) is made. Based on this assumed HP, the flow rates Q1, Q2, and Q3 are calculated. The calculated flow rates are then checked for continuity (Q1 = Q2 + Q3). If continuity is not met, the assumed HP is adjusted, and the calculations are repeated until the continuity equation is satisfied within an acceptable tolerance.

After several iterations, the correct piezometric head at the junction is determined, leading to the final calculated discharge values for Q1, Q2, and Q3. The presenter emphasizes the importance of accuracy in these calculations and the application of the problem in practical engineering scenarios.

The video explains how to determine the piezometric head at the junction (HP) by subtracting the head loss in pipe 1 from the elevation of reservoir 1. This step is crucial for understanding the energy distribution within the pipe network and for further calculations involving other pipe sections.