Question

In a 51-story building, water enters through an underground pipe with an internal diameter of 10.0 cm. A 1.0 cm diameter tube goes to the 51st floor, 215.0 m above. It is estimated that the minimum pressure necessary at the most unfavorable point (highest and furthest) of the installation must be about 50 kPa. If the flow rate in the output pipe is 6 m/s, calculate the speed of the input water, the necessary input pressure in units of atm, and the volume flow of water per second in units of liters/second.

Answer 1

To find the requested parameters, we need to apply the continuity equation and Bernoulli's equation to solve for the speed of input water, the required input pressure, and the volume flow rate in liters per second for the building's water system.

Step-by-step explanation:

The student is asking to calculate the speed of input water, necessary input pressure in units of atm, and the volume flow of water per second in liters/second for a 51-story building's water system. The flow rate at the output pipe, internal diameters of the pipes, and required minimum pressure at the highest point are given. The answer involves the application of the principles of fluid dynamics, specifically the Bernoulli's equation and the equation of continuity.

We first use the continuity equation A1v1 = A2v2 to find the speed of input water, where A1 and A2 are the cross-sectional areas of the input and output pipes, respectively, and v1 and v2 are the corresponding flow speeds. Then we use Bernoulli's equation to calculate the necessary input pressure, taking into account the height difference and the minimum pressure requirement at the output. Finally, we calculate the volume flow rate using the area of the input pipe and the input speed obtained from the continuity equation.