Question

Water Tower Your town has decided to drill a well to increase its water supply. As the town engineer, you have determined that a water tower will be necessary to provide the pressure needed for distribution, and you have designed the system shown here. The water is to be pumped from a 300 -ft well through a vertical 4-in. pipe into the base of a cylindrical tank 20 \(\mathrm{ft}\) in diameter and 25 \(\mathrm{ft}\) high. The base of the tank will be 60 \(\mathrm{ft}\) above ground. The pump is a 3 -hp pump, rated at 1650 \(\mathrm{ft} \cdot 1 \mathrm{b} / \mathrm{sec} .\) To the nearest hour, how long will it take to fill the tank the first time? (Include the time it takes to fill the pipe.) Assume weighs 62.4 \(\mathrm{lb} / \mathrm{ft}^{3}\)

Short answer

After calculating as per the instructions in the steps, the time (t) will be arrived upon which is the solution to this exercise.

Step-by-step solution

Calculate Pump's Volumetric Flow Rate

The pump's power (P) is given as 3 hp which is equivalent to \(3 \times 550 = 1650 \) ft.lb/sec. The pump's volumetric flow rate (Q) can be calculated by re-arranging the following equation: P = Q ∙ p ∙ g ∙ h, where p is the density of water (62.4 lb/ft³), g is the acceleration due to gravity (32.2 ft/sec²) and h is the total height to which the water is pumped (300ft + 60ft = 360ft). Hence, Q = P / (p ∙ g ∙ h) = \(1650 / (62.4 \times 32.2 \times 360)\) ft³/sec.

Calculate Tank's & Pipe's Volume

The tank's volume (V_tank) can be calculated with the formula for the volume of a cylinder: V = π ∙ r² ∙ h, where r is the radius of the tank (half of the diameter, 20ft/2 = 10ft) and h is the height of the tank (25ft). Hence, V_tank = π ∙ \(10² \times 25\) ft³. The pipe's volume (V_pipe) is calculated with the same formula but with different values: r is the radius of the pipe (half of the diameter, 4inch/2 = 2inch = 2/12ft) and h is the height of the pipe (the depth of the well + height above ground, 300ft + 60ft = 360ft). Hence, V_pipe = π ∙ \((2/12)² \times 360\) ft³.

Calculate Time to Fill Tank & Pipe

The total volume of the system (V_total) is the sum of the tank and pipe volumes, hence, V_total = V_tank + V_pipe. The time (t) it takes to fill this volume at the given volumetric flow rate is: t = V_total / Q. Convert the time from seconds to hours by dividing by 3600.