Question

Consider a 300 -L storage tank of a solar water heating system initially filled with warm water at \(45^{\circ} \mathrm{C}\). Warm water is withdrawn from the tank through a 2 -cm diameter hose at an average velocity of \(0.5 \mathrm{m} / \mathrm{s}\) while cold water enters the \(\tan k\) at \(20^{\circ} \mathrm{C}\) at a rate of \(15 \mathrm{L} / \mathrm{min}\). Determine the amount of water in the tank after a 20 -minute period. Assume the pressure in the tank remains constant at 1 atm.

Short answer

Answer: Approximately 412 L.

Step-by-step solution

Convert the given rates and time to consistent units

We need to convert the cold water flow rate (15 L/min) into an equivalent volumetric flow rate in cubic meters per second, and the hose diameter (2 cm) into meters. We also need to convert the given time interval (20 minutes) into seconds to perform the calculations. 1 L = 0.001 m³, and 1 minute = 60 seconds. Cold water flow rate = 15 L/min * 0.001 m³/L * 1 min/60 s = 0.00025 m³/s Hose diameter = 2 cm * 0.01 m/cm = 0.02 m Time interval = 20 min * 60 s/min = 1200 s

Calculate the flow rate of warm water leaving the tank

We are given the average velocity of warm water in the hose (0.5 m/s) and its diameter (0.02 m), and we will use this information to calculate the volumetric flow rate of warm water leaving the tank. Volumetric flow rate (Q) = Area of the hose (A) * Velocity of water (v) A = (pi/4) * (d^2) Q = ((pi/4) * (0.02 m)^2) * (0.5 m/s) Q ≈ 0.000157 m³/s

Calculate the changes in volume over the 20-minute period

Now that we have the volumetric flow rates of cold water entering and warm water leaving the tank, we can use the time interval (1200 s) to calculate the changes in volume over the 20-minute period. Change in volume of cold water = flow rate * time interval Change in volume of warm water = flow rate * time interval Change in volume of cold water = (0.00025 m³/s) * (1200 s) = 0.30 m³ Change in volume of warm water = (0.000157 m³/s) * (1200 s) ≈ 0.188 m³

Determine the amount of water in the tank after 20 minutes

To find the final amount of water in the tank, we subtract the warm water volume that has left the tank from the initial volume and add the cold water volume that has entered the tank. Initial volume = 300 L = 0.3 m³ Final volume = Initial volume - Change in volume of warm water + Change in volume of cold water Final volume = 0.3 m³ - 0.188 m³ + 0.3 m³ ≈ 0.412 m³ To convert the final volume back to liters: 0.412 m³ * 1000 L/m³ ≈ 412 L Therefore, the amount of water in the tank after a 20-minute period is approximately 412 L.