1. **Problem Statement:** We want to model the water flow in an Automated Water Management System (AWMS) to detect abnormal water loss such as leakage. 2. **Formula Used:** The volumetric flow rate $Q$ through a pipe outlet is given by $$Q = C_d A \sqrt{2gh}$$ where $C_d$ is the discharge coefficient, $A$ is the cross-sectional area of the pipe, $g$ is gravitational acceleration, and $h$ is the effective water head. 3. **Step 1: Calculate Water Velocity at Tank Outlet** Using Torricelli's law with velocity coefficient $C_v$: $$V_{wp} = C_v \sqrt{2gh}$$ Given $C_v=0.97$, $g=9.8$, $h=1$ m: $$V_{wp} = 0.97 \sqrt{19.6} \approx 4.294\, \text{m/s}$$ 4. **Step 2: Calculate Cross-Sectional Area of Pipe** Pipe diameter $d=0.0127$ m, so radius $r=\frac{d}{2}=0.00635$ m: $$A = \pi r^2 = \pi (0.00635)^2 \approx 1.267 \times 10^{-4}\, \text{m}^2$$ 5. **Step 3: Calculate Volumetric Flow Rate** Using $C_d=0.6014$: $$Q = C_d A \sqrt{2gh} = 0.6014 \times 1.267 \times 10^{-4} \times \sqrt{19.6} \approx 3.374 \times 10^{-4}\, \text{m}^3/\text{s}$$ Convert to liters per second: $$Q \approx 0.3374\, \text{L/s}$$ Convert to cubic meters per hour: $$Q_w = Q \times 3600 \approx 1.2146\, \text{m}^3/\text{h}$$ 6. **Step 4: Calculate Time to Empty Tank** Tank volume $V_{tank} = 1$ m$^3$: $$t = \frac{V_{tank}}{Q} = \frac{1}{3.374 \times 10^{-4}} \approx 2964\, \text{s} \approx 49.4\, \text{minutes}$$ 7. **Step 5: General Flow Rate Expression** The flow rate formula can be expressed as: $$Q = C_d \left( \frac{\pi d^2}{4} \right) \sqrt{2gh}$$ 8. **Leakage Detection Criterion:** If the measured flow rate $Q_{measured}$ exceeds a threshold $Q_{threshold}$ (typically 110% of expected $Q$) for a sustained time (e.g., 5-10 seconds), an alert for abnormal water flow is triggered. This model helps detect leaks by comparing real-time flow against expected baseline flow. **Final answer:** The baseline volumetric flow rate is approximately $0.3374$ L/s, and the tank empties in about $49.4$ minutes under normal conditions.