Let's break down the problem step by step: ### **Given Data** - **Flow Rate (Q):** 100 L/min = 0.100 m³/min = 0.00167 m³/s - **Small Tube (upstream):** OD = 25 mm, wall = 1.5 mm → **ID = 25 - 2×1.5 = 22 mm = 0.022 m** - **Large Tube (downstream):** OD = 80 mm, wall = 2.8 mm → **ID = 80 - 2×2.8 = 74.4 mm = 0.0744 m** - **Loss Coefficient (K):** 0.740 - **Water density (\(\rho\)):** 1000 kg/m³ (assume standard) - **Water dynamic viscosity (\(\mu\)):** not needed unless asked for Reynolds number. - **Enlargement is horizontal** (ignore elevation effects). --- ### **1. Calculate Velocities in Each Pipe** **Small Tube (V₁):** \[ A_1 = \frac{\pi}{4} d_1^2 = \frac{\pi}{4} (0.022)^2 = 3.801 \times 10^{-4} \text{ m}^2 \] \[ V_1 = \frac{Q}{A_1} = \frac{0.00167}{3.801 \times 10^{-4}} = 4.39 \text{ m/s} \] **Large Tube (V₂):** \[ A_2 = \frac{\pi}{4} d_2^2 = \frac{\pi}{4} (0.0744)^2 = 4.346 \times 10^{-3} \text{ m}^2 \] \[ V_2 = \frac{Q}{A_2} = \frac{0.00167}{4.346 \times 10^{-3}} = 0.384 \text{ m/s} \] --- ### **2. **Energy Loss Due to Sudden Enlargement** The head loss due to sudden enlargement: \[ h_L = K \frac{V_1^2}{2g} \] Where: - \(K = 0.740\) - \(V_1 = 4.39\) m/s - \(g = 9.81\) m/s² \[ h_L = 0.740 \times \frac{(4.39)^2}{2 \times 9.81} = 0.740 \times \frac{19.26}{19.62} = 0.740 \times 0.982 = 0.726 \text{ m} \] --- ### **3. **Pressure Difference Across the Enlargement** Apply the energy equation (Bernoulli) between the two sections, considering the loss term: \[ P_1/\rho g + \frac{V_1^2}{2g} = P_2/\rho g + \frac{V_2^2}{2g} + h_L \] So, \[ \frac{P_1 - P_2}{\rho g} = \frac{V_2^2 - V_1^2}{2g} + h_L \] \[ P_1 - P_2 = \rho g \left[ \frac{V_2^2 - V_1^2}{2g} + h_L \right] \] Calculate \(\frac{V_2^2 - V_1^2}{2g}\): \[ \frac{(0.384)^2 - (4.39)^2}{2 \times 9.81} = \frac{0.147 - 19.26}{19.62} = \frac{-19.113}{19.62} = -0.974 \text{ m} \] So, \[ P_1 - P_2 = 1000 \times 9.81 \left[ -0.974 + 0.726 \right] \] \[ = 1000 \times 9.81 \times (-0.248) \] \[ = 1000 \times (-2.43) \] \[ = -2430 \text{ Pa} \] ### **Summary:** 1. **Energy loss (head loss) due to sudden enlargement:** \(\boxed{0.73 \ \text{m}}\) 2. **Pressure difference (P₁ - P₂):** \(\boxed{-2430 \ \text{Pa}}\) (P₁ is less than P₂ by 2.43 kPa) --- **Let me know if you need a step-by-step explanation or further clarification!**