Let's break down the solution into 3 steps, explaining each step, and provide the final answer at the end. --- ### **Step 1: Calculate the velocities in the duct and the room** - **Given:** - Discharge pipe area, \( A_1 = 150 \text{ cm}^2 = 0.015 \text{ m}^2 \) - Ventilating duct area, \( A_2 = 500 \text{ cm}^2 = 0.05 \text{ m}^2 \) - Fan capacity (flow rate), \( Q = 0.46 \text{ m}^3/\text{s} \) - Minimum ventilating rate from room, \( V_{\text{room}} = 0.30 \text{ m}^3/\text{s} \) - **Find velocities:** - Velocity in discharge pipe: \( V_1 = Q / A_1 = 0.46 / 0.015 = 30.67 \text{ m/s} \) - Velocity in ventilating duct: \( V_2 = Q / A_2 = 0.46 / 0.05 = 9.2 \text{ m/s} \) - Velocity in room: \( V_{\text{room}} = 0.30 / 0.05 = 6 \text{ m/s} \) **Explanation:** We first convert all areas to m² and use the flow rate \( Q \) to find the velocities at each section using \( V = Q / A \). --- ### **Step 2: Apply the energy equation (Bernoulli + losses) between the room and the fan outlet** - **Bernoulli equation with local loss:** \[ P_{\text{room}} + \frac{1}{2}\rho V_{\text{room}}^2 + \rho g h = P_2 + \frac{1}{2}\rho V_2^2 + \text{losses} \] - Loss term (for duct entrance): \( 0.5 V_{\text{room}}^2/2g \) - The height difference \( h = L \sin \beta \), if \( L \) is the vertical height. Since the room is vented to atmosphere and discharge is also to atmosphere, \( P_{\text{room}} = P_2 \). \[ \frac{1}{2}\rho V_{\text{room}}^2 + \rho g h = \frac{1}{2}\rho V_2^2 + \frac{1}{2}\rho (0.5 V_{\text{room}}^2) \] \[ g h = \frac{1}{2}(V_2^2 - V_{\text{room}}^2) + 0.25 V_{\text{room}}^2 \] \[ g h = \frac{1}{2}(9.2^2 - 6^2) + 0.25 \times 6^2 = \frac{1}{2}(84.64 - 36) + 9 = \frac{1}{2}(48.64) + 9 = 24.32 + 9 = 33.32 \text{ m}^2/\text{s}^2 \] **Explanation:** We apply Bernoulli’s equation with the given loss term to relate the vertical height (which depends on the installation angle) to the velocities. --- ### **Step 3: Relate height to installation angle \( \beta \) and solve** - **Assume the vertical height \( h = L \sin \beta \), and let \( L \) be the length of the duct.** - But if the problem asks for the angle and the height is not specified, we can express \( \sin \beta \) in terms of the given or required quantities. \[ g h = 33.32 \] If \( h = L \sin \beta \): \[ g L \sin \beta = 33.32 \] If the vertical rise equals the height of the ventilating duct (say, \( h = 2 \) m as per the diagram), then: \[ g h = 9.81 \times 2 = 19.62 \] But this does not match our calculated value, so let's solve for \( h \): \[ h = \frac{33.32}{9.81} = 3.4 \text{ m} \] So, \[ \sin \beta = \frac{h}{L} \] If \( L \) is known, plug in the value, otherwise, the answer is: ### **Final Answer:** **The installation angle for the centrifugal fan is:** \[ \boxed{ \sin \beta = \frac{33.32}{g L} } \] If the vertical height \( h \) is used instead (from the diagram, if \( h = 2 \) m): \[ \beta = \arcsin\left(\frac{3.4}{2}\right) \] But \( \frac{3.4}{2} > 1 \), which is not possible, so check if a parameter is missing or if you need to use the above formula with your actual duct length \( L \). --- **Summary of Steps:** 1. **Calculated velocities** using areas and flow rates. 2. **Applied Bernoulli’s equation** with losses to relate height and velocities. 3. **Solved for the installation angle** in terms of the vertical height and duct length. **Final formula:** \[ \boxed{ \sin \beta = \frac{33.32}{g L} } \] If a specific value of \( L \) is given, plug it in to find \( \beta \).