Let's solve the problem **step-by-step**, focusing on the **volumetric flow rate of water** using the orifice plate data. --- ## **Step 1: Write Down Given Data** - **Pipe diameter (\(D\))** = 10 cm = **.10 m** - **Orifice diameter (\(d\))** = 4.6 cm = **.046 m** - **Manometer reading (\(h\))** = 18 cm = **.18 m** - **Discharge coefficient (\(C_d\))** = .61 - **Fluid** = Water - **Reynolds Number** is high (\(Re > 30,000\)), so use the given \(C_d\). - **Manometer fluid**: Mercury (assume standard properties) - **Densities**: - Water (\(\rho_w\)) ≈ **100 kg/m³** - Mercury (\(\rho_m\)) ≈ **13,600 kg/m³** --- ## **Step 2: Calculate the Beta Ratio (\(\beta\))** \[ \beta = \frac{d}{D} = \frac{.046}{.10} = .46 \] --- ## **Step 3: Find the Pressure Difference (\(\Delta P\))** The **manometer** reading gives the difference in height (\(h\)) of mercury, so the pressure difference is: \[ \Delta P = (\rho_m - \rho_w) \cdot g \cdot h \] Plug in values: \[ \Delta P = (13,600 - 1,000)\ \text{kg/m}^3 \cdot 9.81\ \text{m/s}^2 \cdot .18\ \text{m} \] \[ \Delta P = 12,600 \cdot 9.81 \cdot .18 \] \[ \Delta P \approx 22,260\ \text{Pa} \] --- ## **Step 4: Use the Orifice Flow Equation** The **volumetric flow rate** (\(Q\)) for an orifice meter is: \[ Q = C_d \cdot A_2 \cdot \sqrt{ \frac{2 \Delta P}{\rho_w (1 - \beta^4)} } \] Where: - \(A_2\) = Area of orifice = \(\frac{\pi}{4} d^2\) - \(\beta\) = \(d/D\) - \(\rho_w\) = Density of water --- ### **Calculate Orifice Area (\(A_2\)):** \[ A_2 = \frac{\pi}{4} d^2 = \frac{\pi}{4} (.046)^2 \approx .001662\ \text{m}^2 \] --- ### **Calculate \(1 - \beta^4\):** \[ \beta^4 = (.46)^4 = .0447 \] \[ 1 - \beta^4 = 1 - .0447 = .9553 \] --- ### **Plug In All Values:** \[ Q = .61 \times .001662 \times \sqrt{ \frac{2 \times 22,260}{100 \times .9553}} \] Calculate inside the square root: \[ \frac{2 \times 22,260}{100 \times .9553} = \frac{44,520}{955.3} \approx 46.6 \] \[ \sqrt{46.6} \approx 6.83 \] Now, multiply: \[ Q = .61 \times .001662 \times 6.83 \] \[ Q \approx .001015 \times 6.83 \] \[ Q \approx .00693\ \text{m}^3/\text{s} \] --- ## **Step 5: Final Answer** ### **The volumetric flow rate of water in the pipe is:** \[ \boxed{.00693\ \text{m}^3/\text{s} \quad \text{or} \quad 6.93\ \text{L/s}} \] --- ### **Summary Table** | Parameter | Value | |-----------|-------| | Pipe diameter (\(D\)) | .10 m | | Orifice diameter (\(d\)) | .046 m | | Beta (\(\beta\)) | .46 | | Area of orifice (\(A_2\)) | .001662 m² | | Pressure difference (\(\Delta P\)) | 22,260 Pa | | Discharge coefficient (\(C_d\)) | .61 | | Flow rate (\(Q\)) | **.00693 m³/s** | --- **Let me know if you need more clarification or a breakdown of any step!**