1. **Problem Statement:** Calculate the settling velocity of particles in water using Stokes' Law and Newton's Law, then find Reynolds number and drag coefficient, and finally determine which velocity is more appropriate. 2. **Given Data:** - Particle diameter, $d = 200\ \mu m = 200 \times 10^{-6} = 2 \times 10^{-4}\ m$ - Particle density, $\rho_s = 2650\ kg/m^3$ - Water density, $\rho = 999.1\ kg/m^3$ - Gravity, $g = 9.81\ m/s^2$ - Dynamic viscosity, $\mu = 1.139 \times 10^{-3}\ N\cdot s/m^2$ 3. **Formulas:** - Settling velocity by Stokes' Law: $$v = \frac{g (\rho_s - \rho) d^2}{18 \mu}$$ - Reynolds number: $$N_R = \frac{v d \rho}{\mu}$$ - Drag coefficient: $$C_D = \frac{24}{N_R} + \frac{3}{\sqrt{N_R}} + 0.34$$ - Settling velocity by Newton's Law: $$v = \sqrt{\frac{4 g (\rho_s - \rho) d}{3 C_D \rho}}$$ 4. **Step A: Settling velocity by Stokes' Law** Calculate $v$: $$v = \frac{9.81 \times (2650 - 999.1) \times (2 \times 10^{-4})^2}{18 \times 1.139 \times 10^{-3}}$$ Calculate numerator: $$9.81 \times 1650.9 \times 4 \times 10^{-8} = 9.81 \times 1650.9 \times 4 \times 10^{-8} = 6.48 \times 10^{-4}$$ Calculate denominator: $$18 \times 1.139 \times 10^{-3} = 0.0205$$ Thus, $$v = \frac{6.48 \times 10^{-4}}{0.0205} = 0.0316\ m/s$$ 5. **Step B: Reynolds number and Drag coefficient** Calculate Reynolds number: $$N_R = \frac{0.0316 \times 2 \times 10^{-4} \times 999.1}{1.139 \times 10^{-3}} = \frac{6.32 \times 10^{-3}}{1.139 \times 10^{-3}} = 5.55$$ Calculate Drag coefficient: $$C_D = \frac{24}{5.55} + \frac{3}{\sqrt{5.55}} + 0.34 = 4.32 + 1.27 + 0.34 = 5.93$$ 6. **Step C: Settling velocity by Newton's Law** Calculate $v$: $$v = \sqrt{\frac{4 \times 9.81 \times (2650 - 999.1) \times 2 \times 10^{-4}}{3 \times 5.93 \times 999.1}}$$ Calculate numerator: $$4 \times 9.81 \times 1650.9 \times 2 \times 10^{-4} = 12.95$$ Calculate denominator: $$3 \times 5.93 \times 999.1 = 17775.5$$ Thus, $$v = \sqrt{\frac{12.95}{17775.5}} = \sqrt{7.28 \times 10^{-4}} = 0.0270\ m/s$$ 7. **Step D: Explanation of appropriate settling velocity** - Stokes' Law assumes laminar flow and low Reynolds number ($N_R < 1$), which is not the case here ($N_R = 5.55$). - Newton's Law is more appropriate for higher Reynolds numbers where inertial forces dominate. - Since $N_R$ is greater than 1, Newton's Law settling velocity ($0.0270\ m/s$) is more accurate for this scenario. **Final answers:** - Settling velocity by Stokes' Law: $0.0316\ m/s$ - Reynolds number: $5.55$ - Drag coefficient: $5.93$ - Settling velocity by Newton's Law: $0.0270\ m/s$ - More appropriate velocity: Newton's Law velocity