Let's solve the problem step by step: --- ## **Step 1: Understanding the Problem** - **Given:** - Void ratio, \( e = .48 \) - Effective grain size, \( D_{10} = 230 \ \mu\text{m} = 230 \times 10^{-6} \ \text{m} \) - Tube is filled with clay and placed in water. - **Required:** 1. Height of capillary rise (\( h_c \)) 2. Capillary pressure at \( h_c \) in kN/m². --- ## **Step 2: Capillary Rise Formula** The empirical formula for capillary rise in soils is: \[ h_c = \frac{C}{e \cdot D_{10}} \] - \( h_c \) = capillary rise (m) - \( C \) = empirical constant (usually taken as .3 for SI units) - \( e \) = void ratio (dimensionless) - \( D_{10} \) = effective grain size (m) --- ## **Step 3: Substitute the Given Values** Given: \( C = .3 \), \( e = .48 \), \( D_{10} = 230 \times 10^{-6} \) m \[ h_c = \frac{.3}{.48 \times 230 \times 10^{-6}} \] Calculate denominator: \[ .48 \times 230 \times 10^{-6} = 110.4 \times 10^{-6} = 1.104 \times 10^{-4} \] Now substitute: \[ h_c = \frac{.3}{1.104 \times 10^{-4}} \] \[ h_c = 2717.39 \ \text{m} \] This seems extremely high, which suggests we should check our constant. For clayey soils, the capillary rise can be over a meter, but not thousands of meters. Some references use \( C = .15 \). Let's use \( C = .15 \): \[ h_c = \frac{.15}{1.104 \times 10^{-4}} \] \[ h_c = 1358.7 \ \text{m} \] Still very high. Let's check the formula. Some texts use: \[ h_c = \frac{.3}{e \cdot D_{10} (\text{in mm})} \] So, let's use \( D_{10} \) in mm: \[ D_{10} = .23 \ \text{mm} \] \[ h_c = \frac{.3}{.48 \times .23} \] \[ .48 \times .23 = .1104 \] \[ h_c = \frac{.3}{.1104} = 2.72 \ \text{m} \] --- ## **Step 4: Capillary Pressure Calculation** Capillary pressure (\( p_c \)) at height \( h_c \): \[ p_c = -\gamma_w \cdot h_c \] - \( \gamma_w \) = unit weight of water (typically \( 9.81 \ \text{kN/m}^3 \)) \[ p_c = -9.81 \times 2.72 = -26.7 \ \text{kN/m}^2 \] --- ## **Step 5: Final Answers** ### **1. Capillary Rise, \( h_c \):** \[ \boxed{2.72 \ \text{m}} \] ### **2. Capillary Pressure at \( h_c \):** \[ \boxed{-26.7 \ \text{kN/m}^2} \] --- ## **Summary Table** | Quantity | Value | |----------------------|--------------------| | Capillary rise, \( h_c \) | 2.72 m | | Capillary pressure, \( p_c \) | -26.7 kN/m² | --- ### **Explanation** - The capillary rise is calculated using the empirical formula for soils, using \( D_{10} \) in mm. - Capillary pressure is the negative pore water pressure at the top of the capillary rise and is calculated as the product of the unit weight of water and the capillary rise height.