Question

Two soils of permeabilities \( k_1 \) and \( k_2 \) are placed in a horizontal flow apparatus, as shown in the figure. For Soil 1, \( L_1 = 50 \, {cm} \), and \( k_1 = 0.055 \, {cm/s} \); for Soil 2, \( L_2 = 30 \, {cm} \), and \( k_2 = 0.035 \, {cm/s} \). The cross-sectional area of the horizontal pipe is 100 cm², and the head difference (\( \Delta h \)) is 150 cm. The discharge (in cm³/s) through the soils is ........ (rounded off to 2 decimal places). 

Hint:

For horizontal flow in series arrangements, calculate the effective permeability first and then use it to find the discharge. The permeability depends on the length and conductivity of each soil layer.

Answer 1

For series arrangement of Soil 1 and Soil 2, the discharge is given by: \[ q = \frac{K_{{eff}} \Delta h}{L} \] where \( K_{{eff}} \) is the effective permeability and is given by: \[ K_{{eff}} = \frac{\sum \left( \frac{L}{k} \right)}{L_1 + L_2} \] Substituting the values for \( L_1 \), \( L_2 \), \( k_1 \), and \( k_2 \), we get: \[ K_{{eff}} = \frac{50 + 30}{0.055 + 0.035} = \frac{80}{0.09} = 888.89 \, {cm/s}. \] Now, the discharge through the soils is: \[ q = \frac{K_{{eff}} \Delta h}{L} = \frac{888.89 \times 150}{100} = 8.49 \, {cm³/s}. \] Thus, the discharge is \( \boxed{8.49} \, {cm³/s} \) (rounded to two decimal places).