Coefficient of permeability for a concrete sample with cross-sectional area 100 cm2 , thickness 20 cm, drop in hydraulic head 2 cm and rate of flow 100 cm3 /sec will be
- A10 m2 /s
- B10 m/s
- C0.1 m2 /s
- D0.1 m/sCorrect
Solution
Concept
Calculation of the coefficient of permeability of concrete using Darcy's law based on steady-state flow under a hydraulic gradient.
Formula
k = (q × L) / (A × h)
Step-by-step solution
1. Identify the given parameters: Cross-sectional area A = 100 cm² = 0.01 m²; 2. Thickness L = 20 cm = 0.2 m; 3. Hydraulic head drop h = 2 cm = 0.02 m; 4. Rate of flow q = 100 cm³/s = 100 × 10⁻⁶ m³/s = 10⁻⁴ m³/s; 5. Apply Darcy's Law: q = k × (h / L) × A; 6. Rearrange for k: k = (q × L) / (A × h) = (10⁻⁴ × 0.2) / (0.01 × 0.02) = 0.1 m/s.
Explanation
Darcy's law for permeability is given by q = k × (h / L) × A, where q is the rate of flow, k is the coefficient of permeability, h is the head loss, L is the thickness, and A is the cross-sectional area. Substituting the given values in consistent SI units (A = 100 cm² = 0.01 m², L = 20 cm = 0.2 m, h = 2 cm = 0.02 m, q = 100 cm³/s = 1e-4 m³/s), we solve for k. k = (q × L) / (A × h) = (1e-4 × 0.2) / (0.01 × 0.02) = 0.1 m/s.
Why the other options are wrong
A: Dimensional error in unit or calculation. B: Incorrect calculation magnitude. C: Incorrect unit for permeability in this context.
Shortcut
Convert all units carefully to meters before calculation to avoid unit conversion errors.
Common mistake
Mixing up cm and meters, leading to incorrect powers of 10 in the final result.
Correct answer: D) 0.1 m/s