The relation between coefficient of consolidation Cᵥ, time factor Tᵥ, drainage path d and time t is given by:
Try it first: pick an option, then read the explanation.
Correct answer: A. Cᵥ = d²Tᵥ / t
Asked in SSC JE 2010
Detailed Explanation
The time factor is defined as
\[ T_v = \frac{c_v\,t}{d^2} \]
where d is the length of the longest drainage path (half the layer thickness for double drainage, the full thickness for single drainage).
Rearranging for \(c_v\):
\[ c_v = \frac{T_v\,d^2}{t} \]
Why “Cᵥ = d²Tᵥ / t” Is Correct
Check with units: Tv is dimensionless, d² is length², and t is time. So cv has units of length²/time (for example cm²/s), which is correct. Only option (a) has this form.
Why the Other Options Are Wrong
- B. Cᵥ = d²t / Tᵥ: d²t/Tv has units of length² × time, which is wrong for cv.
- C. Cᵥ = tTᵥ / d²: tTv/d² has units of time/length², the inverse of what is needed.
- D. Cᵥ = Tᵥ / (t·d²): Tv/(t·d²) has units of 1/(time × length²), which is also wrong.
Key Point
Tv = cv·t/d² ⇔ cv = Tv·d²/t ⇔ t = Tv·d²/cv. Use units to check.
Key Concepts for Students
- In the lab: cv = 0.197 d²/t50 or cv = 0.848 d²/t90.
- For a 20 mm oedometer sample with double drainage, d = 10 mm.
- Field time = lab time × (d_field/d_lab)² for the same clay and degree of consolidation.
Practice more: this is question 19 in Soil Consolidation MCQs – Part 2. Try the whole set and check every answer instantly.