In a hydrometer analysis, the corrected hydrometer reading in a 1000 ml uniform soil suspension at the start of sedimentation was 28. After a lapse of 30 minutes, the corrected hydrometer reading was 12 and the corresponding effective depth 10.5 cm. the specific gravity of the solids was 2.68. Assuming the viscosity and unit weight of water at the temperature of the test as 0.001 Ns/m2 and 9.81 kN/m3 respectively. Determine the weight of solids mixed in the suspension, the effective diameter corresponding to the 30 minutes reading and the percentage of particle finer than this size.
We can use the following formula to calculate the weight of solids in the suspension:
W = V * (Gs / (1 + e)) * (H - H') * ρw
where:
W = weight of solids in the suspension
V = volume of suspension (1000 ml = 1 L)
Gs = specific gravity of solids (2.68)
e = void ratio (assumed to be zero for a uniform soil suspension)
H = initial depth of suspension (1000 mm)
H' = final depth of suspension (10.5 cm = 0.105 m)
ρw = unit weight of water (9.81 kN/m3)
Plugging in the values, we get:
W = 1 * (2.68 / (1 + 0)) * (1 - 0.105/1000) * 9.81
W = 0.239 kN
Next, we can use the following formula to calculate the effective diameter at the 30 minute mark:
d10 = K * (H - H') / log10(H / H')
where:
d10 = effective diameter corresponding to the 30 minute reading
K = constant based on the hydrometer used (assumed to be 0.1 for this problem)
H = initial depth of suspension (1000 mm)
H' = final depth of suspension (10.5 cm = 0.105 m)
Plugging in the values, we get:
d10 = 0.1 * (1000 - 0.105) / log10(1000 / 0.105)
d10 = 0.026 mm
Finally, we can use the following formula to calculate the percentage of particles finer than the effective diameter:
% finer = (1 - (d10 / D50)^2) * 100
where:
D50 = diameter corresponding to 50% finer on the particle size distribution curve (assumed to be 10 times the effective diameter for this problem)
Plugging in the values, we get:
D50 = 10 * d10
D50 = 0.26 mm
% finer = (1 - (0.026 / 0.26)^2) * 100
% finer = 0.99%
Therefore, the weight of solids in the suspension is 0.239 kN, the effective diameter corresponding to the 30 minute reading is 0.026 mm, and the percentage of particles finer than this size is 0.99%.
To solve this problem, we can use the Stokes' Law equation to calculate the weight of solids mixed in the suspension and the effective diameter corresponding to the 30 minutes reading. Then, we can use the relationship between sedimentation and particle size to determine the percentage of particles finer than this size.
1. Calculate the weight of solids mixed in the suspension:
- Given: Specific gravity of solids (Gs) = 2.68, Unit weight of water (γw) = 9.81 kN/m^3
- First, convert the unit weight of water to N/m^3:
γw = 9.81 kN/m^3 = 9.81 × 1000 N/m^3 = 9810 N/m^3
- The weight of solids (Ws) can be calculated using the formula:
Ws = (V × (Gs - 1) × γw) / Gs
where V is the volume of the soil suspension.
- Given: V = 1000 ml = 1000 cm^3
- Convert V to m^3:
V = 1000 cm^3 = 1000 × 10^-6 m^3 = 0.001 m^3
- Calculate Ws:
Ws = (0.001 × (2.68 - 1) × 9810) / 2.68
Ws = 0.001 × 1.613 × 9810
Ws ≈ 16.13 N
2. Calculate the effective diameter corresponding to the 30 minutes reading:
- Given: Initial corrected hydrometer reading (H0) = 28, Final corrected hydrometer reading (H30) = 12, Effective depth (d) = 10.5 cm
- The time taken for sedimentation (t) can be calculated using the equation:
t = (H0 - H30) / (2.3 + 0.033 × (H0 - H30))
- Calculate t:
t = (28 - 12) / (2.3 + 0.033 × (28 - 12))
t = 16 / (2.3 + 0.033 × 16)
t = 16 / (2.3 + 0.528)
t ≈ 16 / 2.828
t ≈ 5.665 minutes
- Convert t to seconds:
t = 5.665 minutes × 60 seconds/minute
t ≈ 339.9 seconds
- Use Stokes' Law to find the effective diameter (D):
D = (18 × μ × t) / (Gs × γw × d^2)
where μ is the viscosity of water and d is the effective depth.
- Given: μ = 0.001 Ns/m^2, d = 10.5 cm = 0.105 m
- Calculate D:
D = (18 × 0.001 × 339.9) / (2.68 × 9810 × (0.105)^2)
D ≈ 0.00122 m = 1.22 mm
3. Calculate the percentage of particles finer than the effective diameter:
- The formula to calculate the percentage finer than a given size is:
% Finer = (Vo / Vt) × 100
where Vo is the volume of water at the final reading and Vt is the total volume of the suspension.
- Given: Vo = 1000 ml = 1000 cm^3
- Convert Vo to m^3:
Vo = 1000 cm^3 = 1000 × 10^-6 m^3 = 0.001 m^3
- Calculate Vt:
Vt = 1000 cm^3 = 1000 × 10^-6 m^3 = 0.001 m^3
- Calculate % Finer:
% Finer = (0.001 / 0.001) × 100 = 100%
Therefore, the weight of solids mixed in the suspension is approximately 16.13 N, the effective diameter corresponding to the 30 minutes reading is approximately 1.22 mm, and the percentage of particles finer than this size is 100%.