Calculating Particle Removal in Sedimentation Tanks

Understanding how sedimentation tanks work is crucial for water treatment processes. This question asks us to determine the approximate percentage of particles removed in an ideal horizontal flow sedimentation tank, given the overflow rate and the settling velocity of the particles.

Understanding Ideal Sedimentation Tanks

An ideal sedimentation tank is a theoretical model used to simplify the analysis of particle removal by gravity. It assumes:

• The water flows horizontally through the tank like a plug flow, with uniform velocity.
• Particles are uniformly distributed vertically at the inlet.
• Particles settle downwards at a constant velocity (\(V_s\)) independently of other particles.
• Once a particle reaches the tank bottom, it is considered removed.
• There is no turbulence or short-circuiting.

In such an ideal tank, a particle is removed if its settling velocity (\(V_s\)) is greater than or equal to the tank's overflow rate (\(V_o\)). The overflow rate is essentially the minimum settling velocity that a particle must have to be removed in the tank. If a particle's settling velocity (\(V_s\)) is less than the overflow rate (\(V_o\)), only a fraction of these particles will be removed. The fraction removed is directly proportional to the ratio of the particle's settling velocity to the overflow rate.

The removal efficiency for particles with settling velocity \(V_s < V_o\) in an ideal tank is given by:

Removal Efficiency \( = \left(\frac{V_s}{V_o}\right) \times 100\%\)

Given Information

We are provided with the following values:

• Overflow Rate (\(V_o\)) = 17 m\textsuperscript{3}/day/m\textsuperscript{2}
• Particle Settling Velocity (\(V_s\)) = 0.1 mm/second

Note that \(V_o\) given as m\textsuperscript{3}/day/m\textsuperscript{2} simplifies to m/day, representing a velocity.

Step-by-Step Calculation of Particle Removal Percentage

To calculate the removal percentage, we must ensure that both the overflow rate and the settling velocity are expressed in the same units. Let's convert both to meters per second (m/s).

Convert Overflow Rate (\(V_o\)) to m/s

The overflow rate is given as 17 m/day. We need to convert days to seconds:

\(1 \text{ day} = 24 \text{ hours} \times 60 \text{ minutes/hour} \times 60 \text{ seconds/minute} = 86400 \text{ seconds}\)

Now, convert the overflow rate:

\(V_o = \frac{17 \text{ m}}{1 \text{ day}} = \frac{17 \text{ m}}{86400 \text{ seconds}}\)

Calculating the value:

\(V_o \approx 0.0001965 \text{ m/s}\)

Convert Particle Settling Velocity (\(V_s\)) to m/s

The settling velocity is given as 0.1 mm/second. We need to convert millimeters to meters:

\(1 \text{ meter} = 1000 \text{ mm}\)

Now, convert the settling velocity:

\(V_s = 0.1 \frac{\text{mm}}{\text{second}} \times \frac{1 \text{ m}}{1000 \text{ mm}}\)

Calculating the value:

\(V_s = 0.0001 \text{ m/s}\)

Calculate Percentage Removal

We compare the settling velocity (\(V_s = 0.0001 \text{ m/s}\)) with the overflow rate (\(V_o \approx 0.0001965 \text{ m/s}\)). Since \(V_s < V_o\), the percentage of particles removed is given by the ratio \(V_s/V_o\) multiplied by 100%.

Removal Percentage \( = \left(\frac{V_s}{V_o}\right) \times 100\%\)

Substitute the calculated values:

Removal Percentage \( = \left(\frac{0.0001 \text{ m/s}}{0.0001965 \text{ m/s}}\right) \times 100\%\)

Removal Percentage \( \approx 0.5089 \times 100\%\)

Removal Percentage \( \approx 50.89\%\)

Conclusion

The calculated removal percentage is approximately 50.89%. Comparing this value to the given options, it is closest to 50%.

Revision Table: Sedimentation Tank Parameters

Additional Information: Real Sedimentation Tanks

While the ideal tank model provides a fundamental understanding, real sedimentation tanks in water or wastewater treatment plants face practical limitations that affect performance:

• Non-uniform Flow: Flow patterns are rarely perfectly horizontal or uniform. Inlet and outlet designs, as well as tank geometry, influence flow distribution.
• Turbulence and Eddies: These can keep smaller particles suspended or re-suspend settled particles, reducing removal efficiency, especially for particles with settling velocities close to the overflow rate.
• Short-Circuiting: Some water may flow through the tank faster than the theoretical detention time, reducing the effective settling time available.
• Effect of Particle Concentration: At high concentrations, particles may interact (hindered settling or flocculation), which isn't accounted for in the single-particle settling assumption of the ideal model.

Therefore, actual removal efficiency in a real tank is often less than that predicted by the ideal tank model, especially for particles with settling velocities significantly less than the overflow rate.