Gravity sedimentation and thickener design (Kynch theory)
Clarifiers and thickeners, the batch settling test, Kynch's assumption and tangent construction, and Coe–Clevenger thickener area.
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Why it matters
Thickeners and clarifiers are the largest pieces of equipment in many mineral, water-treatment and chemical plants, and they are sized almost entirely from a simple batch settling test in a measuring cylinder. Kynch's theory is what turns that one test into the cross-sectional area of a continuous thickener; an undersized unit lets solids escape in the overflow, an oversized one wastes capital.
Key ideas
Clarifier versus thickener. Both are large, shallow tanks fed near the centre, with clear liquid overflowing at the rim and settled sludge raked to a central underflow cone. A clarifier treats dilute suspensions and its aim is a clear overflow; it is sized so that the liquid upflow velocity is less than the settling velocity of the smallest particle to be removed. A thickener treats concentrated slurries and its aim is a dense underflow; its area is limited by how fast solids can pass through the zones of intermediate concentration.
The batch settling test. A slurry of uniform concentration C₀ and height H₀ is allowed to settle in a cylinder, and the height Z of the interface between clear liquid and suspension is plotted against time t. Distinct zones appear: clear liquid at the top; a zone of constant (initial) concentration settling at constant rate; a transition zone of rising concentration; and at the bottom a compression zone where particles rest on each other and liquid is squeezed out slowly. The Z–t curve is therefore straight at first (constant-rate settling), then curves as the rising concentration zones reach the interface, and finally flattens in compression.
Kynch's assumption. The local settling velocity of the solids depends only on the local solids concentration (v decreases as C increases), and concentration is uniform across any horizontal layer. With this assumption, layers of constant concentration rise from the bottom at constant speed, and one batch curve contains the settling velocity for every concentration between C₀ and the compression concentration.
Kynch construction. Draw a tangent to the Z–t curve at any time t. Its intercept on the Z axis is H_i and its slope (magnitude) is the settling velocity v_L of the layer of concentration C_L now at the interface. A solids balance gives C_L·H_i = C₀·H₀.
Continuous thickener area (Coe–Clevenger approach). Solids must pass through every concentration layer between feed and underflow. For a layer of concentration C with settling velocity v, the solids flux it can transmit, given underflow concentration C_u, sets a required area. The design area is the largest value over all C between the feed and underflow concentrations; that layer is the limiting layer. In practice a safety factor is added and the depth is fixed separately from the compression-zone residence time (take design depth allowances from your data book).
Formulas
v_L = (H_i − Z) / t (slope of tangent to the batch curve)
C_L = C₀·H₀ / H_i
- Z: interface height at time t (m); H_i: intercept of the tangent on the height axis (m); H₀, C₀: initial height (m) and concentration (kg/m³); v_L: settling velocity of layer of concentration C_L (m/s).
A = S·(1/C − 1/C_u) / v, design A = maximum over C of this expression
- A: thickener area (m²); S = Q_F·C_F: solids feed rate (kg/s); C: concentration of a layer (kg/m³); C_u: underflow concentration (kg/m³); v: settling velocity of that layer (m/s).
Clarifier: A = Q_o / v_t
- Q_o: overflow (clarified liquid) rate (m³/s); v_t: settling velocity of the smallest particle to be removed (m/s).
Underflow volume (solids balance, no solids in overflow): Q_U = S / C_u
Worked examples
Example 1 (standard): Coe–Clevenger area. A thickener receives 10 kg/s of solids; the underflow must be 500 kg/m³. Batch tests give: C (kg/m³): 100, 150, 200, 250, 300; v (mm/s): 0.40, 0.18, 0.10, 0.060, 0.045.
- Compute (1/C − 1/C_u)/v for each layer:
- C = 100: (0.0100 − 0.0020)/0.40 × 10⁻³ = 20.0 m²·s/kg
- C = 150: 0.004667/0.18 × 10⁻³ = 25.9
- C = 200: 0.0030/0.10 × 10⁻³ = 30.0
- C = 250: 0.0020/0.060 × 10⁻³ = 33.3
- C = 300: 0.001333/0.045 × 10⁻³ = 29.6
- The maximum is at C = 250 kg/m³ (the limiting layer).
- A = 10 × 33.3 = 333 m² (before any safety factor), i.e. a diameter of about 20.6 m.
Example 2 (GATE level): Kynch construction. A slurry with C₀ = 40 kg/m³ is settled from H₀ = 0.80 m. At t = 1 h the interface is at Z = 0.40 m, and the tangent drawn there intercepts the height axis at H_i = 0.64 m. Find the concentration and settling velocity of that layer, and the area needed for 5 kg/s of solids with C_u = 200 kg/m³, assuming this layer is limiting.
- C_L = C₀·H₀/H_i = 40 × 0.80/0.64 = 50 kg/m³.
- v_L = (H_i − Z)/t = (0.64 − 0.40)/3600 = 6.67 × 10⁻⁵ m/s.
- A = S(1/C_L − 1/C_u)/v_L = 5 × (0.02 − 0.005)/6.67 × 10⁻⁵ = 5 × 225 = 1125 m².
Common mistakes
- Sizing a thickener on the initial constant-rate settling velocity only; the limiting layer is usually at a higher concentration.
- Taking the minimum instead of the maximum of (1/C − 1/C_u)/v.
- Reading v_L as Z/t rather than the tangent slope (H_i − Z)/t.
- Mixing concentration units (kg/m³ versus mass fraction) between C and C_u.
- Using the feed volumetric rate where the overflow rate belongs in clarifier sizing.
For GATE CH
Expect Kynch-construction numericals (C_L from the tangent intercept, v_L from the slope), thickener area from a settling table, underflow rate from a solids balance, and conceptual questions on the zones of a batch test and Kynch's assumption. Practise reading tangents from a sketched Z–t curve and running the Coe–Clevenger table quickly.
Quick check
- What is Kynch's central assumption?
- A tangent to the batch curve intercepts at H_i = H₀/2. What is C_L?
- In Coe–Clevenger design, is the area the maximum or minimum of the computed values?
- Which zone of the batch test settles at constant rate? Answers: 1. Settling velocity depends only on local solids concentration. 2. 2C₀. 3. The maximum. 4. The zone at the initial concentration C₀.
Interview questions
All Mechanical Operations interview questionsTry answering each one aloud before you open it.
1.What is gravity sedimentation in the context of mechanical operations?Concept
Gravity sedimentation is a process where particles suspended in a fluid settle out of the fluid under the influence of gravity. This process is used to separate solid particles from liquids in various industrial applications.
2.Explain the Kynch theory of sedimentation.Concept
Kynch assumed that the settling velocity of solids at any point depends only on the local solids concentration, decreasing as concentration rises, and that concentration is uniform across each horizontal layer. It follows that layers of constant concentration propagate upward from the bottom at constant speed, so a single batch settling curve contains the settling velocity for every concentration from the feed value up to compression. A tangent to the interface-height curve at time t has intercept H_i and slope v_L; the layer at the interface then has concentration C_L = C₀H₀/H_i and settles at v_L. These (C, v) pairs are what is used to size a continuous thickener.
3.What is a thickener, and how is it used in gravity sedimentation?Concept
A thickener is a large tank used to separate solids from liquids by gravity sedimentation. It allows the solid particles to settle at the bottom, forming a thick sludge, while the clarified liquid overflows from the top. Thickeners are commonly used in mineral processing, wastewater treatment, and chemical industries.
4.Why is the Kynch theory important in the design of thickeners?Application
The Kynch theory is important in thickener design because it helps predict the settling behavior of particles in a suspension. By understanding how particles settle, engineers can design thickeners that efficiently separate solids from liquids, optimizing the process for specific industrial applications.
5.What factors affect the settling velocity of particles in a suspension?Application
The settling velocity of particles in a suspension is affected by factors such as particle size, shape, and density, as well as the viscosity and density of the fluid. Additionally, the concentration of particles in the suspension can influence the settling velocity, as described by the Kynch theory.
6.How does particle concentration influence the sedimentation process according to the Kynch theory?Application
According to the Kynch theory, particle concentration influences the sedimentation process by affecting the settling velocity. As the concentration of particles increases, the interactions between particles can slow down the settling velocity, leading to a more gradual sedimentation process.
7.What happens if the feed rate to a thickener is increased beyond its design capacity?Application
If the feed rate to a thickener is increased beyond its design capacity, the thickener may become overloaded. This can lead to insufficient settling time for particles, resulting in poor separation efficiency and potentially causing the overflow of solids with the clarified liquid.
8.Why is it important to control the underflow rate in a thickener?Application
Controlling the underflow rate in a thickener is important to ensure that the settled sludge is removed at a consistent rate. This helps maintain the desired concentration of solids in the underflow and prevents the thickener from becoming overloaded, which could compromise its efficiency.
9.A thickener receives 100 m³/h of slurry containing 10% solids by volume. If the underflow is 50% solids by volume and the overflow is clear, what is the underflow rate?Numerical
With no solids in the overflow, a solids balance gives Q_F·φ_F = Q_U·φ_U, so Q_U = 100 × 0.10/0.50 = 20 m³/h. The remaining 80 m³/h leaves as clear overflow. The same balance written in kg/m³ is Q_U = S/C_u, where S is the solids feed rate.
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