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**Stoke’s Law and Settling Particles**

Lecture 12 – MINE

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**Terminal Velocity of Settling Particle**

Rate at which discrete particles settle in a fluid at constant temperature is given by Newton’s equation: vs = [(4g(s - )dp) / (3Cd )] 0.5 where vs = terminal settling velocity (m/s) g = gravitational constant (m/s2) s = density of the particle (kg/m3) = density of the fluid (kg/m3) dp = particle diameter (m) Cd = Drag Coefficient (dimensionless) The terminal settling velocity is derived by balancing drag, buoyant, and gravitational forces that act on the particle.

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Reynolds Number In fluid mechanics, the Reynolds Number, Re (or NR), is a dimensionless number that is the ratio of inertial forces to viscous forces. It quantifies the relative importance of these two types of forces for a given set of flow conditions. where: v = mean velocity of an object relative to a fluid (m/s) L = characteristic dimension, (length of fluid; particle diameter) (m) μ = dynamic viscosity of fluid (kg/(m·s)) ν = kinematic viscosity (ν = μ/ρ) (m²/s) ρ = fluid density (kg/m³)

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**Drag Coefficient and Reynolds Number**

Cd is determined from Stokes Law which relates drag to Reynolds Number

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**Drag Coefficient and Reynolds Number**

Cd is determined from Stokes Law which relates drag to Reynolds Number

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**Drag Coefficient and Reynolds Number**

Cd is determined from Stokes Law which relates drag to Reynolds Number

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**Drag Coefficient and Reynolds Number**

Cd is determined from Stokes Law which relates drag to Reynolds Number

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**Drag Coefficient and Reynolds Number**

Cd is determined from Stokes Law which relates drag to Reynolds Number

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**Drag Coefficient and Reynolds Number**

Cd is determined from Stokes Law which relates drag to Reynolds Number

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**Terminal Velocity of Settling Particle**

Terminal velocity is affected by: Temperature Fluid Density ü Particle Density ü Particle Size ü Particle Shape Degree of Turbulence ü Volume fraction of solids Solid surface charge and pulp chemistry Magnetic and electric field strength Vertical velocity of fluid

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**Drag Coefficient of Settling Particle**

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**Terminal Velocity of Settling Particle**

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**Type I Free-Settling Velocity**

Particle Settling in a Laminar (or Quiescent Liquid) Momentum Balance

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**Type I Free-Settling Velocity**

Particle Settling in a Laminar (or Quiescent Liquid)

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**Type I Free-Settling Velocity**

Integrating gives the steady state solution: For a sphere:

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**Terminal Velocity of Settling Particle**

Type I Settling of Spheres

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**Terminal Velocity of Settling Particle**

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**Terminal Velocity under Hindered Settling Conditions**

McGhee’s (1991) equation estimates velocity for spherical particles under hindered settling conditions for Re < 0.3: Vh/V = (1 - Cv)4.65 where Vh = hindered settling velocity V = free settling velocity Cv = volume fraction of solid particles For Re > 1,000, the exponent is only 2.33 McGhee, T.J., Water Resources and Environmental Engineering. Sixth Edition. McGraw-Hill, New York.

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**Terminal Velocity under Hindered Settling Conditions**

McGhee, T.J., Water Resources and Environmental Engineering. Sixth Edition. McGraw-Hill, New York.

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**Relationship between Cv and Weight%**

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Effect of Alum on IEP

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**Ideal Rectangular Settling Vessel**

Side view

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**Ideal Rectangular Settling Vessel**

Model Assumptions 1. Homogeneous feed is distributed uniformly over tank cross-sectional area 2. Liquid in settling zone moves in plug flow at constant velocity 3. Particles settle according to Type I settling (free settling) 4. Particles are small enough to settle according to Stoke's Law 5. When particles enter sludge region, they exit the suspension

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**Ideal Rectangular Settling Vessel**

Side view u = average (constant) velocity of fluid flowing across vessel vs = settling velocity of a particular particle vo = critical settling velocity of finest particle recovered at 100%

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**Average time spent in the vessel by an element of the suspension**

Retention Time Average time spent in the vessel by an element of the suspension and W, H, L are the vessel dimensions; u is the constant velocity

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**Critical Settling Velocity**

If to is the residence time of liquid in the tank, then all particles with a settling velocity equal to or greater than the critical settling velocity, vo, will settle out at or prior to to, i.e., So all particles with a settling velocity equal to or greater than v0 will be separated in the tank from the fluid

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**Critical Settling Velocity**

Since Note: this expression for vo has no H term. This defines the overflow rate or surface-loading rate - Key parameter to design ideal Type I settling clarifiers - Cross-sectional area A is calculated to get desired v0

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**The Significance of “H”**

Side view The value of H can be used to estimate the fractional recovery of particles with a settling velocity below vo

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**The Significance of “H”**

Only a fraction of particles with a settling velocity vx (less than vo) will settle out. The fraction Fx of particles dx (with velocity vx) that settle out is:

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**The Significance of “H”**

Only a fraction of particles with a settling velocity vx (less than vo) will settle out. The fraction Fx of particles dx (with velocity vx) that settle out is:

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**Cumulative Distribution Curve for Particle Velocities**

settling velocity vs (mm/sec) with a velocity below vs Fraction of particles ò Total Fraction Removed:

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**Ideal Circular Settling Vessel**

Side view

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**Ideal Circular Settling Vessel**

At any particular time and distance ò

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**Ideal Circular Settling Vessel**

In an interval dt, a particle having a diameter below do will have moved vertically and horizontally as follows: For particles with a diameter dx (below do), the fractional removal is given by: ò

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**Sedimentation Thickener/Clarifier**

Top view Side view

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**Plate or Lamella Thickener/Clarifier**

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**Continuous Thickener (Type III)**

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**Thickener (Type III) Control System**

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**Continuous Thickener (Type III)**

Solid Flux Analysis

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**Continuous Thickener (Type III)**

Solid Movement in Thickener

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**Continuous Thickener (Type III)**

Experimental Determination of Solids Settling Velocity

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**Continuous Thickener (Type III)**

Solids Settling Velocity in Hindered Settling

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**Continuous Thickener (Type III)**

Solids Gravity Flux

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**Continuous Thickener (Type III)**

Bulk Velocity where ub = bulk velocity of slurry Qu = volumetric flow rate of thickener underflow A = Surface area of thickener

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**Continuous Thickener (Type III)**

Total Solids Flux

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**Continuous Thickener (Type III)**

Limiting Solids Flux, GL – Dick’s Method

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**Continuous Thickener (Type III)**

Limiting Solids Flux, GL – Dick’s Method - In hindered settling zone, solids concentration ranges from feed concentration to underflow concentration Xu - Within this range, a concentration exists that gives smallest (or limiting) value, GL, of the solid flux G - If thickener is designed for a G value such that G > GL, solids builds up in the clarifying zone and will overflow

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**Continuous Thickener (Type III)**

Limiting Solids Flux, GL – Dick’s Method - The point where the total gravity flux curve is minimum gives GL and XL - GL is highest flux allowed in the thickener - At bottom of thickener, there is no gravity flux as all solid material is removed via bulk flux, i.e.,

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**Mass Balance in a Thickener**

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**Thickener Cross-Sectional Area**

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**Thickener Cross-Sectional Area**

Talmadge – Fitch Method

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**Thickener Cross-Sectional Area**

Talmadge – Fitch Method - Obtain settling rate data from experiment (determine interface height of settling solids (H) vs. time (t) - Construct curve of H vs. t Determine point where hindered settling changes to compression settling - intersection of tangents - construct a bisecting line through this point - draw tangent to curve where bisecting line intersects the curve

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**Thickener Cross-Sectional Area**

Talmadge – Fitch Method - Draw horizontal line for H = Hu that corresponds to the underflow concentration Xu, where - Determine tu by drawing vertical line at point where horizontal line at Hu intersects the bisecting tangent line

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**Thickener Cross-Sectional Area**

Talmadge – Fitch Method - Obtain cross-sectional area required from: - Compute the minimum area of the clarifying section using a particle settling velocity of the settling velocity at t = 0 in the column test. - Choose the largest of these two values

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**Thickener Cross-Sectional Area**

Coe – Clevenger Method - Developed in 1916 and still in use today: where A = cross-sectional area (m2) F = feed pulp liquid/solids ratio L = underflow pulp liquid/solid ratio ρs = solids density (g/cm3) Vm = settling velocity (m/hr) dw/dt = dry solids production rate (g/hr)

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**Thickener Depth and Residece Time**

- Equations describing solids settling do not include tank depth. So it is determined arbitrarily by the designer - Specifying depth is equivalent to specifying residence time for a given flow rate and cross-sectional area - In practice, residence time is of the order of 1-2 hours to reduce impact of non-ideal behaviour

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Typical Settling Test

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**Type II Settling (flocculant)**

- Coalescence of particles occurs during settling (large particles with high velocities overtake small particles with low velocities) - Collision frequency proportional to solids concentration - Collision frequency proportional to level of turbulence (but too high an intensity will promote break-up) - Cumulative number of collisions increases with time

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**Type II Settling (flocculant)**

- Particle agglomerates have higher settling velocities - Rate of particle settling increases with time - Longer residence times and deeper tanks promote coalescence - Fractional removal is function of overflow rate and residence time. - With Type I settling, only overflow rate is significant

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**Primary Thickener Design**

- Typical design is for Type II characteristics - Underflow densities are typically 50-65% solids Safety factors are applied to reduce effect of uncertainties regarding flocculant settling velocities 1.5 to 2.0 x calculated retention time 0.6 to 0.8 x surface loading (overflow rate)

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**Primary Thickener Design**

Non-ideal conditions Turbulence Hydraulic short-circuiting Bottom scouring velocity (re-suspension) All cause shorter residence time of fluid and/or particles

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**Primary Thickener Design Parameters**

Depth (m) m Diameter (m) m Bottom Slope to 0.16 (3.5° to 10°) Rotation Speed of rake arm rpm

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**Hindered (or Zone) Settling (Type III)**

- solids concentration is high such that particle interactions are significant - cohesive forces are so strong that movement of particles is restricted - particles settle together establishing a distinct interface between clarified fluid and settling particles

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**Compression Settling (Type IV)**

- When solids density is very high, particles provide partial mechanical support for those above - particles undergo mechanical compression as they settle - Type IV settling is extremely slow (measured in days)

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