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# Stoke’s Law and Settling Particles

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Stoke’s Law and Settling Particles
Lecture 12 – MINE

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.

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)

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

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

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

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

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

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

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

Drag Coefficient of Settling Particle

Terminal Velocity of Settling Particle

Type I Free-Settling Velocity
Particle Settling in a Laminar (or Quiescent Liquid) Momentum Balance

Type I Free-Settling Velocity
Particle Settling in a Laminar (or Quiescent Liquid)

Type I Free-Settling Velocity
Integrating gives the steady state solution: For a sphere:

Terminal Velocity of Settling Particle
Type I Settling of Spheres

Terminal Velocity of Settling Particle

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.

Terminal Velocity under Hindered Settling Conditions
McGhee, T.J., Water Resources and Environmental Engineering. Sixth Edition. McGraw-Hill, New York.

Relationship between Cv and Weight%

Effect of Alum on IEP

Ideal Rectangular Settling Vessel
Side view

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

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%

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

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

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

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

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:

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:

Cumulative Distribution Curve for Particle Velocities
settling velocity vs (mm/sec) with a velocity below vs Fraction of particles ò Total Fraction Removed:

Ideal Circular Settling Vessel
Side view

Ideal Circular Settling Vessel
At any particular time and distance ò

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

Sedimentation Thickener/Clarifier
Top view Side view

Plate or Lamella Thickener/Clarifier

Continuous Thickener (Type III)

Thickener (Type III) Control System

Continuous Thickener (Type III)
Solid Flux Analysis

Continuous Thickener (Type III)
Solid Movement in Thickener

Continuous Thickener (Type III)
Experimental Determination of Solids Settling Velocity

Continuous Thickener (Type III)
Solids Settling Velocity in Hindered Settling

Continuous Thickener (Type III)
Solids Gravity Flux

Continuous Thickener (Type III)
Bulk Velocity where ub = bulk velocity of slurry Qu = volumetric flow rate of thickener underflow A = Surface area of thickener

Continuous Thickener (Type III)
Total Solids Flux

Continuous Thickener (Type III)
Limiting Solids Flux, GL – Dick’s Method

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

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.,

Mass Balance in a Thickener

Thickener Cross-Sectional Area

Thickener Cross-Sectional Area
Talmadge – Fitch Method

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

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

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

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)

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

Typical Settling Test

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

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

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)

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

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

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

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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