stokes law presentation
TRANSCRIPT
Lecture 12 – MINE 292 - 2012Lecture 12 – MINE 292 - 2012
Terminal Velocity of Settling ParticleTerminal Velocity of Settling ParticleRate 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 NumberReynolds NumberIn 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 NumberDrag Coefficient and Reynolds NumberCd is determined from Stokes Law which relates drag to Reynolds Number
Drag Coefficient and Reynolds NumberDrag Coefficient and Reynolds NumberCd is determined from Stokes Law which relates drag to Reynolds Number
Drag Coefficient and Reynolds NumberDrag Coefficient and Reynolds NumberCd is determined from Stokes Law which relates drag to Reynolds Number
Drag Coefficient and Reynolds NumberDrag Coefficient and Reynolds NumberCd is determined from Stokes Law which relates drag to Reynolds Number
Drag Coefficient and Reynolds NumberDrag Coefficient and Reynolds NumberCd is determined from Stokes Law which relates drag to Reynolds Number
Drag Coefficient and Reynolds NumberDrag Coefficient and Reynolds NumberCd is determined from Stokes Law which relates drag to Reynolds Number
Terminal Velocity of Settling ParticleTerminal Velocity of Settling ParticleTerminal velocity is affected by:
TemperatureFluid Density Particle Density Particle SizeParticle ShapeDegree of TurbulenceVolume fraction of solidsSolid surface charge and pulp chemistryMagnetic and electric field strengthVertical velocity of fluid
Drag Coefficient of Settling ParticleDrag Coefficient of Settling Particle
Terminal Velocity of Settling ParticleTerminal Velocity of Settling Particle
Type I Free-Settling VelocityType I Free-Settling Velocity
Particle Settling in a Laminar (or Quiescent Liquid)
Momentum Balance
Type I Free-Settling VelocityType I Free-Settling Velocity
Particle Settling in a Laminar (or Quiescent Liquid)
Type I Free-Settling VelocityType I Free-Settling Velocity
Integrating gives the steady state solution:
For a sphere:
Terminal Velocity of Settling ParticleTerminal Velocity of Settling ParticleType I Settling of Spheres
Terminal Velocity of Settling ParticleTerminal Velocity of Settling Particle
Terminal Velocity under Terminal Velocity under Hindered Settling ConditionsHindered 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., 1991. Water Resources and Environmental Engineering. Sixth Edition. McGraw-Hill,
New York.
Terminal Velocity under Terminal Velocity under Hindered Settling ConditionsHindered Settling Conditions
McGhee, T.J., 1991. Water Resources and Environmental Engineering. Sixth Edition. McGraw-Hill, New York.
Relationship between CRelationship between Cvv and Weight% and Weight%
Effect of Alum on IEPEffect of Alum on IEP
Ideal Rectangular Settling VesselIdeal Rectangular Settling VesselSide view
Ideal Rectangular Settling VesselIdeal Rectangular Settling VesselModel 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 VesselIdeal Rectangular Settling VesselSide view
u = average (constant) velocity of fluid flowing across vesselvs = settling velocity of a particular particlevo = critical settling velocity of finest particle recovered at 100%
Retention TimeRetention TimeAverage 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 VelocityCritical 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 VelocityCritical Settling Velocity
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
Since
Ideal Circular Settling VesselIdeal Circular Settling VesselSide view
Ideal Circular Settling VesselIdeal Circular Settling VesselAt any particular time and distance
Ideal Circular Settling VesselIdeal 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:
Sedimentation Thickener/ClarifierSedimentation Thickener/ClarifierTop view
Side view
Plate or Lamella Thickener/ClarifierPlate or Lamella Thickener/Clarifier
Continuous Thickener (Type III)Continuous Thickener (Type III)
Thickener (Type III) Control SystemThickener (Type III) Control System
Continuous Thickener (Type III)Continuous Thickener (Type III)Solid Flux Analysis
Continuous Thickener (Type III)Continuous Thickener (Type III)Solid Movement in Thickener
Continuous Thickener (Type III)Continuous Thickener (Type III)Experimental Determination of Solids Settling Velocity
Continuous Thickener (Type III)Continuous Thickener (Type III)Solids Settling Velocity in Hindered Settling
Continuous Thickener (Type III)Continuous Thickener (Type III)Solids Gravity Flux
Continuous Thickener (Type III)Continuous Thickener (Type III)Bulk Velocity
where
ub = bulk velocity of slurry
Qu = volumetric flow rate of thickener underflow
A = Surface area of thickener
Mass Balance in a ThickenerMass Balance in a Thickener
Thickener Cross-Sectional AreaThickener Cross-Sectional Area
Thickener Cross-Sectional AreaThickener Cross-Sectional Area
Talmadge – Fitch Method
Thickener Cross-Sectional AreaThickener 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 AreaThickener 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 AreaThickener 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 AreaThickener Cross-Sectional AreaCoe – 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 TimeThickener 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 TestTypical Settling Test
Type II Settling (flocculant)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)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 DesignPrimary 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 DesignPrimary 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 ParametersPrimary Thickener Design Parameters
Depth (m) 3 - 5 m
Diameter (m) 3 - 170 m
Bottom Slope 0.06 to 0.16 (3.5° to 10°)
Rotation Speedof rake arm 0.02 - 0.05 rpm
Hindered (or Zone) Settling (Type III)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)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)