SETTLING VELOCITY OF PARTICLES...โขParticles of sphalerite (sp. Gr. 4.00) are settling under the...
Transcript of SETTLING VELOCITY OF PARTICLES...โขParticles of sphalerite (sp. Gr. 4.00) are settling under the...
Equation for one-dimensional motion of
particle through fluidโข Expression for acceleration of a particle settling in a fluid:
๐๐๐ข
๐๐ก= ๐น๐ โ ๐น๐ โ ๐น๐ท
Where , ๐น๐= ๐๐๐โข acceleration in external field = ๐๐=g in gravity settling, and ฯ2๐in centrifugal field
โข ๐น๐ =Buoyant force =๐
๐๐๐
โข ๐น๐ท= Drag force, ๐น๐ท =๐ถ๐ท๐๐ข
2๐ด๐
2โข CD = Drag coefficient, Ap=Projected areaโข ๐น๐ท always increases with velocity, and soon acceleration
becomes 0.โข Terminal velocity is the constant velocity the particle attains
when acceleration becomes 0.
DRAG COEFFICIENT &TERMINAL VELOCITY
โข ๐ช๐ซ = ๐ญ๐ซ ๐จ๐
๐๐๐๐ ๐
โข ๐๐ ๐
๐ ๐= ๐ญ๐ โ ๐ญ๐ โ ๐ญ๐ซ = ๐๐๐ โ
๐
๐๐๐๐๐ โ
๐ช๐ซ๐๐๐๐จ๐
๐
โข๐ ๐
๐ ๐= ๐๐ โ
๐
๐๐๐๐๐ โ
๐ช๐ซ๐๐๐๐จ๐
๐๐
โข ๐๐ ๐ โ๐
๐๐=๐ช๐ซ๐๐
๐๐จ๐
๐๐
โข ๐๐ =๐๐๐ ๐๐โ๐
๐ช๐ซ๐๐๐๐จ๐=๐๐
๐๐ซ๐๐๐๐ ๐ ๐๐โ๐
๐ช๐ซ๐๐๐๐ ๐ซ๐๐
๐
=๐๐ซ๐๐ ๐๐โ๐
๐๐ช๐ซ๐
โข ๐ =๐๐ซ๐๐ ๐๐โ๐
๐๐ช๐ซ๐
Trial and error method for determination of terminal velocity
๐ =๐๐ซ๐๐ ๐๐ โ ๐
๐๐ช๐ซ๐โฆ . (๐)
1. Assume a value of u
2. Calculate Reynolds number of particle, ๐๐ ๐,๐
3. ๐๐ ๐,๐=๐ท๐๐ข๐
๐
โข ๐ท๐ = ๐ท๐๐๐๐๐ก๐๐ ๐๐ ๐๐๐๐ก๐๐๐๐, ๐ข = ๐ฃ๐๐๐๐๐๐ก๐ฆ ๐๐ ๐๐๐๐ก๐๐๐๐,โข ๐ = density of fluid, ๐= viscosity of fluid
4. Determine ๐ช๐ซ from chart of ๐ช๐ซ ๐ฃ๐ ๐๐ ๐,๐5. Calculate u from equation 16. Compare Calculated u with Assumed u, if error is not within limit
restart from step 1 for second trial
Terminal velocity in Stokes Law range and Newtons law range
๐ข๐ก =4๐ ๐๐ โ ๐ ๐ท๐
3๐ถ๐ท๐
โข Stokes Law range, particle Reynolds number less than 1.0
๐ถ๐ท =24
๐๐ ๐,๐, where๐๐ ๐,๐ =
๐ท๐๐ข๐๐
๐
๐ข๐ก =๐๐ท๐
2 ๐๐ โ ๐
18๐
โข Newtons Law Range: 1000<๐๐ ๐,๐<200,000, ๐ถ๐ท = 0.44
๐ข๐ก = 1.75๐๐ท๐ ๐๐ โ ๐
๐
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โข ๐ข๐ก =4๐ ๐๐โ๐ ๐ท๐
3๐ถ๐ท๐
โข ๐ข๐ก =4๐ ๐๐โ๐ ๐ท๐
324๐ท๐๐ข๐๐
๐
๐=๐ท๐2๐ ๐๐โ๐ ๐ข๐ก
18๐
โข ๐ข๐ก =๐ท๐2๐ ๐๐โ๐
18๐
Determination of Range of settling
โข Determine the value of K
๐พ = ๐ท๐๐๐ ๐๐ โ ๐
๐2
1 3
โข Stokes Law range K<2.6
โข Newtons Law range K>68.9
โข Intermediate range K greater than 2.6 and less than 68.9
โข For Stokes Law range:
โข ๐ ๐๐ =๐ท๐๐ข๐ก๐
๐=๐ท๐๐
๐
๐ท๐2๐ ๐๐โ๐
18๐=๐ท๐3๐๐ ๐๐โ๐
18๐2< 1
โข Let ๐พ = ๐ท๐๐๐ ๐๐โ๐
๐2
1/3
โข1
18๐พ3 < 1, ๐๐ ๐พ < 181/3 = 2.6
โข For Newtons law range
โข ๐ ๐๐ =๐ท๐๐ข๐ก๐
๐=๐ท๐๐
๐1.75
๐๐ท๐ ๐๐โ๐
๐= 1.75๐ท๐
1.5 ๐๐ ๐๐โ๐
๐2
3
2.1
3=
1.75๐พ1.5
โข 1.75๐พ1.5 > 1000
โข ๐พ > 68.9
โข Estimate the terminal velocity of 80/100 mesh particles of limestone (density 2800kg/m3) settling in water at 30oC. Viscosity =0.801cP
โข Determine K, to find range of settling,โ If K is not much larger than 2.6 Stokes law can be
assumed and recheckedโ If K is not much less than 68.9 Newtons law can be
assumedโ If K is in Intermediate range, trial and erro method to
be followed
Hindered settling velocity
โข ๐ข๐ = ๐ข๐ก ๐๐ where ๐ = ๐๐๐๐๐ ๐๐ก๐ฆ
โข Particles of sphalerite (sp. Gr. 4.00) are settling under the force of gravity in the carbon tetrachloride (CCl4) at 20oC(sp.gr. 1.594). The diameter of the sphalerite particles is 0.1 mm. The free settling terminal velocity is 0.015m/s. The volume fraction of sphalerite in carbon tetrachloride is 0.2. What is the settling velocity?
Types of sedimentation tanks
โข Circular Radial flow
โข Conical Vertical flow
โข Rectangular horizontal flow
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Circular Radial Flow Gravity Thickener
โข 1 = Raw wastewater inlet pipe2 = Inlet stilling well (baffle)3 = Clarified water overflow weir4 = Primary sludge outlet pipe
โข
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โข After the removal of inorganic screenings and grit, the next step in wastewater treatment is the removal of the grosser suspended solids from the raw sewage. This is achieved via the process of primary settling or primary sedimentation. The wastewater flow is directed into one or more settling tanks, also known as primary clarifiers. Large mechanically-raked circular tanks are generally used at the larger works,.
โข The overall volume of the primary settling tank is designed to ensure the incoming wastewater will take a certain amount of time to flow completely through the structure. This is known as the retention time, and must be sufficiently long enough to allow suitable settling of the solids to take place, yet short enough to prevent the anaerobic breakdown of the settled solids from occurring (i.e. the settled sludge turning septic).
โข The retention time can vary from 1 to 3 hours, depending on the plant loading and incoming wastewater characteristics. In a tank with a retention time of 2 hours, 50 to 70% of the suspended solids may be removed by settling and flotation (scum removal). Removal of these solids will also reduce the BOD by 30 to 35%.
โข Of equal importance in the design of the primary settling tank is the surface loading rate. This will dictate the overall diameter of a circular settling tank. This is effectively the upward velocity of the incoming flow from the base of the tank to the top of the overflow weirs. Upflowvelocities of around 1.0m per hour are generally desired.
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VERTICAL SETTLING TANKDortmund Type
โข For small plants conical Dortmund-type settling tanks are preferred. Occasionally rectangular structures are used, with complex sludge and scum scraper mechanisms.
โข Upward flow tank suited small treatment plants
โข Steep floor slope, large lower hopper volume enable large sludge storage
โข No need for scarping.โข Depth of hopper at least
equal to the top dimension which is less than 6m
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RECTANGULAR HORIZONTAL FLOW
โข .
โข Rectangular are for large primary settling tank
โข Sensitive depth to width /length ratio
โข Occasionally rectangular structures are used, with complex sludge and scum scraper mechanisms.
โข Rectangular are for large primary settling tank
โข Sensitive depth to width /length ratio
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THICKENERS AND CLARIFIERSSETTLING OF FLOCCULATED PARTICLES
โข Fine particles form agglomerates entrapping water within
โข Flocculating agents โ strong electrolytes, polymeric -polyacrylamide
โข Inexpensive water treatment materials lime alumina, sodium silicates, ferrous sulphate, ferric chloride etc. form loose agglomerates and removes fine particles.
Batch Sedimentation test
โข A= Clear liquid
โข B = Uniform initial conc.
โข C=Transition zone
โข D= Settled solid
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gffd
โข Plot settling rate curve from the following settling experimental data, and report (a)Hindered settling velocity (b) settling in compression zone (c) critical composition.
Time min Interface height m
0 2.5
20 1.56
40 0.97
60 0.75
80 0.63
100 0.53
120 0.44
140 0.39
160 0.36
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SETTLING TANK DESIGNRef: McCabe Smith
โข There are two ways the solids can move to the bottom:
1. Under the influence of their settling velocity, downward
2. Due to the continuous removal of sludge at the bottom as underflow
Total downward flux of solids, ๐พ๐ ๐2โ๐, consists of two parts the 1. Transport Flux, ๐บ๐ก2. Settling flux ๐บ๐
Feed overflow๐๐, ๐ถ๐ ๐๐
Underflow
๐๐ , ๐๐, ๐๐ข=Flowrate of inlet, overflow and underflow
๐ถ๐ , ๐ถ๐ข=Concentration of solids in inlet and underflow
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โข Settling flux, Gs, also called the batch settling flux, this is due to the settlement of solids and as expected is a function of solids concentration and the settling velocity of particles.
๐บ๐ =๐๐ป
๐๐ก๐
๐ถ๐
๐๐ป
๐๐ก ๐, ๐ถ๐= velocity and concentration at a layer
"i" in the thickener. It passes through a maxima, as at very low concentration settling rate is constant, and at high concentration the rate decreases rapidly.
โข Transport flux, Gt,Fluxof solids carried by liquid, is independent of the solids settling in the thickener, whether the solids settle or not, there pumping of the sludge from the bottom of the tank.
๐บ๐ก = ๐ข๐ถ๐where, u= the velocity created by the underflow
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SETTLING TANKAREA
โข Total flux curve has a minimum value between the influent solids concentration (Co) and underflow solids concentration (Cu). This minimum flux is the maximum allowable solids loading for the thickener to work successfully. When flux is at minimum, we calculate for the maximum area required. If this limit is exceeded, the solids will overflow in the effluent. So the design of a thickener is thus reduced to the point of determining this flux.
โข ๐บ = ๐บ๐ก + ๐บ๐
= ๐ข๐ถ๐ +๐๐ป
๐๐ก๐
๐ถ๐
As Solid introduced = solid removed๐๐ถ๐ = ๐๐๐ก๐๐ ๐๐๐ข๐ฅ ๐ฅ ๐๐๐๐
๐ด =๐๐ถ๐
๐บ๐ก + ๐บ๐ ๐๐๐
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KYNCH Method to obtain settling flux from one batch settling data
โข Locate any time, ti
โข Draw tangent at the point to cut y axis t Hiโ and x axis at tiโ
โข Determine Ci: ๐ป0๐ถ0 = ๐ป๐โฒ๐ถ๐ ๐๐๐ ๐ถ๐ =๐ป0๐ถ0
๐ป๐โฒ[conc. at top of settling zone]
โข Determine settling velocity, ๐๐ป
๐๐ก ๐=๐ป๐โฒ
๐ก๐โฒ
โข Determine settling Flux : ๐บ๐ ๐ =๐๐ป
๐๐ก ๐๐ถ๐
โข Plot Gsi vs Ci
๐๐ ๐๐โฒ ๐ฏ๐โฒ ๐ถ๐ ๐๐ป
๐๐ก๐
๐บ๐ ๐
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โข Following is the results of a settling experiment, with initial concentration of 60 g/l of calcium carbonate. Data is to be used to design a settling tank to handle 0.03m3/second of slurry, with an underflow velocity of 0.05m/h.
โข Plot (a) Settling flux vs concentration, (b) transport flux vs concentration,(c) total flux vs concentration
โข Determine minimum total flux, and cross sectional area of the tank.
Time,min Height of interface, mm
0 250
10 175
20 123
30 103
40 86
50 75
60 65
70 57
80 52
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Tubular Centrifuge โข Centrifugal liquid liquid
separation
โข Removal of solids from lubricating oil, ink, beverages etc.
โข 10 to 15 cm dia
โข 15,000rpm
โข Solid/dirt accumulate inside the bow, and periodically removed
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Disk Centrifugeโข Liquid liquid separationโข Removal of solids from lubricating
oil, ink., beverages etc.โข Short wide bowl 20to 50cmdiaโข Bowl filled with discs- cones of
sheet metal set one above the other, with matching holes in the middle, which form the liquid passage
โข Heavy liquid moves outwards and light liquid inwards.
โข Soon Heavy Liquid touches lower side of a cone and separates out
โข Shearing helps break emulsions โข Solid/dirt accumulate inside the
bowl, and periodically removed
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Nozzle discharge Centrifuge
โข Modified disk centrifuge
โข 3 mm dia holes at the periphery
โข Dilute slurry leaves through hole
โข Alternatively the holes are plugged most of the time and occasionally opened and thick slurry removed.
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HELICAL CONVEYER CENTRIFUGEโข Bowl diameter โ 10cm to 140 cm
โข Solid removal rate โ 1 to 2 tons/h to 50tons/h
โข May not clarify 100% may require subsequent clarifier
โข Separate free particles from water, like crystals, dewater etc
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Sedimenting Centrifuge
โข ๐2 = radius of the bowlโข ๐1 = radius at liquid surfaceโข ๐ =depth of bowl
โข ๐ท๐ = ๐๐๐๐ก๐๐๐๐ ๐ ๐๐ง๐
โข ๐๐ด= position of particle at inletโข ๐๐ต = position of particle at outlet
โข ๐ข๐ก =๐2๐๐ท๐
2 ๐๐โ๐
18๐
โข Since ๐ข๐ก = ๐๐ ๐๐ก
Time required for a particle of size Dp to travel from radius rA to rB.
โข 0๐ก๐ ๐๐ก = ๐๐ด
๐๐ต 18๐
๐2 ๐๐โ๐ ๐ท๐2
๐๐
๐
โข ๐ก๐ =18๐
๐2 ๐๐โ๐ ๐ท๐2 ๐๐๐๐ต
๐๐ด
โข Residence time
โข ๐ก๐ =๐๐๐๐ข๐๐
๐=๐๐ ๐2
2โ๐12
๐
โข ๐ =๐๐๐2 ๐๐โ๐ ๐ท๐
2
18๐
๐22โ๐1
2
๐๐๐๐ต๐๐ด
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CUT DIAMETER
โข Cut point diameter particles that travels half the distance between ๐2 ๐๐๐ ๐1in the available time
โข For the particles to be removed ๐๐ด =๐1+๐2
2and
๐๐ต = ๐2
โข ๐ =๐๐๐2 ๐๐โ๐ ๐ท๐
2
18๐
๐22โ๐1
2
๐๐๐๐ต๐๐ด
โข ๐๐ =๐๐๐2 ๐๐โ๐ ๐ท๐๐
2
18๐
๐22โ๐1
2
๐๐2๐2๐1+๐2
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Special case: very thin liquid layer ๐1 โ ๐2
โข ๐ข๐ก =๐2๐2๐ท๐
2 ๐๐โ๐
18๐
โข Let the thickness = s
โข If Cut dia is ๐ท๐๐ the particles have to travel a distance s/2
โข ๐ข๐ก =๐
2๐ก๐
โข Residence time ๐ก๐ =๐
๐๐
๐๐ =๐
๐ก๐=2๐๐ข๐ก๐ =2๐๐2๐2๐ท๐
2 ๐๐ โ ๐
18๐๐
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Sigma Value: a parameter for scale up
๐๐ =2๐๐2๐2๐ท๐
2 ๐๐ โ ๐
18๐๐ Let re and se be average value of radius and liquid layer thickness
๐๐ =2๐๐2๐๐๐ท๐
2 ๐๐ โ ๐
18๐๐ ๐
=2๐๐2๐๐๐๐ ๐
๐ท๐2๐ ๐๐ โ ๐
18๐= 2 ๐๐บ
๐๐๐๐๐ ๐ฃ๐๐๐ข๐, =๐๐2๐๐
๐๐ ๐, is a characteristics of centrifuge .
It is the cross sectional area of a gravity settling tank of the same separation capacity as the centrifuge.
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โข What is the capacity in cubic meters per hour of a clarifying centrifuge operating under the following conditions?
โ Diameter of bowl, 600mm
โ Thickness of liquid layer, 75mm
โ Sp.gr of liquid 1.2 sp.gr of solid 1.6
โ Depth of bow 400 mm Viscosity of liquid 2 cP
โ Speed, 1,200rpm Cut size 30๐๐
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