SETTLING VELOCITY OF PARTICLES Equation for one-dimensional motion of particle through fluid • Expression for acceleration of a particle settling in a fluid: 푑푢 푚 = 퐹 − 퐹 − 퐹 푑푡 푒 푏 퐷 Where , 퐹푒= 푚푎푒 • acceleration in external field = 푎푒 =g in gravity settling, and ω2푟in centrifugal field 푚 • 퐹푏 =Buoyant force = 휌 휌푝 퐶 휌푢2퐴 • 퐹 = Drag force, 퐹 = 퐷 푝 퐷 퐷 2 • CD = Drag coefficient, Ap=Projected area • 퐹퐷 always increases with velocity, and soon acceleration becomes 0. • Terminal velocity is the constant velocity the particle attains when acceleration becomes 0. DRAG COEFFICIENT & TERMINAL VELOCITY

푭푫 푨풑 • 푪푫 = ퟐ 흆풖ퟎ ퟐ ퟐ 풅풖 풎 푪푫흆풖 푨풑 • 풎 = 푭풆 − 푭풃 − 푭푫 = 풎풂풆 − 흆풂풆 − 풅풕 흆풑 ퟐ ퟐ 풅풖 ퟏ 푪푫흆풖 푨풑 • = 풂풆 − 흆풂풆 − 풅풕 흆풑 ퟐ풎 ퟐ 흆 푪푫흆풖 푨풑 • 풂풆 ퟏ − = 흆풑 ퟐ풎

흅 ퟐ 푫 ퟑ흆 품 흆 −흆 ퟐ ퟐ풎품 흆풑−흆 ퟔ 풑 풑 풑 ퟒ푫풑품 흆풑−흆 • 풖 = = ퟐ = 푪 흆흆 푨 흅푫풑 ퟑ푪 흆 푫 풑 풑 푪 흆흆 푫 푫 풑 ퟒ

ퟒ푫 품 흆 −흆 • 풖 = 풑 풑 ퟑ푪푫흆

Drag Coefficients for spheres Trial and error method for determination of terminal velocity

ퟒ푫풑품 흆풑 − 흆 풖 = … . (ퟏ) ퟑ푪푫흆

1. Assume a value of u 2. Calculate Reynolds number of particle, 푁푅푒,푝 퐷 푢휌 3. 푁 = 푝 푅푒,푝 휇 • 퐷푝 = 퐷𝑖푎푚푒푡푒푟 표푓 푝푎푟푡𝑖푐푙푒, 푢 = 푣푒푙표푐𝑖푡푦 표푓 푝푎푟푡𝑖푐푙푒, • 휌 = density of fluid, 휇= viscosity of fluid 4. Determine 푪푫 from chart of 푪푫 푣푠 푁푅푒,푝 5. Calculate u from equation 1 6. 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

4푔 휌푝 − 휌 퐷푝 푢푡 = 3퐶퐷휌 • Stokes Law range, particle Reynolds number less than 1.0 24 퐷푝푢표휌 퐶퐷 = , where푁푅푒,푝 = 푁푅푒,푝 휇 2 푔퐷푝 휌푝 − 휌 푢 = 푡 18휇

• Newtons Law Range: 1000<푁푅푒,푝<200,000, 퐶퐷 = 0.44

푔퐷푝 휌푝 − 휌 푢 = 1.75 푡 휌

7 4푔 휌푝−휌 퐷푝 • 푢푡 = 3퐶퐷휌

2 4푔 휌푝−휌 퐷푝 퐷푝 푔 휌푝−휌 푢푡 • 푢푡 = 24 = 3 휌 18휇 퐷푝푢표휌 휇 퐷 2푔 휌 −휌 • 푢 = 푝 푝 푡 18휇 Determination of Range of settling

• Determine the value of K 1 3 푔휌 휌푝 − 휌 퐾 = 퐷 푝 휇2 • Stokes Law range K<2.6 • Newtons Law range K>68.9 • Intermediate range K greater than 2.6 and less than 68.9 • For Stokes Law range: 퐷 푢 휌 퐷 휌 퐷 2푔 휌 −휌 퐷 3휌푔 휌 −휌 • 푅푒 = 푝 푡 = 푝 푝 푝 = 푝 푝 < 1 푝 휇 휇 18휇 18휇2 1/3 휌푔 휌 −휌 • Let 퐾 = 퐷 푝 푝 휇2 1 • 퐾3 < 1, 표푟 퐾 < 181/3 = 2.6 18 • For Newtons law range 3 1 . 퐷 푢 휌 퐷 휌 푔퐷 휌 −휌 휌푔 휌 −휌 2 3 • 푅푒 = 푝 푡 = 푝 1.75 푝 푝 = 1.75퐷 1.5 푝 = 푝 휇 휇 휌 푝 휇2 1.75퐾1.5 • 1.75퐾1.5 > 1000 • 퐾 > 68.9 • Estimate the terminal velocity of 80/100 mesh particles of limestone (density 2800kg/m3) settling in water at 30oC. Viscosity =0.801cP

• Determine K, to find range of settling, – If K is not much larger than 2.6 Stokes law can be assumed and rechecked – If K is not much less than 68.9 Newtons law can be assumed – If K is in Intermediate range, trial and erro method to be followed Hindered settling velocity

푛 • 푢푠 = 푢푡 휀 where 휀 = 푝표푟표푠𝑖푡푦 • 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? Solid Liquid Separation Settling – Gravity and Centrifugal

Reference McCabe Smith

14 GRAVITY SETTLING

15 16 Types of sedimentation tanks

• Circular Radial flow • Conical Vertical flow • Rectangular horizontal flow

17 Circular Radial Flow Gravity Thickener

18 DORR THICKENER

19 Circular Radial Flow Gravity Thickener

• 1 = Raw wastewater inlet pipe 2 = Inlet stilling well (baffle) 3 = Clarified water overflow weir 4 = Primary sludge outlet pipe •

20 • 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,. • 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). • 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%. • 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. Upflow velocities of around 1.0m per hour are generally desired. 21 VERTICAL SETTLING TANK Dortmund Type • For small plants conical Dortmund-type settling tanks are preferred. Occasionally rectangular structures are used, with complex sludge and scum scraper mechanisms. • Upward flow tank suited small treatment plants • Steep floor slope, large lower hopper volume enable large sludge storage • No need for scarping. • Depth of hopper at least equal to the top dimension which is less than 6m 22 RECTANGULAR HORIZONTAL FLOW

• .

• Rectangular are for large primary settling tank • Sensitive depth to width /length ratio • Occasionally rectangular structures are used, with complex sludge and scum scraper mechanisms. • Rectangular are for large primary settling tank • Sensitive depth to width /length ratio

23 THICKENERS AND CLARIFIERS SETTLING OF FLOCCULATED PARTICLES • Fine particles form agglomerates entrapping Batch Sedimentation test water within • A= Clear liquid • Flocculating agents – strong • B = Uniform initial conc. electrolytes, polymeric - • C=Transition zone polyacrylamide • D= Settled solid • Inexpensive water treatment materials lime alumina, sodium silicates, ferrous sulphate, ferric chloride etc. form loose agglomerates and removes fine particles.

24 Time Height of clear liquid interface 0 Ho

25 gffd

• Plot settling rate curve Time min Interface height m from the following 0 2.5 settling experimental 20 1.56 data, and report 40 0.97 (a)Hindered settling 60 0.75 velocity (b) settling in 80 0.63 compression zone (c) 100 0.53 120 0.44 critical composition. 140 0.39 160 0.36

26 SETTLING TANK DESIGN Ref: McCabe Smith • There are two ways the solids can move to the bottom: Feed overflow 1. Under the influence of their 푄푖, 퐶표 푄표 settling velocity, downward 2. Due to the continuous removal of sludge at the bottom as underflow

Total downward flux of solids, 2 퐾푔 푚 ℎ푟, consists of two parts the Underflow 1. Transport Flux, 퐺푡 푄푖, 푄표, 푄푢=Flowrate of inlet, 2. Settling flux 퐺푠 overflow and underflow 퐶표, 퐶푢=Concentration of solids in inlet and underflow 27 • 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. 푑퐻 퐺 = 퐶 푠 푑푡 푖 푖 푑퐻 , 퐶푖= velocity and concentration at a layer 푑푡 푖 "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.

• 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. 퐺푡 = 푢퐶푖 where, u= the velocity created by the underflow 28 • Total flux curve has a minimum value SETTLING TANK between the influent solids concentration AREA (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.

• 퐺 = 퐺푡 + 퐺푠 푑퐻 = 푢퐶 + 퐶 푖 푑푡 푖 푖 As Solid introduced = solid removed 푄퐶표 = 푇표푡푎푙 푓푙푢푥 푥 푎푟푒푎 푄퐶 퐴 = 표 퐺푡 + 퐺푆 푚푖푛

29 KYNCH Method to obtain settling flux from one batch settling data • Locate any time, ti • Draw tangent at the point to cut y axis t Hi’ and x axis at ti’

퐻0퐶0 • Determine Ci: 퐻0퐶0 = 퐻푖′퐶푖 푎푛푑 퐶푖 = [conc. at top of settling zone] 퐻푖′ 푑퐻 퐻 ′ • Determine settling velocity, = 푖 푑푡 푖 푡푖′ 푑퐻 • Determine settling Flux : 퐺푠 = 퐶푖 푖 푑푡 푖 • Plot Gsi vs Ci

풕풊 풕풊′ 푯풊′ 퐶푖 푑퐻 퐺푠푖 푑푡 푖

30 • Following is the results of a Time,min Height of settling experiment, with interface, mm initial concentration of 60 g/l 0 250 of calcium carbonate. Data is to be used to design a settling 10 175 tank to handle 0.03m3/second 20 123 of slurry, with an underflow velocity of 0.05m/h. 30 103 • Plot (a) Settling flux vs 40 86 concentration, (b) transport 50 75 flux vs concentration,(c) total flux vs concentration 60 65 • Determine minimum total flux, 70 57 and cross sectional area of the 80 52 tank.

31 CENTRIFUGAL SETTLING

32 Tubular Centrifuge • Centrifugal liquid liquid separation • Removal of solids from lubricating oil, ink, beverages etc. • 10 to 15 cm dia • 15,000rpm • Solid/dirt accumulate inside the bow, and periodically removed

33 Disk Centrifuge • Liquid liquid separation • Removal of solids from lubricating oil, ink., beverages etc. • Short wide bowl 20to 50cmdia • Bowl filled with discs- cones of sheet metal set one above the other, with matching holes in the middle, which form the liquid passage • Heavy liquid moves outwards and light liquid inwards. • Soon Heavy Liquid touches lower side of a cone and separates out • Shearing helps break emulsions • Solid/dirt accumulate inside the bowl, and periodically removed

34 Nozzle discharge Centrifuge

• Modified disk centrifuge • 3 mm dia holes at the periphery • Dilute slurry leaves through hole • Alternatively the holes are plugged most of the time and occasionally opened and thick slurry removed.

35 HELICAL CONVEYER CENTRIFUGE • Bowl diameter – 10cm to 140 cm • Solid removal rate – 1 to 2 tons/h to 50tons/h • May not clarify 100% may require subsequent clarifier • Separate free particles from water, like crystals, dewater etc

36 Sedimenting Centrifuge 휔2푟퐷 2 휌 −휌 • 푢 = 푝 푝 푡 18휇

• Since 푢푡 = 푑푟 푑푡 Time required for a particle of size Dp to travel from radius rA to rB.

푡푇 푟퐵 18휇 푑푟 • 푑푡 = 2 2 0 푟퐴 휔 휌푝−휌 퐷푝 푟

18휇 푟퐵 • 푡푇 = 2 2 푙푛 휔 휌푝−휌 퐷푝 푟퐴 • 푟2 = radius of the bowl • 푟1 = radius at liquid surface • Residence time • 푏 =depth of bowl 2 2 푉표푙푢푚푒 휋푏 푟2 −푟1 • 퐷푝 = 푝푎푟푡𝑖푐푙푒 푠𝑖푧푒 • 푡 = = 푇 푞 푞 • 푟퐴= position of particle at inlet • 푟퐵 = position of particle at outlet 2 2 2 2 휋푏휔 휌푝−휌 퐷푝 푟2 −푟1 • 푞 = 푟 18휇 푙푛 퐵 푟퐴 37 CUT DIAMETER

• Cut point diameter particles that travels half the distance between 푟2 푎푛푑 푟1in the available time 푟 +푟 • For the particles to be removed 푟 = 1 2 and 퐴 2 푟퐵 = 푟2 2 2 2 2 휋푏휔 휌푝−휌 퐷푝 푟2 −푟1 • 푞 = 푟 18휇 푙푛 퐵 푟퐴 2 2 2 2 휋푏휔 휌푝−휌 퐷푝푐 푟2 −푟1 • 푞푐 = 2푟 18휇 푙푛 2 푟1+푟2

38 Special case: very thin liquid layer 푟1 ≅ 푟2 휔2푟 퐷 2 휌 −휌 • 푢 = 2 푝 푝 푡 18휇 • Let the thickness = s

• If Cut dia is 퐷푝푐 the particles have to travel a distance s/2 푠 • 푢푡 = 2푡푇 푉 • Residence time 푡푇 = 푞푐 2 2 푉 2푉푢푡 2푉휔 푟2퐷푝 휌푝 − 휌 푞푐 = = = 푡푇 푠 18휇푠

39 Sigma Value: a parameter for scale up 2 2 2푉휔 푟2퐷푝 휌푝 − 휌 푞 = 푐 18휇푠 Let re and se be average value of radius and liquid layer thickness 2 2 2푉휔 푟푒퐷푝 휌푝 − 휌 푞푐 = 18휇푠푒 2 2 2푉휔 푟푒 퐷푝 푔 휌푝 − 휌 = = 2 푈퐺 푔푠푒 18휇 푉휔2푟 푆𝑖푔푚푎 푣푎푙푢푒, = 푒, is a characteristics of centrifuge . 푔푠푒 It is the cross sectional area of a gravity settling tank of the same separation capacity as the centrifuge.

40 41 • What is the capacity in cubic meters per hour of a clarifying centrifuge operating under the following conditions? – Diameter of bowl, 600mm – Thickness of liquid layer, 75mm – Sp.gr of liquid 1.2 sp.gr of solid 1.6 – Depth of bow 400 mm Viscosity of liquid 2 cP – Speed, 1,200rpm Cut size 30휇푚

42