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Settling Velocity of Particles

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Settling Velocity of Particles 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 = 퐷푎푚푒푡푒푟 표푓 푝푎푟푡푐푙푒, 푢 = 푣푒푙표푐푡푦 표푓 푝푎푟푡푐푙푒, • 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 = 푝표푟표푠푡푦 • 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.
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