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PARTICLE TECHNOLOGY Industrial Screening Equipment Grizzly: It has a plane screening surface Introduction to Particle Technology composed of longitudinal bars up to 3 m long,...

PARTICLE TECHNOLOGY Industrial Screening Equipment Grizzly: It has a plane screening surface Introduction to Particle Technology composed of longitudinal bars up to 3 m long, fixed in a rectangular framework. It is usually Particle Technology refers to the field of science inclined at an angle to the horizontal and the and technology associated with the greater the angle then the greater is the characterization, formation, processing, and throughput although the screening efficiency is utilization of particles. It is concerned with the reduced. Used for very coarse feed. systems in which one or more of the components are in the form of particles. Vibrating Screen: These are mechanically operated screens which are vibrated by means In the field of particle technology, there exists a of an electromagnetic device. These screens complex relationship between properties of are sometimes mounted in a multi-deck fashion individual particles (such as size and shape with the coarsest screen on top, either distribution, density, roughness, surface area) and horizontally or inclined at angles up to 45° their bulk behavior that is observed during particle flow, storage, mixing, fluidization, compression, and Trommel: It consists of a slowly rotating agglomeration perforated cylinder with its axis at a slight angle to the horizontal. The material to be screened is SCREENING fed in at the top and gradually moves down the screen and passes over apertures of gradually Screening is a method of separating particles increasing size, with the result that all the according to size alone material has to pass over the finest screen. Screening are used on a large scale for the separation of particles according to their sizes Gyrating Screens: Usually gyrated at the feed and for the production of closely graded end in a horizontal plane. The discharge end materials in carrying out size analyses on a reciprocates but does not gyrate. This small scale combination stratifies the feed, so that fine particles travel downward to the screen surface, Screening Equipment where they are pushed through by the larger particles on top Screening equipment is commonly used to size ○ Gyrate - to move or cause to move in a and separate material throughout the circle or spiral, especially quickly. production process. Screens used ahead of a primary crusher that ○ Stratification - This phenomenon occurs as can remove fine material, like abrasive stone or vibration is passed through a bed of sand, which can cause wear and tear on the material. This causes coarse (larger) crusher’s liners. material to rise and finer (smaller) material They can also keep material that is already too to descend within the bed. The material in large from entering the crusher feed, which contact with screen cloth either falls through reduces unnecessary power consumption and a slot or blinds the slot or contacts the cloth wear on the machine. material and is thrown from the cloth to fall to the next lower level. Laboratory Screening Equipment Centrifugal Screens: Material is fed into the Test Sieves: Test sieves are the fundamental feed inlet and redirected into the cylindrical equipment in sieve analysis. They are typically sifting chamber by means of a feed screw. made of wire mesh stretched over a frame. The Rotating, helical paddles within the chamber mesh size varies, allowing for the separation of continuously propel the material against the particles based on their ability to pass through screen, while the resultant, centrifugal force on the openings. the particles accelerates them through the apertures. These rotating paddles, which never Sieve Shaker: A sieve shaker is a mechanical make contact with the screen, also serve to device used to agitate the sieves during the break up soft agglomerates. analysis process. This shaking action helps in the uniform distribution of the sample across Air classifiers: These are used to separate the sieve and facilitates the passage of particles materials based on their density. The machine through the mesh. Sieve shakers are designed uses air flow to separate the material, with the to operate simply and can handle samples lighter particles being carried away by the air rapidly, making them efficient for particle size while the heavier particles fall to the bottom. analysis up to 50 µm.  Mesh Screens - Notation 14/20 means, through 14 mesh and on 20 mesh Mesh screens are made of a thin textile material with many small holes in it - comparable to a fishnet. It is suitable for front projections as well as rear projections. The term gauze screen is used as a synonym for mesh screen. Mesh - no. of opening per linear inch Aperture - clear space between the individual wire openings. Sieve Scales Differential Analysis Sieve scales - essential tools in particle technology 2. Cumulative Analysis - obtained from a for characterizing the particle size distribution of differential analysis by adding cumulatively. granular materials. - The cumulative analysis is a relation between the average particle size Theory of Sieve Scales retained on the screen i (Dpi) and the Operate on the principle of mechanical mass fraction of the sample that sieving consists of particles smaller than Dpi Particles are passed through a series of (Φ) sieves with progressively smaller mesh Φ = 𝑥1 + 𝑥2 +..... + 𝑥𝑛 = Σ𝑥𝑖 sizes. Plus (+) material or Oversize - particles that are retained on a sieve are larger than the mesh opening, Minus (-) material or Undersize - materials that pass through the screen, and are smaller than the mesh opening. By measuring the mass of particles retained on each sieve, particle size distribution can be determined. Particle size Cumulative Analysis In general “diameter” may be specified for Screen Effectiveness any equidimensional particles, however, most solid particles in the industry are not equidimensional, The effectiveness of a screen (screen efficiency) therefore, can not be specified by a single is a measure of the success of the screen in closely dimension. separating undersize and oversize materials. In the Equivalent spheres - where the particle size case of a perfectly functioned screen, all the is defined by the diameter of an equivalent oversize materials would be in overflow and all the sphere having the same property as the undersize materials would be in underflow. actual particle. In a sample of uniform particles of diameter Dp, the The screen effectiveness based on oversize total number of particle in the sample (N) is: material is the ratio of the amount of oversize 𝑚 material that is actually present in the overflow to 𝑁= ρ𝑝𝑣𝑝 the amount of oversize material entering with the Where: feed. m = mass of the sample 𝑣𝑝 = volume of particle ρ𝑝 = density of particle Let x = mass fraction of the mass material 𝑃𝑥𝑃 𝑅𝑒𝑐𝑜𝑣𝑒𝑟𝑦 = The total surface area of the particle is: 𝐹𝑥𝐹 6𝑚 𝐴 = 𝑁𝑆𝑝 = Φ𝑠ρ𝑝𝑣𝑝 ( ) 𝑃 1−𝑥𝑝 𝑅𝑒𝑗𝑒𝑐𝑡𝑖𝑜𝑛 = 1 − 𝑅𝑒𝑐𝑜𝑣𝑒𝑟𝑦 𝑜𝑓 𝑈𝑛𝑑𝑒𝑠𝑖𝑟𝑒𝑑 𝑀𝑎𝑡𝑒𝑟𝑖𝑎𝑙1 − ( ) 𝐹 1−𝑥𝑓 Particle Size Analysis ( ) 𝑃 1−𝑥𝑝 𝑅𝑒𝑗𝑒𝑐𝑡𝑖𝑜𝑛 = 1 − ( ) 𝐹 1−𝑥𝑓 1. Differential Analysis - particle size analysis is tabulated to show the mass 𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒𝑛𝑒𝑠𝑠 (𝐸) = (𝑅𝑒𝑐𝑜𝑣𝑒𝑟𝑦)𝑥(𝑅𝑒𝑗𝑒𝑐𝑡𝑖𝑜𝑛) fraction in each size increment as a 𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒𝑛𝑒𝑠𝑠 (𝐸) = 𝑃𝑥𝑃 ⎡1 − 𝑃(1−𝑥𝑃) ⎤ function of average particle size. 𝐹𝑥𝐹 ⎢ 𝐹(1−𝑥𝐹) ⎥ ⎣ ⎦  In terms of mass fraction: Sphericity Overall Material Balance: F = P+R Sphericity is a measure of how spherical an object Component Material Balance: is. The sphericity of a particle is defined as the ratio of the surface area of an equal-volume sphere to 𝐹𝑥𝐹 = 𝑃𝑥𝑃 + 𝑅𝑥𝑅 the actual surface area of the particle. Solving, 𝑃 (𝑥𝐹−𝑥𝑅) = 𝐹 (𝑥𝑃−𝑥𝑅) 𝑥 (𝑥 −𝑥 ) (1−𝑥 )(𝑥 −𝑥 ) 𝐸 = 𝑥𝑃 𝑥𝐹−𝑥𝑅 ⎡⎢1 − 1−𝑥𝑃 𝑥𝐹−𝑥𝑅 ⎤⎥ Where: 𝐹( 𝑃 𝑅) ⎣ ( 𝐹)( 𝑃 𝑅) ⎦ - v is the volume of one particle - D is the equivalent diameter of one particle Screen Capacity - S is the surface area of one particle The capacity of a screen refers to the mass of For Crushed Particles: material that can be fed per unit of time to a unit Sphericity = 0.6 < Φ < 0.8 area of the screen. This can be controlled by varying the feed rate to the equipment. For Abrasion Particles: Sphericity = Φ < 0.95 If a low efficiency or effectiveness may be tolerated, then the screen may be operated at Sphericities of some common solids: high capacity. Particle Sphericity In dry screening, the greater the amount of moisture or dampness in any particular Crushed coal 0.75 material, the lower the capacity of the screen. Crushed sandstone 0.8–0.9 Thus: ↑ moisture, ↓ screen capacity Round sand 0.92–0.98 Crushed glass 0.65 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑓𝑒𝑒𝑑 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑓𝑒𝑒𝑑 𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦 = = (𝑡𝑖𝑚𝑒)(𝑠𝑐𝑟𝑒𝑒𝑛 𝑎𝑟𝑒𝑎) (𝑡𝑖𝑚𝑒)(𝑚𝑚 𝑠𝑐𝑟𝑒𝑒𝑛 𝑎𝑝𝑒𝑟𝑡𝑢𝑟𝑒)(𝑎𝑟𝑒𝑎) Mica flakes 0.28 Sillimanite 0.75 The ratio of the open area of the screen to the total area is an important factor in determining Common salt 0.84 the screen capacity. Approximate capacity of screens for dense Crusher, Mills materials such as ores There are three stages of size reduction: Coarse, Type of screen Capacity range, Intermediate, and Fine. (tons/ft2/mm/24 h) Grizzlies 1-6 Coarse: - can be classified as soft or hard. Stationary screens 1-5 - 2.96 inches or more. Vibrating screens 5-20 - Compression Shaking and oscillating 2-8 - Ex: Jaw Crusher, Gyratory Crusher, screens Hammer Mill Trommels 0.3-2 Intermediate: - 1-3 inches SIZE REDUCTION - Ex: Cone Crusher - Crushing Roll Particle size reduction is the process in which Fine: large, solid particles are broken down into - 0.25 to 0.5 inches or less smaller particle sizes. This process is often - Impact achieved by the use of high shear forces. - Ex: Ball Mill, Pebble Mill, Rod Mill Particle size reduction is a key process used to transform bulky or randomly sized substances Very Fine/Ultrafine into uniform, small particles suitable for a wide - Attrition range of applications. When done correctly, particle size reduction favorably alters certain Shape key physicochemical properties of the raw - Cutting material Taggarts Rule: 𝑇 = 0. 65𝐿𝑆 Where: 𝐿1 E = Kkfcln 𝐿2 T = Capacity (tons/hr) L = Length of Feed Opening (in) Where: S = Greatest Discharge Opening (in) - Kk is the Kick’s constant Note: - fc is the crushing strength of the material 𝑇 < 0. 115, 𝐽𝑎𝑤 𝐶𝑟𝑢𝑠ℎ𝑒𝑟 - L1 is the initial particle size 2 𝑆 - L2 is the final particle size All crushers and mills are available in Perry's Handbook (SECTION 21) Bond’s Law Bond’s Law states that the work required to form particles from very large feed is proportional to the square root of the surface-to-volume ratio. 𝑃 1 1 𝑇 = 1. 46𝐸𝑖( − ) 𝑥2 𝑥1 Where: P = Power (hp) Ei = work index (kW-h/ton) T = Flow rate (tons/min) x1, x2 = size (ft) 𝑃 1 1 𝑇 = 0. 3162𝐸𝑖( − ) 𝑥2 𝑥1 Where: P = Power (kW) Ei = work index (kW-h/ton) T = Flow rate (tons/hr) x1, x2 = size (mm) Note: Bond’s Law Formula can also be seen in Perry’s HB page 21-69 Rittinger’s Law Rittinger’s Law states that the work required in crushing is proportional to the new surface created. 𝑃 𝐴𝑤𝑏−𝐴𝑤𝑎 𝑇 = 𝑁 Where: P = theoretical power T = flow rate of the feed 𝐴𝑤𝑎 = 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝐴𝑤𝑎 = 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑒𝑒𝑑 𝑁 = 𝑅𝑖𝑡𝑡𝑖𝑛𝑔𝑒𝑟'𝑠 𝑁𝑢𝑚𝑏𝑒𝑟 Kick’s Law Kick's law states that the energy needed to reduce the size of particles is directly proportional to the ratio of the initial size of a typical dimension to the final size of that dimension. This relation is based on stress analysis of plastic deformation within the elastic limit. In practice, it has been found that Kick’s law provides reasonably accurate results for coarse grinding processes, particularly when there is a relatively small increase in surface area per unit mass. Mathematically, Kick’s law is expressed as:  REVIEW QUESTIONS 8. What factor determines the capacity of the screen? 1. It is a method of separating particles according a. Feed rate to size alone b. Moisture content a. Particle technology c. Screen material b. Screening equipment d. Screen size c. Screening 9. In dry screening, what is the relationship d. Microparticles between moisture content in a material and the capacity of the screen? 2. It is used ahead of a primary crusher that can a. The higher the moisture content, the remove fine material lower the screen's capacity. a. Screening equipment b. The higher the moisture content, the higher b. Particle technology the screen's capacity. c. Microparticles c. There is no relationship between moisture d. Macroparticles content and screen capacity. d. The capacity of the screen remains 3. These are mechanically operated screens unaffected by moisture content. which are vibrated by means of an electromagnetic device. 10. It is a measure of how spherical an object is. a. Trommel a. Particle size b. Gyrating screen b. Surface area c. Vibrating screen c. Diameter d. Centrifugal screen d. Sphericity 4. What do you call the particles that are larger 11. Which among these solids have a sphericity of than the mesh opening? 0.28 a. Undersize a. Round sand b. Plus Material b. Mica flakes c. Minus Material c. Common salt d. None of the above d. Sillimanite 5. “Diameter” is the only dimension specified for 12. For the preliminary breaking of hard rock, we the measurement of all materials, may it be use equidimensional or not. a. Gyratory Crusher a. The statement is TRUE b. Tube Mill b. The statement is FALSE c. Ball Mill c. The statement does not make sense d. Squirrel-Cage Disintegrator 6. What type of screen analysis is tabulated in this 13. Feed size of >25cm can be accepted by manner? a. Jaw Crusher b. Rod Mill c. Ball Mill d. Fluid Energy Mill 14. Size reduction mechanism used in jaw crushers is a. Attrition b. Compression c. Cutting d. Impact a. Integral Analysis b. Normal Distribution Analysis 15. To get fine talc powder from its granules, the c. Differential Analysis equipment used d. Cumulative Analysis a. Roller Crusher b. Ball Mill 7. What are the factors affecting screen c. Gyratory Crusher effectiveness? d. Jaw Crusher a. The fraction of the total surface represented 16. Work required if 80% of feed passes from mesh by openings size Dpa and 80% of product of mesh size Dpb. b. The ratio of the diameter of the particle to a. Kick’s Law the width of an opening in the screen b. Rittinger’s Law c. The number of contacts between the c. Bond’s Law particle and the screen surface d. Crushing Law d. All of the above 17. What is Kick’s law? a. Amount of energy required to break the given material b. Amount of energy required to compress the given material c. Amount of energy required to crush the given material d. Amount of energy required to tear the given material 18. Kick’s law is based on which of the following? a. Cumulative analysis b. Frequency analysis c. Screen analysis d. Stress analysis  PROBLEM SOLVING Given: Xf , feed mass fraction = 0.2810 Total Number of Particles and Total Surface Xr , oversize mass fraction = 0.0660 Area: Xp , Undersize mass fraction = 0.3440 What is the total number of particles and the total ( 𝑥𝑃 𝑥𝐹−𝑥𝑅)⎡ (1−𝑥 )(𝑥 −𝑥 ) surface area of a uniform particulate material 𝐸= ⎢1 − 1−𝑥𝑃 𝑥𝐹−𝑥𝑅 ⎤⎥ sample containing cylindrical particles of diameter ( 𝑥𝐹 𝑥𝑃−𝑥𝑅)⎣ ( 𝑓)( 𝑃 𝑅) ⎦ and height 1.5 mm each and density 2650 kg/m3 if the sample mass is 50 kg? 𝐸= 0.3440(0.2810−0.0660) ⎡1 − (1−0.3440)(0.2810−0.0660) ⎤ 0.2810(0.3440−0.0660) ⎣ (1−0.2810)(0.3440−0.0660) ⎦ a. N = 1.07x107; A = 113.2075 m2 b. N = 1.06x107; A = 103.2169 m2 𝐸 = 0. 27871 𝑜𝑟 27. 87 % c. N = 1.07x10-7; A = 113.2169 m2 d. N = 1.06x10-7; A = 103.2075 m2 Screen Capacity: Given: Limestone is crushed by six units operating in D = 1.5 mm ρ𝑝 = 2650 kg/m3 parallel and the products separated by six 35-mesh screens are also in parallel, into two fractions. The h = 1.5 mm ms = 50 kg effective dimension of each screen is 6 ft by 20 ft. The common undersize from the screen comes out Solution: at the rate of 50 tons/hr. Assume no losses. Calculate the Volume of the Particle 4 3 4 1.5𝑚𝑚 1𝑚 3 𝑣𝑝 = 3 π𝑟 = 3 π( 2 * 1000𝑚𝑚 ) Mesh Feed (F) Oversize (R) Undersize (P) −9 𝑣𝑝 = 1. 7671 × 10 𝑚 6/8 0.075 0.080 0.020 Calculate the Total Number of Particles 8/10 0.125 0.145 0.055 𝑚 50𝑘𝑔 𝑁= ρ𝑝𝑣𝑝 𝑁= −9 (2650 𝑘𝑔 𝑚 3 )(1.76×10 𝑚) 10/20 0.100 0.170 0.090 7 𝑁 = 1. 07204 × 10 20/28 0.125 0.150 0.085 7 28/35 0.125 0.280 0.100 𝑁 = 1. 07204 × 10 35/48 0.175 0.175 0.150 Calculate the Surface Area of the Particle 48/65 0.225 0.150 𝑑 𝑑 2 𝑆𝑝 = 2π( 2 )ℎ + 2π( 2 ) 65/100 0.050 0.250 1.5𝑚𝑚 1.5𝑚𝑚 1.5𝑚𝑚 2 𝑆𝑝 = 2π( )( ) + 2π( ) 2×1000𝑚𝑚 1000𝑚𝑚 2×1000𝑚𝑚 100/150 0.100 −5 𝑆𝑝 = 1. 0603 × 10 𝑚 Determine the capacity of each screen in lb/24 hour Calculate the Total Surface Area of the per square foot. Uniform Particulate material 6𝑚 𝑙𝑏 𝐴 = 𝑁𝑆𝑝 = Φ𝑠ρ𝑝𝑣𝑝 a. 1439.3933 2 𝑓𝑡 24ℎ𝑟 7 −5 𝑙𝑏 𝐴 = (1. 07204 × 10 )(1. 0603 × 10 ) b. 3333.3333 2 𝑓𝑡 24ℎ𝑟 𝑙𝑏 2 c. 4739.5739 2 𝑓𝑡 24ℎ𝑟 𝐴 = 113. 2075𝑚 𝑙𝑏 d. 5757.5733 2 𝑓𝑡 24ℎ𝑟 Screen Effectiveness: GIVEN: Fine silica is fed at 1500 lbs/hr to a 6 screens/ 35 mesh double-deck vibrating screen combination to 6 ft x 20 ft obtain a 48/65 mess (tyler) product. The silica P = 50 tons/hr feed is introduced into the upper screen of the 48 mesh and the product is discharged off the 𝑙𝑏 REQUIRED: Capacity in =? surface of the lower screen of the 65 mesh. 2 𝑓𝑡 24ℎ𝑟 What is the screening efficiency at mesh 48/65 using the following data obtained? SOLUTION: 1. Solve for XF, XR, and XP: 𝑋𝐹 = 1 − (0. 125 + 0. 125 + 0. 1 + 0. 125 + 0. 075) 𝑋𝐹 = 0. 45 𝑋𝑅 = 1 − (0. 28 + 0. 15 + 0. 17 + 0. 145 + 0. 08) 𝑋𝑅 = 0. 175 𝑋𝑃 = 1 − (0. 1 + 0. 085 + 0. 09 + 0. 055 + 0. 02) 𝑋𝑃 = 0. 65 2. Solve for values of F and R: 2 rS = 2 (0.8255) 𝑂𝑀𝐵: 𝐹 = 𝑃 + 𝑅 = 1.651 cm 𝑂𝑀𝐵: 𝐹 = 50 + 𝑅 → EQ 1 Surface particle of particle: FXF = PXP + RXR 2 π(rc) (h + rc) = 2 (π) (0.5) (3 + 0.5) = 10.996 cm2 F(0.45) = 50(0.65) + R(0.175) → EQ 2 6𝑉 6 (2.356) EQ 1 TO EQ 2: Φ = 𝐷𝑆 = 1.651 (10.9966) (50 + R)(0.45) = 50(0.65) + R(0.175) R = 36.3636 tons/hr Φ = 0.779 F = 86.3636 tons/hr Bond’s Law: 3. Solve for the capacity of the screen (lb/ft2.24h): A hematite ore was ground from a feed size of F = 86.3636 tons/hr 80% passing 3.5 mesh to a product size of 𝐹 ṁ= (𝑠𝑐𝑟𝑒𝑒𝑛𝑠)(𝑎𝑟𝑒𝑎)(𝑡𝑖𝑚𝑒) 80% passing 100 mesh for flotation treatment, and the power consumption in the process was 86.3636 𝑡𝑜𝑛 ℎ𝑟 2000 𝑙𝑏 ( 1 𝑡𝑜𝑛 ) verified to be 8.77 kWhr per ton of ore ground. ṁ= (6 1 ℎ𝑟 𝑠𝑐𝑟𝑒𝑒𝑛𝑠)(6'×20')( 24 ℎ𝑟 ) Because of the changing nature of the ore with the increased depth of mining, the metallurgical 𝑙𝑏 ṁ = 5757. 5733 2 per screen recovery can only be maintained by finer 𝑓𝑡 24ℎ𝑟 grinding. Test work has indicated that by Sphericity: crushing the feed to 80% passing 5 mesh and grinding in a ball mill to 80% passing 400 Calculate the sphericity of a cylinder of diameter of mesh, recoveries will be satisfactory. Using the 1 cm and height 3 cm. Bond equation, compute the energy required to a. 0.779 grind two tons of this ore. Refer to the following b. 0.881 table for mesh aperture sizes in micrometers: c. 0.668 d. 0.992 Method 1: Volume of particle π (rc2)(h) = π (0.52) (3) = 2.356 cm3 Radius of sphere of volume 2.356 cm3: a. 58.53 kWh 3 2.356 = 4π(𝑟𝑠 ) b. 19.08 kWh 3 c. 26.92 kWh rS = 0.8255 d. 38.16 kWh 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑠𝑝ℎ𝑒𝑟𝑒 𝑜𝑓 𝑠𝑎𝑚𝑒 𝑣𝑜𝑙𝑢𝑚𝑒 𝑎𝑠 𝑡ℎ𝑒 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 Φ = 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 Solution: Calculate the Work Index, since sizes surface area of sphere of same volume as the particle: are in micrometer we can convert them 4π(rS2) = 4 (π) (0.82552) = 8.563 cm2 into mm so we can use the eq 2: 𝑃 1 1 𝑇 = 0. 3162𝐸𝑖( − ) 𝑥2 𝑥1 surface particle of particle: 1 1 2 π(rc) (h + rc) = 2 (π) (0.5) (3 + 0.5) = 10.996 cm2 8. 77 = 0. 3162𝐸𝑖( −3 − −3 ) 150×10 5600×10 8.563 Φ = 10.996 𝑘𝑊ℎ 𝐸𝑖 = 12. 8429 𝑇 Φ = 0.779 Calculate the Energy for the Second Process Method 2: 𝑃 1 1 𝑇 = 0. 3162𝐸𝑖( − ) Volume of particle 𝑥2 𝑥1 π (rc2)(h) = π (0.52) (3) = 2.356 cm3 𝑃 = 0. 3162(12. 8429)( 1 − 1 ) 𝑇 −3 −3 37×10 4000×10 Radius of sphere of volume 2.356 cm3: 𝑊 = 19. 0813 𝑘𝑊ℎ/𝑡 3 4π(𝑟𝑠 ) 2.356 = 3 Calculate the energy required to grind rS = 0.8255 two tons of the ore in question 𝐸 = 2𝑊 = 2(19. 0813 𝑘𝑊ℎ/𝑡) Diameter of particle  𝑊 = 38. 1627 𝑘𝑊ℎ REQUIRED: E2 = ? FORMULA: Rittinger’s Law 𝐿1 A material is crushed in a Blake jaw crusher E = Kkfcln 𝐿2 such that the average size of particle is SOLUTION: reduced from 50 mm to 10 mm with the (Kkfc)case 1 = (Kkfc)case 2 consumption of energy of 13.0 kW/(Kg/s). what would be the consumption of energy needed to 𝐸1 𝐿1 = 𝐸2 𝐿1 crush the same material of average size 75mm 𝑙𝑛 𝐿2 𝑙𝑛 𝐿2 to an average of 25 mm, assuming Rittinger’s 13 𝑘𝑊/(𝑘𝑔/𝑠) 𝐸2 law applies to it? 50 𝑚𝑚 = 75 𝑚𝑚 𝑙𝑛 10 𝑚𝑚 𝑙𝑛 25 𝑚𝑚 Given: P/m ,power per unit mass flow rate = 13.0 E2 = 8.8739 kW/(kg/s) kW/(kg/s) 𝐾𝑟 ,Rittinger’s constant = ? 𝐷𝑓 , final particle size = 10 mm 𝐷𝑖 , initial particle size = 50 mm E= 𝑃 𝑚 = 𝐾𝑟 ( 1 𝐷𝑓 − 1 𝐷𝑖 ) Solve for 𝐾𝑟 : 𝑃 𝑚 = 𝐾𝑟 ( 1 𝐷𝑓 − 1 𝐷𝑖 ) 13 𝑘𝑊 𝑘𝑔/𝑠 = 𝐾𝑟 ( 1 10 𝑚𝑚 − 1 50 𝑚𝑚 ) 𝑘𝑊 𝐾𝑟 = 162. 5 𝑘𝑔 𝑚𝑚 Using the obtained 𝐾𝑟, solve for the energy required consumption needed to crush the material from an average diameter of 75 mm to an average of 25 mm, assuming Rittinger’s law applies 𝐸 = 𝐾𝑟 ( 1 𝐷𝑓 − 1 𝐷𝑖 ) 𝐸 = 162. 5 𝑘𝑊 𝑘𝑔 𝑚𝑚 ( 1 25 𝑚𝑚 − 1 75 𝑚𝑚 ) 𝑘𝑊 𝐸 = 4. 33 𝑘𝑔/𝑠 Kick’s Law: A material is crushed in a Blake jaw crusher such that the average size of particle is reduced from 50 mm to 10 mm with the consumption of energy of 13.0 kW/(kg/s). What would be the consumption of energy needed to crush the same material of average size 75 mm to an average size of 25 mm assuming Kick’s law applies? e. 5.52 kW/(kg/s) f. 6.07 kW/(kg/s) g. 7.24 kW/(kg/s) h. 8.87 kW/(kg/s) GIVEN: Case 1: Case 2: E1 = 13.0 kW/(kg/s) L1 = 75 mm L1 = 50 mm L2 = 25 mm L2 = 10 mm  MOMENTUM TRANSFER applicable only to the model gas, it may be extended to real gases, liquids, and solids. Introduction to Transport Processes A simplified kinetic theory of gases postulates the Transport Phenomena is the subject that deals following model: with the movement of different physical quantities in any chemical or mechanical a. The gas is made up of molecules each of which process and describes the basic principles and is a perfect sphere of diameter 𝜎. laws of transport. It also describes the relations b. No attractive or repulsive forces exist between and similarities among different types of gas molecules. transport that may occur in any system. c. The actual volume of the molecules is negligible compared to the volume between molecules. Molecular transport of mass, heat, and d. All collisions between molecules are perfectly momentum may occur in a solid, liquid, or gas. elastic. The transport mechanism may be developed e. Each molecule is in random motion at a mean from the kinetic theory of gasses and liquids or speed, 𝑐, in a random direction. from a consideration of the physics of the solid f. Each molecule will move a distance 𝑙 between state. collisions with other molecules. The distance 𝑙 is called the mean free path. Transport in a chemical or mechanical process can g. The time required for a molecule to travel a be classified into three types: mean free path traveling at the mean speed is Heat transfer. In this fundamental process, we the mean time between collisions 𝜃. That is, 𝜃 = are concerned with the transfer of energy in the 𝑙/𝑐. form of heat from one place to another. It h. The number of molecules is large enough that occurs in the separation processes of drying, statistically average values of properties can be evaporation, distillation, and many others. used to describe all the molecules. Molecular transport of heat is called conduction. This is a highly idealized molecular picture of a gas. Mass transfer. Here, material (or mass) is Molecules of real gasses are not spherical, and transferred from one phase to another distinct there may be strong attractive or repulsive forces phase; the basic mechanism is the same among molecules. Furthermore, the molecules will whether the phases are gas, solid, or liquid. move at various speeds for various distances Separation processes dependent on mass between collisions. transfer include distillation, absorption, liquid-liquid extraction, membrane separation, A more rigorous treatment of a real gas involves adsorption, crystallization, and leaching. The complex physical and mathematical concepts. transport of mass by individual molecular Since the molecules are in random motion, they will motion is usually referred to as molecular move in all possible directions. To simplify the diffusion. situation, the derivation will consider that the molecules move in directions parallel with the Momentum transfer. This is concerned with coordinate axes 𝑥, 𝑦, and 𝑧. the transfer of momentum that occurs in moving media, such as in the separation processes of In nature, the trained observer perceives those fluid flow, sedimentation, mixing, and filtration. changes to occur in response to imbalances or Momentum transfer is commonly called fluid driving forces. For example, heat (energy in motion) mechanics in other disciplines. Molecular flows from one point to another under the influence momentum transport occurs in laminar flow. of a temperature difference. It is seen in other What is momentum transfer? examples in such diverse cases as the flow of mass, momentum, electrons, and neutrons. It can Momentum transport in a fluid depends upon be said that a flux occurs when there is a driving the transfer of the macroscopic momentum of force. molecules of the system. If a fluid is in motion, the molecules will possess a macroscopic Furthermore, the flux is related to a gradient by momentum in the direction of flow. If there is a some characteristic of the system itself—the variation in flow velocity, the faster-moving system or transport coefficient. molecules possess a greater momentum in the direction of flow and can transfer the excess momentum to their slower-moving neighbors. Principles of Fluid Mechanics (Fluid Statics) General Molecular-Transport Equation The general rate equation for molecular transport may be derived using a simple physical model of a gas. Although the resultant equation is strictly A. Force, Units, and Dimensions Another important type of pressure commonly used For a static fluid, an important property is the in chemical engineering calculations is known as pressure in the fluid. Pressure can be thought of as gage pressure. Gage (or gauge) pressure is the the surface force exerted by a fluid against the pressure relative to atmospheric pressure and thus walls of its container. Also, pressure exists at any is essentially the pressure determined from a piece point in a volume of fluid. of equipment or pressure sensor that already takes into account the atmospheric pressure of the In order to understand pressure, which is defined system. as the force exerted per unit area, Newton’s law of gravitation must be discussed. Newton’s law of gravitation is used to calculate the force exerted by C. Head of a Fluid a mass under the influence of gravity and is given by. F = mg A common method of expressing pressures is in terms of “head” in units of m or ft of a particular fluid. This height or head in meters or feet of the B. Pressure in a Fluid given fluid will exert the same pressure as the pressures it represents. Since pressure is defined as force per unit area. The force equation can be extended to calculate Using the previous equation, which relates the pressure in a fluid by taking into account the pressure 𝑃 and height ℎ of a fluid, the height or area. The forces at any given horizontal point in a “head” of the given fluid can be expressed as stationary or static fluid must be the same in all directions. Also, for a fluid at rest, the force/unit area, or pressure, is the same at all points with the same elevation. To calculate the mass of the fluid, its density (ρ) D. Devices Used to Measure Pressure and and volume (𝑉) that it occupies must be known. Pressure Differences Therefore, the mass of fluid occupying volume 𝑉 1. Simple U-tube Manometer. A U-shaped glass can be calculated by tube that is open at both ends and partially filled with liquid. The first arm is connected to a m = ρV pressurized system while the opposite arm is exposed to air pressure or an alternative reference The volume that it occupies is ℎ2𝐴. Therefore, the pressure. total mass of fluid that consists of height ℎ2 m and density 𝜌 kg/m3 can be calculated by ΔP = ρgh m = ρh2A 2. Two-fluid U-tube. It is a sensitive device for measuring very small heads or pressure This yields to the new equation of force: differences. As the name suggests it is composed of two different liquids typically with different F/A = ρgh densities, such as mercury and water or oil and water. This expression can be generalized for the pressure of the fluid at any depth ℎ in the fluid, Δp = (ρa - ρb) gh known as the hydrostatic pressure: 3. Bourdon pressure gauge. A coiled hollow tube F = ρgh in the gage tends to straighten out when subjected to internal pressure, and the degree of To calculate the total pressure on the fluid, it is straightening depends on the pressure difference necessary to take into account the atmospheric between the inside and outside pressures. The pressure or sometimes a greater external pressure tube is connected to a pointer on a calibrated dial. acting on the fluid. By taking into account the atmospheric pressure at the top of the fluid, P0, the 4. Gravity separator for two immiscible liquids. total pressure P2 can be calculated by Although gravity separators do not explicitly measure pressure, the device is used to separate P2 = ρgh + P0 liquids based on the principles of hydrostatics. A continuous gravity separator (decanter) is shown It is necessary to know the difference in pressure for the separation of two immiscible liquids, 𝐴 between two depths (vertical points) in a fluid. For (heavy liquid) and 𝐵 (light liquid). example, the pressure difference between points 2 and 1 is The feed mixture of the two liquids enters at one end of the separator vessel, and the liquids flow P2 - P1 = ρg(h2 - h1) slowly to the other end and separate into two velocity of this plate. The layer just above is at a distinct layers. Each liquid flows through a separate slightly slower velocity and each layer moves at a overflow line. Assuming the frictional resistance to slower velocity as we progress up in the 𝑦 direction. the flow of the liquids is essentially negligible, the principles of fluid statics can be used to analyze the performance. At steady state a constant force 𝐹 is needed. In this situation Since the vessel and the overflow lines are both vented to the atmosphere, a hydrostatic balance gives where 𝜇 = fluid’s viscosity Hence the 𝐹/𝐴 term is the flux of momentum (because force= 𝑑(momentum)/𝑑𝑡. If we use the Substituting ℎ𝐵 = ℎ𝑇 − ℎ𝐴1 differential form (converting F/𝐴 to a shear stress 𝜏), then we obtain This expression is known as Newton’s Law of Viscosity of Fluids (Newton’s Law of Viscosity) Viscosity. Note that the shear stress is subscripted with two letters. The reason for this is that Viscosity is the property of a fluid that gives rise to momentum transfer is not a vector (three forces that resist the relative movement of adjacent components) but rather a tensor (nine layers in the fluid. Sometimes, the property of components). viscosity is conceptualized as the fluid’s “resistance” to flow, or deformation. Classification of Fluid Behavior Consider this situation: Many fluids obey Newton’s Law of Viscosity. In these fluids, which are called Newtonian, the viscosity is a property of the system. It depends on the substance or substances in the system, temperature, and pressure but not on the velocity gradient, which is the rate of shear or on the time parameter. There are other cases where other factors influence the flow behavior and this means that there are a number of categories of fluid behavior. These are: A. Newtonian Fluids. The momentum transport equations developed before are written for fluids with a viscosity that is A liquid at rest between two plates. At a given time constant at constant temperature and independent the bottom plate moves with a velocity 𝑉. This of rate of shear and time of application of shear. causes the fluid in its vicinity to also move. After a Fluids with this property are called “Newtonian" period of time with unsteady flow we attain a linear fluids. All gasses and pure low-molecular-weight velocity profile that is associated with steady-state liquids are Newtonian. Miscible mixtures of flow. low-molecular-weight liquids are also Newtonian. B. Simple non-Newtonian. The other fluids show a decreasing apparent viscosity with increasing shear rate (shear-thinning fluid) or an increasing apparent viscosity with an increasing shear rate (shear-thickening fluid). Both A fluid is contained between two infinite (very long of these fluid types have other popular names. and very wide) parallel plates. Suppose that the Shear-thinning fluids are also called bottom plate is moving parallel to the top plate and pseudoplastic at a constant velocity relative to the top plate Shear-thickening fluids are termed because of a steady force being applied. This force dilatant. is called the viscous drag force and it arises from the viscous forces in the fluid. Each layer of liquid Bingham plastic fluids require exceeding a moves in the 𝑧 direction. The layer immediately threshold stress before flow can occur. adjacent to the bottom plate is carried along at the Rheology is the overall science that considers flow Finally, the uniform flow, the velocity profile, and and deformation of fluids (as well as solids) other properties such as pressure, is uniform across the section of pipe. This profile is often assumed in pipe and channel flow problems since it C. Complex non-Newtonian. approximates the more common turbulent flow so Those in which the time parameter becomes a well. factor are complex non-Newtonian fluids. A way of considering the behavior of these fluids is to first reflect that in a Newtonian fluid, 𝜇 is independent of rate of shear. In nature, some fluids have the characteristics of B. Viscous and Inviscid Flows both the Newtonian fluid and the elastic solid. These are called viscoelastic fluids. Note that the In an inviscid flow, the effects of viscosity can be viscoelastic fluid will behave as a simple completely neglected with no significant effects on non-Newtonian after some time (after the elastic the solution to a problem involving the flow. It is a effects have taken place). Typical viscoelastic fluids viscous flow if the viscous effects cannot be are certain polymer melts or polymer solutions. neglected. Viscous effects are very important in pipe flows and many other kinds of flows inside Two other fluid types in which time is a parameter conduits; they lead to losses and require pumps in are those that have been categorized as long pipelines. time-dependent or more specifically as thixotropic or rheopectic. The apparent viscosity increases Consider an external flow, flow external to a body, with time for the rheopectic fluid and decreases such as the flow around an airfoil or a hydrofoil. If with time for the thixotropic. the airfoil is moving relatively fast (faster than about 1 m/s), the flow away from a thin layer near the boundary can be assumed to have zero viscosity Classification of Fluid Flow with no significant effect on the solution to the flow field (the velocity, pressure, temperature fields). All Understanding the behavior of fluids in motion is the viscous effects are concentrated inside the essential for designing systems such as pipelines, boundary layer and cause the velocity to be zero at pumps, and aerodynamic structures. Fluid flow can the surface of the airfoil, the no-slip condition. be classified based on various criteria, including Since inviscid flows are easier to solve than viscous velocity, density, viscosity, and flow pattern. flows, the recognition that the viscosity can be ignored in the flow away from the surface in many A. Dimensional Flows. flows leads to much simpler solutions. The flow that depends on three space coordinates is a three-dimensional flow; it could be a steady flow if time is not involved. Certain flows can be approximated as two-dimensional flows; flows over a wide weir, in the entrance region of a pipe, and around a sphere C. Laminar and Turbulent Flow are examples that are of special interest. In such two-dimensional flows the dependent variables When a fluid is flowing through a closed channel depend on only two space variables. If the space such as a pipe or between two flat plates, one of coordinates are x and 𝑦, we refer to the flow as a two main types of fluid flow behavior will usually plane flow. occur One-dimensional flows are flows in which the At low velocities, the fluid tends to flow without velocity depends on only one space variable. For lateral mixing, and adjacent layers slide past one flow in a long pipe, the velocity depends on the another. There are no cross-currents perpendicular radius 𝑟, and in a wide channel (parallel plates) it to the direction of flow nor are there eddies, which depends on 𝑦 are swirling packets of fluid. This flow regime or type of flow behavior is called laminar flow. At higher velocities, eddies form, which leads to lateral mixing. This is called turbulent flow. The fluid’s velocity will also increase with increasing stress. The flows shown are also referred to as developed flows; the velocity profiles do not change with A simple display of 𝑉(𝑡) is not sufficient to decide if respect to the downstream coordinate. a particular flow is laminar or turbulent. To be turbulent, the motion has to be random, but it also has to have mixing of fluid particles. It is laminar near the leading edge and undergoes transition to a turbulent flow with sufficient length. There is a quantity, called the Reynolds number, that is used to determine if a flow is laminar or Continuity Equation turbulent. First studied by Osborne Reynolds, it is summarized to the equation: When a fluid is in motion, it must move in such a way that mass is conserved. To see how mass conservation places restrictions on the velocity field, consider the steady flow of fluid through a where: duct (that is, the inlet and outlet flows do not vary - 𝑉 is a characteristic velocity (the average with time). The inflow and outflow are velocity in a pipe or the speed of an airfoil), one-dimensional so that the velocity V and density - 𝐿 is a characteristic length (the diameter of a \rho are constant over the area A. pipe or the distance from the leading edge of a flat plate), - 𝜈 is the kinematic viscosity. For flow in a pipe, assuming the usually rough pipe wall, the critical Reynolds number is usually taken to be 2000 One-dimensional duct showing control volume. If the Reynolds number is larger than a critical Reynolds number, the flow is turbulent; if it is lower Now we apply the principle of mass conservation. than the critical Reynolds number, the flow is Since there is no flow through the side walls of the laminar. duct, what mass comes in over 𝐴1goes out of 𝐴2, (the flow is steady so that there is no mass For a boundary layer on a flat plate with a zero-pressure gradient, it is between 3 ×105 and accumulation). Over a short time interval ∆𝑡. 106, using the distance from the leading edge. 𝑣𝑜𝑙𝑢𝑚𝑒 𝑓𝑙𝑜𝑤 𝑖𝑛 𝑜𝑣𝑒𝑟 𝐴1 = 𝐴1𝑉1∆𝑡 In practice, the characteristic length used to 𝑣𝑜𝑙𝑢𝑚𝑒 𝑓𝑙𝑜𝑤 𝑖𝑛 𝑜𝑣𝑒𝑟 𝐴2 = 𝐴2𝑉2∆𝑡 calculate flow through circular and non-circular tubes in order to examine flow conditions is its Therefore: hydraulic diameter: 𝑚𝑎𝑠𝑠 𝑖𝑛 𝑜𝑣𝑒𝑟 𝐴 = ρ𝐴1𝑉1∆𝑡 𝑚𝑎𝑠𝑠 𝑜𝑢𝑡 𝑜𝑣𝑒𝑟 𝐴 = ρ𝐴2𝑉2∆𝑡 So: where: - 𝐴𝑐 is the cross sectional area ρ𝐴1𝑉1∆𝑡 = ρ𝐴2𝑉2∆𝑡 - 𝑝 is the wetted perimeter. This is a statement of the principle of mass We do not refer to an inviscid flow as laminar or conservation for a steady, one-dimensional flow, turbulent. In an external flow, the inviscid flow is with one inlet and one outlet. This equation is called called a free stream flow. A free stream has the continuity equation for steady one-dimensional disturbances but the disturbances are not flow. For a steady flow through a control volume accompanied by shear stresses, another with many inlets and outlets, the net mass flow requirement of both laminar and turbulent flows. must be zero, where inflows are negative and The free stream can also be irrotational or it can outflows are positive. possess vorticity. Mechanical Energy Balance A boundary layer is a thin layer of fluid that develops on a body due to the viscosity causing the The mechanical energy balance assumes fluid to stick to the boundary; it causes the velocity an incompressible fluid and steady-state to be zero at the wall. flow. An increase in fluid velocity corresponds to The viscous effects in such a layer can actually a decrease in the sum of the static burn up a satellite on reentry. pressure, potential energy and internal energy. For a fluid flowing horizontally, when the velocity increases, the pressure decreases. The Bernoulli equation is used when there are no frictional losses and no shaft work.  A Pitot tube can be used to measure fluid - Depends on the type and size of the velocity by using the Bernoulli equation. fitting or obstruction 𝑂𝐸𝐵: 𝑄 + 𝑊 = ∆𝐾𝐸 + ∆𝑃𝐸 + ∆𝐻 (𝐽/𝑘𝑔 𝑜𝑟 𝐵𝑇𝑈/𝑙𝑏) ' ∆ρ 𝑀𝐸𝐵: 𝑊 = ∆𝐾𝐸 + ∆𝑃𝐸 + ρ + Σ𝐹 B. Factors Affecting Friction Losses ' 𝑊 = 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝑤𝑜𝑟𝑘 1. Fluid Properties 2 2 𝑣2−𝑣1 ∆𝐾𝐸 = 2𝑔𝑐 a. Viscosity: Higher viscosity leads to ∆𝑃𝐸 = ∆𝑧𝑔 greater friction losses. 𝑔𝑐 b. Density: Higher density can increase friction losses, especially Frictional Losses for turbulent flow. Friction - a force that resists the relative motion 2. Flow Velocity - in laminar flow, friction loss between two surfaces in contact. is directly proportional to the flow velocity - In fluid mechanics, friction losses occur due to the interaction between the fluid and 3. Surface Roughness - Rougher surfaces the solid surfaces it flows over. increase friction losses. This effect is also - Result: Decrease in the fluid's kinetic more pronounced for turbulent flow. energy and pressure. 4. Pipe Diameter - Larger diameter pipes generally have lower friction losses. Head Loss - a measure of the reduction in the total head of the fluid as it moves through a fluid system. The head loss for fluid flow is directly proportional - a measure of the energy loss in a fluid to the length of pipe, the square of the fluid velocity, system due to friction. and a term accounting for fluid friction called the - Often expressed in terms of the equivalent friction factor. On the other hand it is inversely height of fluid column that would be required proportional to the diameter of the pipe. to overcome the frictional resistance. 𝐿𝑣 2 - Head losses can be categorized into two 𝐻𝑒𝑎𝑑 𝐿𝑜𝑠𝑠 ∝ 𝑓 𝐷 main types: Major Losses - Occur due to friction Friction Factor - a dimensionless quantity that along the length of the pipe or represents the resistance to flow in a pipe or conduit. conduit due to friction. Minor Losses - Occur due to - Determined to depend on the Reynolds fittings, valves, and other Number for the flow and the degree of obstructions in the flow path. roughness of the pipe’s inner surface (relative roughness). Frictional Loss - part of the total head loss that 𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑅𝑜𝑢𝑔ℎ𝑛𝑒𝑠𝑠 = ε 𝐷 occurs as the fluid flows through straight pipes. (p. 6-10, Perry’s HB) A. Types of Friction Losses Where: ε = average height of surface irregularities 1. Skin Friction D = pipe diameter - Occurs at the fluid-solid interface due to the viscosity of the fluid. Factors affecting Friction Factor - Shear stress at the wall causes a loss of momentum 1. Reynolds number - The friction factor is - Depends on the fluid properties, flow influenced by the Reynolds number, which velocity, and surface roughness. characterizes the flow regime. a. For laminar flow (Re < 2000), the 2. Form Drag friction factor is directly proportional - Occurs due to the shape of the to the Reynolds number. object immersed in the fluid b. For turbulent flow (Re > 4000), the - The fluid creates a wake behind the friction factor is primarily influenced object, resulting in a pressure by surface roughness and to a difference and a net force acting on lesser extent by Reynolds number. the object. 2. Relative Roughness - the ratio of the - Depends on the object's shape, size, average surface roughness to the pipe and orientation relative to the flow diameter. - Rougher surfaces have higher friction 3. Minor Losses factors, especially at higher Reynolds - Occurs due to fittings, valves, and numbers. other obstructions in the flow path. - Are relatively small compared to skin Friction Factor and Reynolds Number friction and form drag  For a Newtonian fluid in a smooth pipe, Nikuradse experiments on fluid flow in dimensional analysis relates the frictional smooth and rough pipes showed that the pressure drop per unit length DP/L to the characteristics of the friction factor were different pipe diameter D, density r, viscosity m, and for laminar and turbulent flow. average velocity V through Fanning friction Laminar Flow [Re ≤ 2100] factor (f) and Reynolds number (Re). (Hagen-Poiseuille equation) 𝐷∆𝑃 16 𝑓= 2 (p. 6-10 eq. 6-31, Perry’s HB) 𝑓= 𝑅𝑒 (p. 6-11 eq. 6-34, Perry’s HB) 2ρ𝑉 𝐿 𝐷𝑉ρ 𝑅𝑒 = µ (p. 6-10 eq. 6-32, Perry’s HB) Turbulent Flow in Smooth Tubes [4000 < In Rough pipe, the relative roughness ε/D Re < 100000] (Blasius equation) also affects the friction factor. A chart plots f 𝑓= 0.079 (p. 6-11 eq. 6-35, Perry’s HB) 0.25 𝑅𝑒 as a function of Re and ε/D. Moody Diagram Darcy-Weisbach Equation The value of the friction factor is usually obtained In the transition region where the friction from the Moody Chart. The Moody diagram is a factor depends on both Reynolds number and the graphical representation of the friction factor as a relative roughness (ε/D), the friction factor of the function of Reynolds number and relative commercial pipe is found to be different from those roughness. obtained from the sand roughness used by Nikuradse There are two common friction factors used in momentum transfer problems, the Turbulent Flow for rough pipes [Re > Darcy-Weisbach factor and Fanning friction 4000] (Colebrook Formula) 1 ε 1.256 factor. =− 4𝑙𝑜𝑔( 3.7𝐷 + ) 𝑓 𝑅𝑒 𝑓 𝑓𝐷 = 4𝑓𝑓 (p. 6-11 eq. 6-37, Perry’s HB) Usually the Moody diagram plots Darcy friction The frictional head loss can be calculated factor, however, it can also plot Fanning friction using a mathematical relationship known as the factor. It depends on the formula indicated in the Darcy’s equation for head loss. graph. 16 Friction Losses associated with pipe length If the formula for laminar flow is 𝑓 = 𝑅𝑒 , it 2 𝐿 𝑣 𝐻𝑓 = 𝑓 is the Fanning friction factor (ff). 𝐷 2𝑔 If the formula for laminar flow is 𝑓 = 64 , it Where: 𝑅𝑒 f = Darcy Friction Factor (unitless) is the Darcy-Weisbach factor (fD) L = length of pipe (ft) D = diameter of pipe (ft) v = fluid velocity (ft/sec) g = gravitational acceleration (ft/s2) Minor Losses of individual fluid system components 2 𝑣 𝐻𝑓 = 𝑘 2𝑔 Where: k = minor loss coefficient v = fluid velocity (ft/sec) g = gravitational acceleration (ft/s2) Equivalent Piping Length. Minor losses may be Moody Diagram (p. 6-10 Fig. 6-9, Perry’s HB) expressed in terms of the equivalent length (Leq) of pipe. This relationship can be found by setting the - Values of ε or surface roughness for various two forms of Darcy's equation equal to each other. 2 2 𝐿 𝑣 𝑣 materials are given in p. 6-11 Table 6-2, 𝑓 =𝑘 𝐷 2𝑔 2𝑔 Perry’s HB Yielding useful equations such as: 𝐷 𝐿𝑒𝑞 𝐿𝑒𝑞 = 𝑘 𝑓 and 𝑘=𝑓 𝐷 Category of Minor Losses 1. Due to Contraction of Pipe 1.1. Sudden Contraction - usually causes a marked drop in pressure in the pipe because of the increase velocity and loss of Fanning Friction Factor energy of turbulence.  2 2 𝑣2 𝑣 𝐻𝑚 = 𝐾𝑐 𝐻𝑚 = 𝐾𝑓 2𝑔 2𝑔 Where: Kf values can be found in p. 6-17 Table 6-5, Kc = sudden contraction coefficient Perry’s HB v2 = fluid velocity in smaller pipe For sudden contraction, the loss 6. Due to Bends and Elbows - dependent on coefficient based of v2 is given for the ratio of the radius of curvature bend (R) turbulent flow approximately by to the pipe diameter (D). 2 𝐴2 𝑣 𝐾𝑐 = 0. 5(1 − ) 𝐻𝑚 = 𝐾𝑏 2𝑔 𝐴1 For smooth pipe bend of 90°, Kb values of (p. 6-16 eq. 6-90, Perry’s HB) various R/D are as follows: 1.2. Gradual Contraction - Head loss from pipe contraction may be greatly reduced by introducing a gradual pipe transition. 2 The friction factor directly affects the head 𝑣2 𝐻𝑚 = 𝐾𝑐' 2𝑔 losses in a pipe or conduit. A higher friction factor results in greater head losses for a given flow rate Where: and pipe length. Therefore, it is essential to Kc’ = gradual contraction coefficient accurately estimate the friction factor to design and v2 = fluid velocity in smaller pipe operate fluid systems efficiently. 2. Entrance Loss - General equation for an Non-Circular conduit entrance head loss is also expressed in terms of velocity head of pipe: 𝑣 2 Non-circular conduits, such as rectangular 𝐻𝑚 = 𝐾𝑒 2𝑔 channels, trapezoidal channels, and elliptical ducts, Approximate values for the entrance loss are commonly encountered in various engineering coefficient for different entrance condition applications, including hydraulic structures, HVAC are given below: systems, and industrial processes. Calculation of frictional pressure drop in non-circular channels depends on whether the flow is laminar or turbulent and whether the channel is full or open. For turbulent flow in ducts running full, D is substituted by Hydraulic Diameter (DH) 4𝐴 3. Due to Enlargement of Pipe 𝐷𝐻 = 𝑃 3.1. Sudden Enlargement (p. 6-12, Perry’s HB) Where: A = Cross-sectional area P = wetted-perimeter 2 (𝑉1−𝑉2) 𝐻𝑚 = 2𝑔 Pumps 3.2. Gradual Enlargement Pumps - a pump is a device that moves fluids by mechanical action, from one place to the other. The shaduf is the first device used for lifting water in several civilizations and thus the earliest form of pump. It is, essentially, the earliest form of (𝑉1−𝑉2) 2 machine, dating back to ancient Egypt. 𝐻𝑚 = 𝐾𝑒' 2𝑔 Pumps are divided into 2 major categories: Ke’ values vary with the diffuser angle Dynamic and Positive Displacement (aka Displacement). The following are some of the pumps under both 4. Exit Loss 2 categories (it is impossible to list all): 𝑣 𝐻𝑚 = 𝐾𝑑 2𝑔 where Kd = 1.0 5. Due to Pipe Fittings - Fittings are installed A. Dynamic Pumps in pipelines to control flow. As with other - maintain a steady flow of the fluid losses in pipes, head loss through fittings - use direct mechanical methods – typically a may also be expressed in terms of velocity propeller or impeller – to move liquid head in pipe: - Examples:  - These pumps are suitable for transferring 1. Centrifugal pumps stormwater, groundwater, sewage, - the most commonly used in the world blackwater, greywater, rainwater, trade - robust, efficient and fairly inexpensive to waste, chemicals, bore water and food. manufacture -a mechanical device designed to move a 5. Fire hydrant systems fluid by means of the transfer of rotational - are also known as fire hydrant boosters, energy from one or more driven rotors, fire pumps and fire pumps. called impellers. Fluid enters the rapidly - These are h