Common mechanical operation
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This document provides equations and definitions for calculating properties related to particle size and shape, including: porosity, specific surface area, average diameter, and median diameter. It then works through examples of estimating shape factors for different particle shapes and calculating specific surface area, average diameter, and other properties from a screen analysis dataset. The key steps shown are: 1) estimating shape factors for different particle geometries, 2) calculating specific surface area from given properties, and 3) determining average diameter and other metrics from a particle size distribution.
Common mechanical operation
- 1. Common Mechanical Operation Section 1 Eng. Kareem H. Mokhtar
- 2. Rules • ℰ= 𝑉 𝑝𝑜𝑟𝑒𝑠 𝑉 𝑏𝑢𝑙𝑘 = 1- 𝑉 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑉 𝑏𝑢𝑙𝑘 • Sp. Area per unit volume = 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑣𝑜𝑙𝑢𝑚𝑒 = 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝐷𝑝 • Sp. Surface area = 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝜌 𝑋𝑖 𝐷 𝑎𝑣 • Mean diameter = 𝐷 𝑝 dφ = 𝑥𝑖 𝐷𝑝 𝑎𝑣 • Average surface volume particle diameter --> 1 𝐷𝑣𝑠 = 𝑋𝑖 𝐷𝑝 𝑎𝑣 • D median it is the diameter at φc = 0.5
- 3. Sheet 1 • 1) Estimate the shape factor of particles of the given shapes: • a) • ℰ= 𝑉 𝑝𝑜𝑟𝑒𝑠 𝑉 𝑏𝑢𝑙𝑘 = 1- 𝑉 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑉 𝑏𝑢𝑙𝑘 • Sp. Area per unit volume = 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑣𝑜𝑙𝑢𝑚𝑒 = 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝐷𝑝 • Sp. Surface area = 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝜌 𝑋𝑖 𝐷 𝑎𝑣 • Mean diameter = 𝐷 𝑝 dφ = 𝑥𝑖 𝐷𝑝 𝑎𝑣 • Average surface volume particle diameter 1 𝐷𝑣𝑠 = 𝑋𝑖 𝐷𝑝 𝑎𝑣 • D median it is the diameter at φc = 0.5
- 4. Sheet 1 Area = 2*4x*5x+2*3x*5x+2*3x*4x = 94x2 Volume = 3*4*5 = 60 x3 S= 94 x2 60x3 = 1.6 𝑥 = 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝐷𝑝 Dp = 4x (second largest ) Shape factor = 1.044 • ℰ= 𝑉 𝑝𝑜𝑟𝑒𝑠 𝑉 𝑏𝑢𝑙𝑘 = 1- 𝑉 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑉 𝑏𝑢𝑙𝑘 • Sp. Area per unit volume = 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑣𝑜𝑙𝑢𝑚𝑒 = 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝐷𝑝 • Sp. Surface area = 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝜌 𝑋𝑖 𝐷 𝑎𝑣 • Mean diameter = 𝐷 𝑝 dφ = 𝑥𝑖 𝐷𝑝 𝑎𝑣 • Average surface volume particle diameter 1 𝐷𝑣𝑠 = 𝑋𝑖 𝐷𝑝 𝑎𝑣 • D median it is the diameter at φc = 0.5
- 5. Sheet 1 • 1) Estimate the shape factor of particles of the given shapes: • b) • ℰ= 𝑉 𝑝𝑜𝑟𝑒𝑠 𝑉 𝑏𝑢𝑙𝑘 = 1- 𝑉 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑉 𝑏𝑢𝑙𝑘 • Sp. Area per unit volume = 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑣𝑜𝑙𝑢𝑚𝑒 = 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝐷𝑝 • Sp. Surface area= 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝜌 𝑋𝑖 𝐷 𝑎𝑣 • Mean diameter = 𝐷 𝑝 dφ = 𝑥𝑖 𝐷𝑝 𝑎𝑣 • Average surface volume particle diameter 1 𝐷𝑣𝑠 = 𝑋𝑖 𝐷𝑝 𝑎𝑣 • D median it is the diameter at φc = 0.5
- 6. Sheet 1 • Area= 7.5*pi * x2 Volume = 1.5 * pi * x3 Shape factor = • 1.667 • ℰ= 𝑉 𝑝𝑜𝑟𝑒𝑠 𝑉 𝑏𝑢𝑙𝑘 = 1- 𝑉 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑉 𝑏𝑢𝑙𝑘 • Sp. Area per unit volume = 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑣𝑜𝑙𝑢𝑚𝑒 = 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝐷𝑝 • Sp. Surface area= 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝜌 𝑋𝑖 𝐷 𝑎𝑣 • Mean diameter = 𝐷 𝑝 dφ = 𝑥𝑖 𝐷𝑝 𝑎𝑣 • Average surface volume particle diameter 1 𝐷𝑣𝑠 = 𝑋𝑖 𝐷𝑝 𝑎𝑣 • D median it is the diameter at φc = 0.5
- 7. Sheet 1 • 1) Estimate the shape factor of particles of the given shapes: • c) • ℰ= 𝑉 𝑝𝑜𝑟𝑒𝑠 𝑉 𝑏𝑢𝑙𝑘 = 1- 𝑉 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑉 𝑏𝑢𝑙𝑘 • Sp. Area per unit volume = 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑣𝑜𝑙𝑢𝑚𝑒 = 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝐷𝑝 • Sp. Surface area= 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝜌 𝑋𝑖 𝐷 𝑎𝑣 • Mean diameter = 𝐷 𝑝 dφ = 𝑥𝑖 𝐷𝑝 𝑎𝑣 • Average surface volume particle diameter 1 𝐷𝑣𝑠 = 𝑋𝑖 𝐷𝑝 𝑎𝑣 • D median it is the diameter at φc = 0.5 Shape factor = 1.5
- 8. Sheet 1 • Given the following screen analysis, calculate: • (a) The specific surface area. • (b) The average particle diameter. • (c) The average surface volume particle diameter. • (d) The median diameter of particles. • Specific gravity = 1.8, sphericity = 0.67 • ℰ= 𝑉 𝑝𝑜𝑟𝑒𝑠 𝑉 𝑏𝑢𝑙𝑘 = 1- 𝑉 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑉 𝑏𝑢𝑙𝑘 • Sp. Area per unit volume = 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑣𝑜𝑙𝑢𝑚𝑒 = 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝐷𝑝 • Sp. Surface area = 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝜌 𝑋𝑖 𝐷 𝑎𝑣 • Mean diameter = 𝐷 𝑝 dφ = 𝑥𝑖 𝐷𝑝 𝑎𝑣 • Average surface volume particle diameter 1 𝐷𝑣𝑠 = 𝑋𝑖 𝐷𝑝 𝑎𝑣 • D median it is the diameter at φc = 0.5
- 9. Sheet 1 • Specific gravity = 1.8 • Sphericity = 0.67 • ℰ= 𝑉 𝑝𝑜𝑟𝑒𝑠 𝑉 𝑏𝑢𝑙𝑘 = 1- 𝑉 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑉 𝑏𝑢𝑙𝑘 • Sp. Area per unit volume = 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑣𝑜𝑙𝑢𝑚𝑒 = 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝐷𝑝 • Sp. Surface area = 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝜌 𝑋𝑖 𝐷 𝑎𝑣 • Mean diameter = 𝐷 𝑝 dφ = 𝑥𝑖 𝐷𝑝 𝑎𝑣 • Average surface volume particle diameter 1 𝐷𝑣𝑠 = 𝑋𝑖 𝐷𝑝 𝑎𝑣 • D median it is the diameter at φc = 0.5 D mm xi 1.651 0 1.168 0.01 0.833 0.1 0.589 0.15 0.417 0.22 0.295 0.3 0.208 0.14 0.147 0.06 0.074 0.02 pan 0
- 10. Sheet 1 • Specific gravity = 1.8, sphericity = 0.67 • ℰ= 𝑉 𝑝𝑜𝑟𝑒𝑠 𝑉 𝑏𝑢𝑙𝑘 = 1- 𝑉 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑉 𝑏𝑢𝑙𝑘 • Sp. Area per unit volume = 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑣𝑜𝑙𝑢𝑚𝑒 = 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝐷𝑝 • Sp. Surface area= 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝜌 𝑋𝑖 𝐷 𝑎𝑣 • Mean diameter = 𝐷 𝑝 dφ = 𝑥𝑖 𝐷𝑝 𝑎𝑣 • Average surface volume particle diameter 1 𝐷𝑣𝑠 = 𝑋𝑖 𝐷𝑝 𝑎𝑣 • D median it is the diameter at φc = 0.5 D mm xi 1.651 0 1.168 0.01 0.833 0.1 0.589 0.15 0.417 0.22 0.295 0.3 0.208 0.14 0.147 0.06 0.074 0.02 pan 0 Dav - 1.4095 1.0005 0.711 0.503 0.356 0.2515 0.1775 0.1105 - xi/Dv 0.01 0.10 0.21 0.44 0.84 0.56 0.34 0.18 2.67 xi * Dav 0.01 0.10 0.11 0.11 0.11 0.04 0.01 0.00 0.49
- 11. Sheet 1 • Specific gravity = 1.8, sphericity = 0.67 • (a) The specific surface area. • Sp. Surface area= 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝐷𝑝 𝑋𝑖 𝐷 𝑎𝑣 = 6 ∗ 1 0.67 1800 * 2.674 mm *( 1000 𝑚𝑚 1 𝑚 ) =13.3 m2/kg D mm xi 1.651 0 1.168 0.01 0.833 0.1 0.589 0.15 0.417 0.22 0.295 0.3 0.208 0.14 0.147 0.06 0.074 0.02 pan 0 Dav - 1.4095 1.0005 0.711 0.503 0.356 0.2515 0.1775 0.1105 - xi/Dv 0.01 0.10 0.21 0.44 0.84 0.56 0.34 0.18 2.674 xi * Dav 0.01 0.10 0.11 0.11 0.11 0.04 0.01 0.00 0.496
- 12. Sheet 1 • Given the following screen analysis, calculate: • (a) The specific surface area. • (b) The average particle diameter. • (c) The average surface volume particle diameter. • (d) The median diameter of particles. • Specific gravity = 1.8, sphericity = 0.67 • ℰ= 𝑉 𝑝𝑜𝑟𝑒𝑠 𝑉 𝑏𝑢𝑙𝑘 = 1- 𝑉 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑉 𝑏𝑢𝑙𝑘 • Sp. Area per unit volume = 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑣𝑜𝑙𝑢𝑚𝑒 = 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝐷𝑝 • Sp. Surface area = 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝜌 𝑋𝑖 𝐷 𝑎𝑣 • Mean diameter = 𝐷 𝑝 dφ = 𝑥𝑖 𝐷𝑝 𝑎𝑣 • Average surface volume particle diameter 1 𝐷𝑣𝑠 = 𝑋𝑖 𝐷𝑝 𝑎𝑣 • D median it is the diameter at φc = 0.5
- 13. Sheet 1 • Specific gravity = 1.8, sphericity = 0.67 • (b) The average particle diameter. • Mean diameter = 𝐷 𝑝 dφ = 𝑥𝑖 𝐷𝑝𝑎 𝑣 = 0.496 D mm xi 1.651 0 1.168 0.01 0.833 0.1 0.589 0.15 0.417 0.22 0.295 0.3 0.208 0.14 0.147 0.06 0.074 0.02 pan 0 Dav - 1.4095 1.0005 0.711 0.503 0.356 0.2515 0.1775 0.1105 - xi/Dv 0.01 0.10 0.21 0.44 0.84 0.56 0.34 0.18 2.674 xi * Dav 0.01 0.10 0.11 0.11 0.11 0.04 0.01 0.00 0.496
- 14. Sheet 1 • Given the following screen analysis, calculate: • (a) The specific surface area. • (b) The average particle diameter. • (c) The average surface volume particle diameter. • (d) The median diameter of particles. • Specific gravity = 1.8, sphericity = 0.67 • ℰ= 𝑉 𝑝𝑜𝑟𝑒𝑠 𝑉 𝑏𝑢𝑙𝑘 = 1- 𝑉 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑉 𝑏𝑢𝑙𝑘 • Sp. Area per unit volume = 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑣𝑜𝑙𝑢𝑚𝑒 = 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝐷𝑝 • Sp. Surface area = 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝜌 𝑋𝑖 𝐷 𝑎𝑣 • Mean diameter = 𝐷 𝑝 dφ = 𝑥𝑖 𝐷𝑝 𝑎𝑣 • Average surface volume particle diameter 1 𝐷𝑣𝑠 = 𝑋𝑖 𝐷𝑝 𝑎𝑣 • D median it is the diameter at φc = 0.5
- 15. Sheet 1 • Specific gravity = 1.8, sphericity = 0.67 • (c) The average surface volume particle diameter. • Average surface volume particle diameter 1 𝐷𝑣𝑠 = 𝑋𝑖 𝐷𝑝 𝑎𝑣 • Dvs = 1 2.674 = 0.374 𝑚𝑚 D mm xi 1.651 0 1.168 0.01 0.833 0.1 0.589 0.15 0.417 0.22 0.295 0.3 0.208 0.14 0.147 0.06 0.074 0.02 pan 0 Dav - 1.4095 1.0005 0.711 0.503 0.356 0.2515 0.1775 0.1105 - xi/Dv 0.01 0.10 0.21 0.44 0.84 0.56 0.34 0.18 2.674 xi * Dav 0.01 0.10 0.11 0.11 0.11 0.04 0.01 0.00 0.496
- 16. Sheet 1 • Given the following screen analysis, calculate: • (a) The specific surface area. • (b) The average particle diameter. • (c) The average surface volume particle diameter. • (d) The median diameter of particles. • Specific gravity = 1.8, sphericity = 0.67 • ℰ= 𝑉 𝑝𝑜𝑟𝑒𝑠 𝑉 𝑏𝑢𝑙𝑘 = 1- 𝑉 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑉 𝑏𝑢𝑙𝑘 • Sp. Area per unit volume = 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑣𝑜𝑙𝑢𝑚𝑒 = 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝐷𝑝 • Sp. Surface area = 6 (𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝜌 𝑋𝑖 𝐷 𝑎𝑣 • Mean diameter = 𝐷 𝑝 dφ = 𝑥𝑖 𝐷𝑝 𝑎𝑣 • Average surface volume particle diameter 1 𝐷𝑣𝑠 = 𝑋𝑖 𝐷𝑝 𝑎𝑣 • D median it is the diameter at φc = 0.5
- 17. Sheet 1 • Specific gravity = 1.8, sphericity = 0.67 D mm xi 1.651 0 1.168 0.01 0.833 0.1 0.589 0.15 0.417 0.22 0.295 0.3 0.208 0.14 0.147 0.06 0.074 0.02 pan 0 φ 0 0.01 0.11 0.26 0.48 0.78 0.92 0.98 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Phi D mm
- 18. Sheet 1 • Calculate the values of the constants (a) and (k) • from the following equation • 𝑑φ 𝑑 𝐷𝑝 = 𝑎 . 𝐷𝑝k • Given that: • Maximum particle size = 2mm • Minimum particle size ≈ 0 mm • Median particle size = 1.13 mm • Then, calculate the mean particle • the average surface volume diameter of particles. • Mean diameter = 𝐷 𝑝 dφ = 𝑥𝑖 𝐷𝑝 𝑎𝑣 • Average surface volume particle diameter 1 𝐷𝑣𝑠 = 𝑋𝑖 𝐷𝑝 𝑎𝑣 • D median it is the diameter at φc = 0.5