2) Calculate the average velocity of the products in the tube using the equation: Where V...

    

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2) Calculate the average velocity of the products in the tube using the equation: Where V = velocity (m/s), F= volumetric flow rate (m³/s) A = cross-sectional area (D²/4m²) F V = A 3) Calculate the Reynolds number (NRe) for each product, using the equation: Dn. V(2-n)p NRe n K.8n-1 3n+1) 4n Where D = diameter (m), n = flow behaviour index (dimensionless) V velocity (m/s) p=density (kg/m³) K = consistency index (Pa.s) You will need the following data: milk, P = 1012 kg/m³ at 4°C, 981.1 kg/m³ at 80°C, K = 0.00136 Pa.s, n = 1 ice cream mix, p = 1100 kg/m³ at 4°C, 1071.8 kg/m³ at 80°C, K = 0.135 Pa.s, n = 0.7 cream (40%), p = 982.6 kg/m³ at 4°C, 931.8 kg/m³ at 80°C, K = 0.00464 Pa.s, n = 1 4) Determine the ratio of average to maximum velocity (efficiency factor) from the attached graph 5) From the calculated velocity of the fastest moving particle (Q2) and the length of the tube (25.6 meters), determine the minimum holding time in seconds for each product. 6) Calculate the apparent viscosity of each product using the following equation and considering a typical shear rate for pipe flow of 300 s¹ Where n = apparent viscosity (mPa.s) K = consistency index (mPa.s) y = shear rate (s1) N= flow behaviour index (dimensionless) n = K.yn-1 Example Holding Time Calculation for Fluid milk (using different specifications than above) Flow rate = 112 s for 50 lbs., length of tube = 5.2 m 50 lbs water/112s x kg/2.205 lbs x m³/971.8 kg = .000208 m³/s For milk, correction factor = 1.05, so vol. flow rate = .000208x1.05=.000219 m³/s Cross-sectional Area = лD²/4 = π x .034852/4 = .000954 m² Milk Velocity = .000219 m³/s /.000954 m² = 0.229 m/s Nre .03485 m x .229 m/s x 981.1 kg/m³ = 5757 .00136 Pa s (Note: 1 Pa = 1 N/m², and 1 N = 1 kg m/s²) V/Vmax = .75 Max. velocity = .229 m/s/.75 = .305m/s Min. holding time = 5.2m/.305m/s = 17.05s FIGURE 1 EFFICIENCY VALUES RATIO OF AVERAGE TO MAXIMUM VELOCITY (V/Vmax) 0.9 0.8 0.7 0.6 90 0.5 T T 102 LAMINAR FLOW TRANSITIONAL FLOW TURBULENT FLOW 103 104 105 REYNOLDS NO. LOG CHART 106 Fig. 1 Effect of Reynolds number on the ratio of average to maximum velocity in smooth tubes DETERMINATION OF MAXIMUM VELOCITY 107 2) Calculate the average velocity of the products in the tube using the equation: Where V = velocity (m/s), F= volumetric flow rate (m³/s) A = cross-sectional area (D²/4m²) F V = A 3) Calculate the Reynolds number (NRe) for each product, using the equation: Dn. V(2-n)p NRe n K.8n-1 3n+1) 4n Where D = diameter (m), n = flow behaviour index (dimensionless) V velocity (m/s) p=density (kg/m³) K = consistency index (Pa.s) You will need the following data: milk, P = 1012 kg/m³ at 4°C, 981.1 kg/m³ at 80°C, K = 0.00136 Pa.s, n = 1 ice cream mix, p = 1100 kg/m³ at 4°C, 1071.8 kg/m³ at 80°C, K = 0.135 Pa.s, n = 0.7 cream (40%), p = 982.6 kg/m³ at 4°C, 931.8 kg/m³ at 80°C, K = 0.00464 Pa.s, n = 1 4) Determine the ratio of average to maximum velocity (efficiency factor) from the attached graph 5) From the calculated velocity of the fastest moving particle (Q2) and the length of the tube (25.6 meters), determine the minimum holding time in seconds for each product. 6) Calculate the apparent viscosity of each product using the following equation and considering a typical shear rate for pipe flow of 300 s¹ Where n = apparent viscosity (mPa.s) K = consistency index (mPa.s) y = shear rate (s1) N= flow behaviour index (dimensionless) n = K.yn-1 Example Holding Time Calculation for Fluid milk (using different specifications than above) Flow rate = 112 s for 50 lbs., length of tube = 5.2 m 50 lbs water/112s x kg/2.205 lbs x m³/971.8 kg = .000208 m³/s For milk, correction factor = 1.05, so vol. flow rate = .000208x1.05=.000219 m³/s Cross-sectional Area = лD²/4 = π x .034852/4 = .000954 m² Milk Velocity = .000219 m³/s /.000954 m² = 0.229 m/s Nre .03485 m x .229 m/s x 981.1 kg/m³ = 5757 .00136 Pa s (Note: 1 Pa = 1 N/m², and 1 N = 1 kg m/s²) V/Vmax = .75 Max. velocity = .229 m/s/.75 = .305m/s Min. holding time = 5.2m/.305m/s = 17.05s FIGURE 1 EFFICIENCY VALUES RATIO OF AVERAGE TO MAXIMUM VELOCITY (V/Vmax) 0.9 0.8 0.7 0.6 90 0.5 T T 102 LAMINAR FLOW TRANSITIONAL FLOW TURBULENT FLOW 103 104 105 REYNOLDS NO. LOG CHART 106 Fig. 1 Effect of Reynolds number on the ratio of average to maximum velocity in smooth tubes DETERMINATION OF MAXIMUM VELOCITY 107