Conveyor Equipment Manufacturers Association

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1 Conveyor Equipment Manufacturers Association 567 Strand Ct., Suite, Naples, Florida Tel: (39) Fax: (39) Web Site: Belt Book, 7th Edition Errata items As of December 11, 014 Page 76 Equation lacks a Square function Pages 164/165 Tables 6.47/ Corrections to Type rubber Pages 167 and 169 Typos (9.4 o C) should read (-9.4 o C) Page 538 Figure Ø1 in the drawing should be Ø (Angle of Incline of the Belt Conveyor) Page 55 Figure Replace: 'Vs=Tangential Velocity, fps, of the cross-sectional area center of gravity of the load shape' with: 'Vs=Velocity of the load cross section used for plotting the trajectory' Page 559 Figure Greek letters are incorrect, there is a font error for phi describing the angle of incline of the conveyor. It is showing the 'capital phi' not in lower case as it should be. Page 559 Figure The following Greek letters are incorrect case the symbols for the angle of incline of the belt and the angle to the discharge point in this figure are the wrong case of Greek letters: φ should be ϕ( places) and ϒ should be γ (1place). Page 560 Between Equation & Equation In the paragraph, the reference to Table 4.4, needs to be corrected to say 'Table 4.6'. Page 560 Figure This should read: Vs = Velocity of the load cross section used for plotting the trajectory: 1 Belt velocity, V, is used as the velocity of the material at its center of mass if the discharge point is at the tangency of the belt-to-discharge pulley (Vs = V) Velocity of the material at its center of mass, Vcg, is used as the velocity of the material for all other conditions of discharge after the point of belt-to-discharge pulley tangency. (Vs= Vcg) THE VOICE OF THE NORTH AMERICAN CONVEYOR INDUSTRY

2 THE VOICE OF THE NORTH AMERICAN CONVEYOR INDUSTRY

3 4 CAPACITIES, BELT WIDTHS AND SPEEDS 100% Full, Edge To Edge, Cross Sectional Area, A f porting the conveyor system should be designed for the dead loads plus the live material load as if the belt f, d f and r schf are introduced for calculations for a we wmc w and b w is used to calculate A f and d f. The s with b we set to zero. w f d f A f ß s b w b c r schf Figure 4.30 A f, Cross sectional area dimensionless nomenclature for 100% full, edge to edge, r schf = ( 1-cos ( ) b c sin( s ) + cos ( ) Equation 4.31 r schf A f = BW r schf s - sin ( s) cos( s ) + b c b w xsin( ) +b sin( )cos w ( ) Equation 4.3 A f d f = b w sin( )+ b c +b w cos ( ) 1 sin( s ) - 1 tan( s ) Equation 4.33 d f 76

4 4 CAPACITIES, BELT WIDTHS AND SPEEDS 100% Full, Edge To Edge, Cross Sectional Area, A f Occasionally the conveyor will be loaded completely full edge to edge or 100% full. The structure supporting the conveyor system should be designed for the dead loads plus the live material load as if the belt were 100% full, rather than the design capacity assuming the material is loaded to the standard edge distance. For clarity and consistency new variables b f, d f and r schf are introduced for calculations for a 100% full belt. For a 100% full belt, b we = 0.0 so, b wmc = b w and b w is used to calculate A f and d f. The values for a 100% full belt could be obtained from the equations for A s with b we set to zero. w f d f A f ß Φ s b w b c r schf Figure 4.30 A f, Cross sectional area dimensionless nomenclature for 100% full, edge to edge, belt loading ( ) b + cos( β) c r schf = 1-cos( β) sin(φ s ) Equation 4.31 r schf, Dimensionless ratio of radius of top of bulk material profile for 100% full, edge to edge, belt loading A f = BW r schf Φ s - sin ( Φ s) cos( Φ s ) + b c b xsin( β) w +b sin( β) cos( β) w Equation 4.3 A f, Cross sectional area of 100 % full, edge to edge, belt loading d f = b w sin( β)+ b c +b cos( β) w 1 sin( Φ s ) - 1 tan( Φ s ) Equation 4.33 d f, Dimensionless ratio for maximum depth of bulk material on 100% full, edge to edge, belt loading 76

5 6 BELT TENSION AND POWER ENGINEERING P = c 1 + [c (x P )] + [c 3 (x P )] + [c 4 (x P 3 )] c 5 + [c 6 (x P )] + x P (dimensionless) x P = - C 1 ( T T 0 ) C + ( T T 0 ) Notes : For constants c i see Table 6.48 Equation 6.46 P, Rubber strain level adjustment + log( v u ) (dimensionless) For constants C 1, C and T 0 see Table 6.47 Where: T = Operating temperature ( o C) v u = Belt speed m s Note: v must be inm u s units in x equation F Temperature and belt speed are reflected in F, while P adjusts the linear viscoelastic properties to the actual loading strain which is commonly in the range of nonlinear stress strain behavior. A full set of constants representative of four types of example cover rubbers are provided in Tables 6.47 and These constants are intended to approximate the performance of commercially available classes of conveyor belt rubber covers. Constant Default Rubber Type 1 Rubber Type Rubber Type 3 Rubber T 0 ( C) E 0 (N/m ) C C a a a a b b b b b b Table 6.47 Constants for Equation 6.44, K bir-s F factor 164

6 6 BELT TENSION AND POWER ENGINEERING P = c 1 + [c (x P )] + [c 3 (x P )] + [c 4 (x P 3 )] c 5 + [c 6 (x P )] + x P (dimensionless) x P = - C 1 ( T T 0 ) C + ( T T 0 ) Notes : For constants c i see Table 6.48 Equation 6.46 P, Rubber strain level adjustment + log( v u ) (dimensionless) For constants C 1, C and T 0 see Table 6.47 Where: T = Operating temperature ( o C) v u = Belt speed m s Note: v must be inm u s units in x equation F Temperature and belt speed are reflected in F, while P adjusts the linear viscoelastic properties to the actual loading strain which is commonly in the range of nonlinear stress strain behavior. A full set of constants representative of four types of example cover rubbers are provided in Tables 6.47 and These constants are intended to approximate the performance of commercially available classes of conveyor belt rubber covers. Constant Default Rubber Type 1 Rubber Type Rubber Type 3 Rubber T 0 ( C) E 0 (N/m ) C C a a a a b b b b b b Table 6.47 Constants for Equation 6.44, K bir-s F factor 164

7 BELT TENSION AND POWER ENGINEERING 6 c i Intervals: Each c i constant is linearly interpolated for w iw, relative to the seven levels of w ref and extrapolated past the last row. Cover Compound w ref (N/m) c 1 c c 3 c 4 c 5 c 6 (dimensionless) Default Type Type Type Table 6.48 Constants for Equation 6.46, K bir-s P factor at several w ref values Type constants are for typical rubber cover compounds while Type 3 constants may also apply to cover compounds for common covers for more conservative designs that operate at lower temperatures. Type 1 constants are for a low rolling resistance rubber cover compound considered for applications where indentation losses are a significant contributor to the belt tension. These, especially, must be specified and verified by the manufacturer to apply to final designs. The default constants are intended for designs 165

8 BELT TENSION AND POWER ENGINEERING 6 c i Intervals: Each c i constant is linearly interpolated for w iw, relative to the seven levels of w ref and extrapolated past the last row. Cover Compound w ref (N/m) c 1 c c 3 c 4 c 5 c 6 (dimensionless) Default Type Type Type Table 6.48 Constants for Equation 6.46, K bir-s P factor at several w ref values Type constants are for typical rubber cover compounds while Type 3 constants may also apply to cover compounds for common covers for more conservative designs that operate at lower temperatures. Type 1 constants are for a low rolling resistance rubber cover compound considered for applications where indentation losses are a significant contributor to the belt tension. These, especially, must be specified and verified by the manufacturer to apply to final designs. The default constants are intended for designs 165

9 BELT TENSION AND POWER ENGINEERING 6 Small Sample Indentation Loss Example The following example of the small sample method for belt cover indentation loss is divided into several Belt Cover Indentation Loss Example Assumptions Design capacity: Q =,500 tph Belt width: BW = 48 in Cover thickness in contact with the rollers: h b = in Unit weight of the belt, W b = 6.3 lbf, fabric belt ft Default rubber cover compound, E 0 = 9,945,456 N lbf = 07,715 m ft Belt speed: V = 600 ft min, v = 3.05 m u s Three equal roll troughing idler angle: = 35deg. Idler roll diameter: D r = 6.0 in Idler spacing S i = 5.0 ft Bulk material surcharge angle: s = 0deg. Material bulk density: m = 90 lbf Typo - should have ft 3 been -9.4 Operating temperature: T F = 15 o F ( o C) Length of the flight: L = 500 ft Constants used in T bin calculation: w max = 85.5 lbf in, X ldref = 5. psi Figure 6.50 bi 167

10 BELT TENSION AND POWER ENGINEERING 6 K bir-s = F P c sd c bc Calculate F: F = b + b (x ) + b (x ) + b 1 F 3 F 4 (x 3 F ) b 5 + b 6 (x F ) + x F x F = C ( T T ) log( v u )- s T 0 = C and V = 600 ft C + ( T T 0 ) min or 3.05 m s s = a 1 + a (x s ) + a 3 (x s ) + a 4 (x s 3 ) Typo - should have been -9.4 v u = 3.05 w iw = D m m + W lbf lbf b BW S = 8.8 in ft 3 + ft 1 1 i 178 in3 48 in ft 3 ft in 5 ft 1 in ft lbf lbf = in = 30.4 in in x s = w iw W max 1/ = /3 = s = a 1 + a (x s ) + a 3 (x s ) + a 4 (x 3 s ) = ( 0.475) ( 0.473) ( 0.473) 3 = = = From Table 6.47 for Default Rubber: C 1 = , C = , T 0 = C b 1 = , b = , b 3 = , b 4 = , b 5 = , b 6 = a 1 = , a = , a 3 = , a 4 = ( ) x F = - C ( T-T ) 1 0 C + ( T-T 0 ) + log v o C - ( o C) ( ) - s = + log( 3.05)- s = u ( 15 o F - 3 o F) ( o C) 1.8 x F = log(3.048) = = F = b + b (x ) + b (x ) + b 1 F 3 F 4 (x 3 F ) = b 5 + b 6 (x F ) + x F = (0.386) (0.386) (0.386) (0.386) + (0.386) = F = = = Figure 6.5 bi, Example F calculation 169

11 B NOMENCLATURE Chapter Twelve Continued T s T Q φ s φ φ i V eb V fs c V y W W s y 1 y V V fs Belt Tension required to remove material from feeder due the bulk material The resistance (belt tension) to shear the bulk material for a simple feeder Surcharge angle Angle of incline of the belt conveyor Angle of internal friction of bulk solid on shear plane Vertical velocity of bulk material as it leaves the discharging chute Consolidated pressure volume The volume of the surcharge material Falling bulk material stream vertical velocity Weight of the bulk material acting at the center of gravity of the discharging load Width between skirtboards Vertical height of skirtboard at the rear of feeder hopper Vertical height of skirtboard at front of feeder hopper Belt speed Consolidated pressure volume Tangential speed of the discharging load center of gravity Chapter Thirteen Symbol Description CF N n P dn rpm rpm AC-Motor SF T accel T bn Time accel WK Conversion Constant Backstop pulley rotational speed Total power at drive location Motor shaft revolutions per minute AC motor shaft speed Back stop service factor Prime mover torque - required load torque Minimum torque rating of the backstop Acceleration time in seconds of the motor Moment of inertia of rotating parts 780

12 B NOMENCLATURE Chapter Twelve Continued T s T Q φ s φ φ i V eb V fs c V y W W s y 1 y V V fs V cg Belt Tension required to remove material from feeder due the bulk material The resistance (belt tension) to shear the bulk material for a simple feeder Surcharge angle Angle of incline of the belt conveyor Angle of internal friction of bulk solid on shear plane Vertical velocity of bulk material as it leaves the discharging chute Consolidated pressure volume The volume of the surcharge material Falling bulk material stream vertical velocity Weight of the bulk material acting at the center of gravity of the discharging load Width between skirtboards Vertical height of skirtboard at the rear of feeder hopper Vertical height of skirtboard at front of feeder hopper Belt speed Consolidated pressure volume Tangential speed of the load at the center of gravity. V cg is used when the material discharges at other than the tangency of the belt and pulley. = V cg = Velocity of the load cross section used for plotting the trajectory when the material diacharges at the tangency of the belt and pulley. = V (belt speed) Chapter Thirteen Symbol Description CF N n P dn rpm rpm AC-Motor SF T accel T bn Time accel WK Conversion Constant Backstop pulley rotational speed Total power at drive location Motor shaft revolutions per minute AC motor shaft speed Back stop service factor Prime mover torque - required load torque Minimum torque rating of the backstop Acceleration time in seconds of the motor Moment of inertia of rotating parts 780

13 1 TRANSFER POINTS φ, can be determined through shear cell testing and hopper geometry. Conservative values for φ i for free flowing bulk solids are 70 to 80 degrees. The shearing force for simple feeder designs handling free flowing bulk solids can be estimated by calculating the volume of surcharge material above the shear plane, calculating the weight and applying the coefficient of internal friction of the material. For complex hopper or feeder designs consult a CEMA Member company for advice. Pressure Volume (see text) h y y 1 Ф h h 1 L h Figure 1.73 Simple belt feeder b b 1 h 1 h h Ф i b 1 b 1 L h L h b Consolidated Load Volume Based on Internal Angle of Friction, Ф i b Unconsolidated Hydrostatic Load Volume Figure 1.74 Simple belt feeder pressure volumes For consolidated materials the volume of the bulk material is: V fs = 1 b 1+b h 1+h L h Equation 1.75 V fs, Consolidated pressure volume Q i = V fs γ m Equation 1.76 Q i, Load on feeder belt 538

14 1 TRANSFER POINTS φ, can be determined through shear cell testing and hopper geometry. Conservative values for φ i for free flowing bulk solids are 70 to 80 degrees. The shearing force for simple feeder designs handling free flowing bulk solids can be estimated by calculating the volume of surcharge material above the shear plane, calculating the weight and applying the coefficient of internal friction of the material. For complex hopper or feeder designs consult a CEMA Member company for advice. Pressure Volume (see text) h y y 1 Фi h h 1 L h Figure 1.73 Simple belt feeder b b 1 h 1 h h Ф i b 1 b 1 L h L h b Consolidated Load Volume Based on Internal Angle of Friction, Ф i b Unconsolidated Hydrostatic Load Volume Figure 1.74 Simple belt feeder pressure volumes For consolidated materials the volume of the bulk material is: V fs = 1 b 1+b h 1+h L h Equation 1.75 V fs, Consolidated pressure volume Q i = V fs γ m Equation 1.76 Q i, Load on feeder belt 538

15 1 TRANSFER POINTS Fundamental Force-Velocity Relationships Fundamentally, if the tangential velocity is., and if g is the acceleration due to gravity; r s is the radial distance from the center of the pulley to the center of mass (i.e., the cross-sectional center of gravity of the material load shape); and W is the gravity weight force of the material acting at the center of mass, then the centrifugal force acting at the center of mass of the material is as follows: Centrifugal Force = W V s g r s Equation 1.9 Centrifugal force When this centrifugal force equals the radial component of the material weight force, W, the material will no longer be supported by the belt and will commence its trajectory. At just what angular position around the pulley this will occur is governed by the slope of the conveyor at the discharge and pulley is described for the following three conditions: horizontal, inclined and declined conveyor trajectories. Equation 1.9 can be re-written to provide and expression used to determine where the material will start its trajectory. Centrifugal Force W = g r s Equation 1.93 Relationship used to determine trajectory starting point, e t If V belt >1.0 then = V, If no then : If g r s V cg <1.0 then: = V cg, If no then: = g r s g r s Figure 1.94 Test used to determine the tangential velocity,, used for plotting the trajectory Belt Trajectory Nomenclature a 1 = Distance from the belt to the center of gravity of the load shape cg = Center of gravity of the cross section of the load shape e t = Point where the material leaves the belt g = Acceleration due to gravity h = Distance from the belt to the top of the load shape r p = Radius of the pulley r s = Radius from the center of pulley to the cross-sectional center of gravity of the load shape t = Thickness of the belt V = Belt speed = Tangential velocity, fps, of the cross-sectional area center of gravity of the load shape W = Weight of bulk material acting at center of gravity γ = Angle between pulley vertical centerline and point e t (degrees) φ = Angle of incline of the belt conveyor to the horizontal (degrees) Figure 1.95 Discharge trajectory nomenclature 55

16 1 TRANSFER POINTS Fundamental Force-Velocity Relationships Fundamentally, if the tangential velocity is., and if g is the acceleration due to gravity; r s is the radial distance from the center of the pulley to the center of mass (i.e., the cross-sectional center of gravity of the material load shape); and W is the gravity weight force of the material acting at the center of mass, then the centrifugal force acting at the center of mass of the material is as follows: Centrifugal Force = W V s g r s Equation 1.9 Centrifugal force When this centrifugal force equals the radial component of the material weight force, W, the material will no longer be supported by the belt and will commence its trajectory. At just what angular position around the pulley this will occur is governed by the slope of the conveyor at the discharge and pulley is described for the following three conditions: horizontal, inclined and declined conveyor trajectories. Equation 1.9 can be re-written to provide and expression used to determine where the material will start its trajectory. Centrifugal Force W = g r s Equation 1.93 Relationship used to determine trajectory starting point, e t If V belt >1.0 then = V, If no then : If g r s V cg <1.0 then: = V cg, If no then: = g r s g r s Figure 1.94 Test used to determine the tangential velocity,, used for plotting the trajectory Belt Trajectory Nomenclature a 1 = Distance from the belt to the center of gravity of the load shape cg = Center of gravity of the cross section of the load shape e t = Point where the material leaves the belt g = Acceleration due to gravity h = Distance from the belt to the top of the load shape r p = Radius of the pulley r s = Radius from the center of pulley to the cross-sectional center of gravity of the load shape t = Thickness of the belt V = Belt speed = Velocity of the load cross section used for plotting the trajectory W = Weight of bulk material acting at center of gravity γ = Angle between pulley vertical centerline and point e t (degrees) φ = Angle of incline of the belt conveyor to the horizontal (degrees) Figure 1.95 Discharge trajectory nomenclature 55

17 TRANSFER POINTS 1 Declined Belt Conveyor Discharge Case 7 If the tangential speed is insufficient to make the material leave the belt at the initial point of tangency of the belt and pulley, the material will follow partly around the pulley. Tangential velocity,, is used for plotting the trajectory. g r s >1.0 Equation Discharge trajectory velocity test for case 7 cg v a 1 h r s φ γ Vs r p et φ t Figure Discharge trajectory case 7 559

18 TRANSFER POINTS 1 Declined Belt Conveyor Discharge Case 7 If the tangential speed is insufficient to make the material leave the belt at the initial point of tangency of the belt and pulley, the material will follow partly around the pulley. Tangential velocity,, is used for plotting the trajectory. g r s >1.0 Equation Discharge trajectory velocity test for case 7 cg v a 1 h r s φ γ Vs r p et φ t Figure Discharge trajectory case 7 559

19 1 TRANSFER POINTS PLOTTING THE TRAJECTORY Before the trajectory of the discharged material can be plotted, it is necessary to calculate the values of. and r s in order to solve the expression If the value of expression 1.94 is less than 1.0 the trajectory starts at the position defined by the point of tangency between the belt and the pulley and the velocity used for plotting is the design belt speed, V. If expression 1.94 is equal to or greater than 1.0 then the trajectory starts at a position other than the point of tangency between the belt and pulley and the velocity at the center of gravity,., is used for plotting the trajectory. a 1 h r p r s t = Distance above the belt surface of the center of gravity of the cross-section shape of the load, at the point where the pulley is tangent to the belt = Distance above the belt surface of the top of the load, at the point where the belt is tangent to the pulley = Radius of the outer surface of the pulley and lagging = Radius from the center of the pulley to the center of gravity of the circular segment load cross section = Thickness of the belt = Velocity at the center of gravity of the load cross section used for plotting the trajectory: 1. Belt velocity is used as the velocity of the material at its center of mass if discharge point is at the tangency of the belt-to-discharge pulley (V). Velocity of the material at its center of mass is used as the velocity of the material for all other conditions of discharge after the point of belt-to-discharge pulley tangency. (V cg ) Figure Trajectory plotting nomenclature r s = a 1 + t + r p Equation r s, Radius from pulley center to load cross section center of gravity The values of a 1 and h have been tabulated for the various belt widths, idler end roll angles, and surcharge angles, for troughed conveyor belts loaded to the standard edge distance [0.055 BW inch (0.055 BW+3 mm )], as listed in Table should never be calculated from the nominal speed of the belt. It is also necessary to find the height of the flattened load of material on the belt, so that the upper limit of the material path can be plotted. The tangential velocity,. in distance per second, should be calculated from the relation: V = π (r p + t) rpm of discharge pulley 60 V cg = π r s rpm of discharge pulley 60 Equation 1.11., Tangential velocity of the center of gravity of the load profile 560

20 1 TRANSFER POINTS PLOTTING THE TRAJECTORY Before the trajectory of the discharged material can be plotted, it is necessary to calculate the values of. and r s in order to solve the expression If the value of expression 1.94 is less than 1.0 the trajectory starts at the position defined by the point of tangency between the belt and the pulley and the velocity used for plotting is the design belt speed, V. If expression 1.94 is equal to or greater than 1.0 then the trajectory starts at a position other than the point of tangency between the belt and pulley and the velocity at the center of gravity,., is used for plotting the trajectory. a 1 h r p r s t = Distance above the belt surface of the center of gravity of the cross-section shape of the load, at the point where the pulley is tangent to the belt = Distance above the belt surface of the top of the load, at the point where the belt is tangent to the pulley = Radius of the outer surface of the pulley and lagging = Radius from the center of the pulley to the center of gravity of the circular segment load cross section = Thickness of the belt = Velocity at the center of gravity of the load cross section used for plotting the trajectory: 1. Belt velocity, V, is used as the velocity of the material at its center of mass if the discharge point is at the tangency of the belt to discharge pulley ( = V). Velocity of the material at its center of mass, V cg, is used as the velocity of the material for all other conditions of discharge after the point of belt to discharge pulley ( ) tangency. = V cg Figure Trajectory plotting nomenclature r s = a 1 + t + r p Equation r s, Radius from pulley center to load cross section center of gravity The values of a 1 and h have been tabulated for the various belt widths, idler end roll angles, and surcharge angles, for troughed conveyor belts loaded to the standard edge distance [0.055 BW inch (0.055 BW+3 mm )], as listed in Table should never be calculated from the nominal speed of the belt. It is also necessary to find the height of the flattened load of material on the belt, so that the upper limit of the material path can be plotted. The tangential velocity,. in distance per second, should be calculated from the relation: V = π (r p + t) rpm of discharge pulley 60 V cg = π r s rpm of discharge pulley 60 Equation 1.11., Tangential velocity of the center of gravity of the load profile 560