Design Aspects of Pipe Conveyors

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Design Aspects of Pipe Conveyors

G. Lodewijks and M.E. Zamiralova

The 12th_{International Conference on Bulk Materials Storage,}

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Challenge the future

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2.

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Rolling resistance force

Rotational inertia of the idlers

Flexural deformation

Indentation

Load forces fro the belt - for open trough belt conveyor 61% [1]

[1] Hager, M., Hintz, A., The Energy-Saving Design of Belts for Long Conveyor Systems, Bulk Solids Handling, 13 (1993) 749.

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Concentrated forces from the material load for pipe belt conveyors [2]_,

modified from the approach for deep trough belt conveyors [3]_:

( )

2 m ( , ) p act d pas d 2 n l F R g C C α α α α α α α α α ρ α α α α +Δ +Δ ⎛ ⎞ ʹ Δ = ⋅ Δ ⋅ ⋅ ⋅ _⎜ + _⎟ ⎝

∫

⎠

( )

2 2

act (cos2 cos ) (cos sin ),

C α = ϕ+ α ⋅ α+ ⋅m α

( )

2 2 pas sin (cos 2 cos ) (cos ).

C

m α

α = ϕ + α ⋅ α +

Load forces determination

[2] Dmitriev, V. G., Sergeeva, N. V.: Tension calculation of pipe conveyors, GIAB M, Moscow State Mining University, 16 (2009) 144-170.

[3] Gushin, V. M.: Determination parameters of the load-carrying surface for the steep inclined conveyors with the deep troughed belt, Mine and quarry transport, Nedra, 1 (1975) 164-166.

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Concentrated forces from the belt. Total normal contact force.

Total normal contact force for each n-th roll:

m b zn n n F ₌ F ₊ F 3 b2 b stif 2 1 2 p ' 1 24 E h F l R α µ µ = ⋅ ⋅

− _{Weight of the belt}_[2]

Stiffness of the belt [4]_:

p b b1 stif 2 6 R m g F F l В π ʹ ʹ = + ⋅ ⋅ p b b2 b6 stif 2 cos 6 3 R m g F F F l B π _π ʹ ʹ = = + ⋅ ⋅ p b b3 b5 stif 2 cos 6 3 R m g F F F l B π _π ʹ ʹ = = − ⋅ ⋅ p b b4 stif 2 6 R m g F F l В π ʹ ʹ = − ⋅ ⋅

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10 Indentation contact model

The stress distribution [5]_:

2 2 0 1 1 1 ( , ) ( ) ( ) 1 exp 2 m i i i i i E E k x a x y a x x a a k R h hR k σ = ⎛ ⎛ ⎛ − ⎞⎞⎞ = − + ⎜_⎜ − + + ⎜_⎜ − ⎜ ⎟⎟_⎟⎟_⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

∑

Parameters ki equal [5] : i i b i b i V k V E η τ = =

[5] Nuttall, A. J. G., Lodewijks, G., Klein Breteler, A. J.: Modeling rolling contact phenomena in a pouch belt conveyor system, Wear, 260 (2006) 1081-1089.

Generalized Maxwell model with multiple parameters

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Deformation of the contact [5]_: 2 2 0 1 2 ( , ) 2 2 x y w x y z R R = − − Winkler foundation

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1) Half length of the contact:

2) Division into strips, y – coordinates.

3) Determination x - coordinates:

The normal contact force on each idler roll:

. 2 0 ) , 0 ( c = ⇒c= R₂z₀ σ y1, y2, y3…yp ( ( ), ) 0 ( ) w a y y = ⇒a y ( ( ), ) 0b y y b y( ) σ − = ⇒ 1 1 [5] ( ) calc 2 0 ( ) 2 ( , )d d 2 ( , )d ( , )d 2 j j a a a y c p z j b y b b y F _σ x y x y _σ x y x y _σ x y x = − − − ⎡_Δ ⎤ ⎢ ⎥ = = + Δ ⎢ ⎥ ⎣

∑

⎦

∫ ∫

∫

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Indentation force.

Where q – number of strips,

- for odd number of strips.

( ) 0 ( ) 2 ( , )d d a y c y b y M x σ x y x y − =

_{∫ ∫}

⋅

Moment My form the asymmetric pressure

distribution [5] _: , d ) , ( 2 d ) , ( 2 1 1 2 1 1 ind ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + = =

_∫

∑

− = a b p j y n x x y x x x y x qR c R M F _σ _σ

Indentation rolling resistance force Find for

each roll of the idler set:

2 1 + = q

p

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14 0 2 4 6 8 10 0 0.005 0.01 0.015 0.02 0.025 0.03 Vb [m/s] f ind [-] Q = 25% Q = 50% Q = 75%

Indentation rolling resistance friction factor:

0 2 4 6 8 10 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Vb [m/s] f ind [-] Q = 25% Q = 50% Q = 75%

For each n-th roll:

n z n n F F f_ind ₌ ind l g m m F f n n ʹ ʹ + ʹ =

∑

= ) ( _m _b 6 1 ind ind DIN:

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0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x 10 -3 Vb [m/s] z 0 [m ] F₁ F₂, F₆ F₃, F₅ F₄ 0 2 4 6 8 10 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Vb [m/s] f ind [-] Pipe Trough Indentation depth.

Comparison with the open trough belt conveyors.

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3.

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Pipe conveyor test rigs: design, application and test results

I.  Static 6-point testing devices

II.  Static test rigs with various frames and supports

III.  Dynamic measurements at the dynamic test rigs or

IV.  Existing pipe conveyor installations

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Main goal is measuring the roll/pipe contact forces that

primarily determine the pipe forming capacity, with a

focus on the effect of the belt’s stiffness.

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Theoretical predication of load distribution

10 20 30 40 50 60 70 80 0 100 200 300 400 500 600 700 m`_b [kg/m3] Re a c ti o n f o rc e R [N] R 1 R₂, R₆ R₃, R₅ R₄ 20 30 40 50 60 70 80 90 100 0 50 100 150 200 250 300 350 400 E₂ [MPa] Re a c ti o n f o rc e R [N] R₁ R₂, R₆ R₃, R₅ R₄

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0 50 100 150 200 250 300 1 2 3 4 5 6 0.00% 5.00% 10.00% 15.00% 20.00% 25.00% 30.00% 1 2 3 4 5 6

Absolute values

Percentile values: F

_n

/Σ|F

_n

|

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i) Static 6-point testing devices

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CDI together with Veyance, USA Taiyuan University of Science and Technology, China

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24 TU Delft tests performed on Phoenix test rig, Germany. Focus on effect overlap on load distribution using parameter B/D. Adjustable plate lengths and pipe diameter.

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26 Test rig from ITA Leibzig

University of Hanover - Hötte

With a static 6 point test device the effects of curves and belt tension can not be studied.

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ii) Static test rigs with various frames

Taiyuan University of Science and Technology, China

Variation in tension, pitch and curve radii. No information available on the tested belts.

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12 m long static test rig – CKIT/FLS/ConvEx/TU Delft

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30 Overlap at the top Overlap at the bottom

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Test rig of the Technical University of Košice, Slovak Republic

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Test rig of Phoenix Conveyor Belt Systems GmbH and the

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34 Typical test results with the Hanover test rig

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iii) Dynamic measurements

Dynamic test rig of the University of Leoben, Department of Conveying Technology and Design Methods, Austria

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Test rig of the Institute of Transport and Automation Technology (ITA) at Leibniz

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Experiment details

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38 Test results for a belt speed of 2 m/s

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Test results for the “Rollgurt” conveyor in the lime and cement plant in Alsen Breitenburg, Germany

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Test section in pipe conveyor in cement plant Hugo Miebach Söhne, Portland-Zementwerk Wittekind, used by Hötte for the field measurements, Institute of Transport and Automation Technology (ITA) a t L e i b n i z U n i ve r s i ty o f Hannover.

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4.

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•  Curve routes in horizontal and

vertical planes

•  Low or unbalanced load from the

material •  Irregular wear of the idler _rolls

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44 Twisting effect

Mass centroid effect

Moment of inertia effect

Shear center effect

Twisting effect.

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Shear center concept.

Open thin-wall structures.

•  The thickness of the walls is small, compared to the

height and width of the cross section

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46 Location of the shear center for doubly and singly symmetric cross sections.

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S 2 z wy yz wz y z yz I I I I y I I I + = − S 2 y wz yz wy y z yz I I I I z I I I − + = − p d wy A I ₌

_∫

w z A p d wz A I =

_∫

w y A

Shear center location for unsymmetrical cross sections.

Shear center coordinates in centroidal coordinate system:

Sectorial liner moments about y and z axes: Sectorial area: p p p 0 d d s s A w =

_∫

r s =

_∫

r s

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48 Shear center concept.

Pipe belt conveyors

•  Lateral load forces •  Geometry

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Lateral load forces inducing the shear center effect

.

Lateral forces from the curves

Load forces from the material

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50 Geometry of the cross section.

_{Assumption about the singly symmetric cross section}

_.

[1] Lodewijks, G., Drenth, K. F., Van der Mel, P. S., 2010, "Belt Conveyor Technology - Rotation of Pipe Conveyors," Bulk solids handling : the international journal of storing and handling bulk materials., 30(3), pp. 144-148.

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Unsymmetrical cross section. Archimedean spiral 0 ( ) 2 r _ϕ r δ _ϕ π = + d ( d ) ( ) d 2 r r ϕ ϕ r ϕ δ ϕ π = + − = 0 d ( ) d d 2 h r _ϕ _ϕ r δ _ϕ _ϕ π ⎛ ⎞ = ⋅ = _⎜ + _⎟ ⎝ ⎠ 0 d d 2 A δ r δ ϕ ϕ π ⎛ ⎞ ≈ _⎜ + _⎟ ⎝ ⎠

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52 0 0.1 0.2 0.3 0.4 0.5 0 20 40 60 80 100 120 140 160 k [-] γ [° ] 0 20 40 60 80 100 120 140 160 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 γ [°] r 0 [m ] Length of overlap: _{Δ =} kB Angle of overlap : γ 2 ( ) ( ) 4 2 B πδ B πδ πδ γ δ − + − Δ + + − Δ + ⋅ Δ = Initial radius r0 : 2 o B r πδ π − − Δ =

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Location of the shear center. 3 3 'sin 'cos ; 2 2 3 3 'cos 'sin . 2 2 x y x y y x π π γ γ π π γ γ ⎧ ⎛ ⎞ ⎛ ⎞ = _⎜ + _⎟+ _⎜ + _⎟ ⎪ ⎪ ⎝ ⎠ ⎝ ⎠ ⎨ ⎛ ⎞ ⎛ ⎞ ⎪ = _⎜ + _⎟− _⎜ + _⎟ ⎪ _⎝ _⎠ _⎝ _⎠ ⎩ ( ) ( ) cos ; ( ) ( )sin . x r y r ϕ ϕ ϕ ϕ ϕ ϕ ʹ = ⎧ ⎨ _ʹ = ⎩ 0 ( ) 2 r ϕ r δ ϕ π = +

Position of any point of the cross section medial line.

Polar coordinate system xy:

Cartesian coordinate system x’y’ :

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54 2 2 2 0 0 d sin d 2 x A S y A r π γ δ δ ϕ ϕ ϕ π + ʹ ⎛ ⎞ ʹ = = _⎜ + _⎟ ⋅ ⎝ ⎠

∫

2 2 2 0 0 d cos d 2 y A S x A r π γ δ δ ϕ ϕ ϕ π + ʹ ⎛ ⎞ ʹ = = _⎜ + _⎟ ⋅ ⎝ ⎠

∫

'C y S X A ʹ = ' C ' Sx Y A =

Position of the centroid.

0 0.1 0.2 0.3 0.4 0.5 -4 -3 -2 -1 0 1 2 3 4 k [-] X C [m m ] 0 0.1 0.2 0.3 0.4 0.5 0 5 10 15 20 25 k [-] Y C [m m ] C C C C C C ' sin ' cos ; ' cos ' sin . X X Y Y X Y γ γ γ γ = − ⎧ ⎨ = + ⎩

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Location of the shear center. Centroidal coordinate system.

C C 0 C C C 0 C 3 cos ; 2 2 3 sin . 2 2 x x X r X y y Y r Y δ π ϕ ϕ γ π δ π ϕ ϕ γ π ⎧ ⎛ ⎞ ⎛ ⎞ = − =_⎜ + _⎟ _⎜ − − _⎟− ⎪ ⎪ ⎝ ⎠ ⎝ ⎠ ⎨ ⎛ ⎞ ⎛ ⎞ ⎪ _{= −} ₌ ₊ ₋ ₋ ₋ ⎜ ⎟ ⎜ ⎟ ⎪ _⎝ _⎠ _⎝ _⎠ ⎩

Position of any point of the cross section medial line in centroidal coordinate system : _{x y}C C

Geometrical moments of inertia:

∫

= A x y A I ( C)2d C

∫

= A C y x A I ( )2d C Product of inertia:

∫

= A y x x y A I C Cd C C

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56 Centroidal coordinate system.

Sectorial liner moments in centroidal coordinate system

Sectorial area in centroidal coordinate system: S 2 p p 0 0 d d 2 A w r l r θ δ ϕ ϕ π ⎛ ⎞ = = _⎜ + _⎟ ⎝ ⎠

∫

= A C wx w y A I C _p d

∫

= A C wy w x A I C _p d

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Position of the shear center.

-error less than 0,1%

C C C C C C C C C S ₂ y wx x y wy x y x y I I I I x I I I − = − C C C C C C C C C S 2 x wy x y wx x y x y I I I I y I I I − + = − 0 0.1 0.2 0.3 0.4 0.5 -5 0 5 10 15 20 25 30 35 k [-] XS [m m ] Analytical ANSYS 0 0.1 0.2 0.3 0.4 0.5 -400 -300 -200 -100 0 100 200 k [-] Y S [m m ] Analytical ANSYS

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Location of the shear center.

ANSYS model.

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Location of the shear center.

Cross sections for various length of overlap.

k = 0,0 _{k = 0,1} k = 0,2

k = 0,3 k = 0,4 k = 0,5

- Shear center - Centroid

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•  Calculation method has been

constructed for the determination position of the shear center.

•  Changing of the geometrical parameters

of the belt can not solve the shear center effect of the twisting of pipe conveyor belt.

Solutions?

•  Increasing lateral friction between the

belt and rolls.

•  Increasing torsional stiffness of the belt

structure.

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From this paper it can be concluded that important design aspects that affect the performance of a pipe conveyor include:

I. the belt’s stiffness. This does not only determine the pipe forming capability but also the indentation rolling resistance. If the pipe contact forces are known then the indentation rolling resistance can be calculated. The test rig to measure these forces however, determines their magnitude and distribution.

II. The ratio between the overlap and the belt width. This ratio has an impact on the location of the shear center which affects the torque generated when a force exerted on the belt does not go through the shear center.

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Design Aspects of Pipe Conveyors