  EQUATIONS

The troughability of a conveyor belt can be estimated by using this equation,

where

m"G = belt mass in kg/m²

B = belt width in m

Sz = carcass thickness in mm

Cq = transverse rigidity factor (polyamide = 18, steel cord = 42)

Test

TROUGHABILITY The modulus of elasticity is calculated by dividing the stress by the strain,

where

M = modulus of elasticity (ISO 9856)

F = force (N)

εelast = elastic elongation at the end of the specified number of cycles (N/mm)

In other words: The higher the modulus the lower the elastic elongation per unit stress. More

MODULUS OF ELASTICITY The modulus of elasticity can be used to calculate the tension force it exerts under a specific extension,

where

T = tension force

λ = modulus of elasticity

A = cross-sectional area

x = extension

l = length (m)

TENSION FORCE The minimum belt tensions for transmitting

the pulley peripheral forces are calculated as follows,

where

Fu = minimum peripheral force

C = coefficient C

f = artificial friction coefficient

L = conveyor length (m)

g = acceleration (m/s²)

qRo mass of revolving idler parts of top strand (kg/m),

qRu mass of revolving idler parts of bottom strand (kg/m),

qB mass of the belt on top strand (kg/m),

qG mass of the belt in bottom strand (kg/m),

FS1 special main resistances,

FS2 special secondary resistances

Chart

MINIMUM PERIPHERAL FORCE The required take-up length is calculated as follows,

where

SSp = take-up length (m)

L = conveyor length (m)

ε = belt elongation, elastic and permanent (%)

As a rough guideline, use 1,5 % elongation for textile belts

and 0,2 % for steel cord belts.

Note: For long-distance conveyors, dynamic start-up calculations

may be required, because not all elements are set in motion simultaneously,

due to the elastic properties of the conveyor belt.

TAKE-UP LENGTH  The coefficient C is a function of the length of the conveyor.

The total resistances without slope and special resistances are divided by the main resistances,

where

C = coefficient C

FH = primary resistances

FN = secondary resistances

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COEFFICIENT C The Arrhenius equation describes the quantitative relation between reaction velocity and temperature

(the speed of chemical reactions increases with rising temperature),

where

k = temperature dependence of the rate constant (of a chemical reaction)

EA = activation energy

T = temperature (in Kelvin)

R = gas constant

Ae = prefactor (frequency factor)

ARRHENIUS EQUATION where

σ = stress

v = period of strain oscillation

δ = phase lag between stress and strain

STRESS IN RUBBER where

ε = strain

ω = period of strain oscillation

t = time

STRAIN IN RUBBER The storage modulus measures the stored energy, representing the elastic portion,

and the energy dissipated as heat, representing the viscous portion,

where

E' = storage modulus

σ = stress

ε = strain

δ = phase lag between stress and strain

STORAGE MODULUS The loss modulus measures the stored energy, representing the elastic portion,

and the energy dissipated as heat, representing the viscous portion,

where

E'' = loss modulus

σ = stress

ε = strain

δ = phase lag between stress and strain

LOSS MODULUS Internal friction is the force resisting motion between the elements making up a solid material

while it undergoes deformation. The tan δ is sometimes used to determine the indentation

loss of a conveyor belt cover (energy saving belts). E' and E'' should be as low as possible.

However, there are a number of misconceptions related to specifying E' and E''.

Where

tan δ = internal friction of a rubber

E' = storage modulus (N/mm²)

E'' = loss modulus (N/mm²)

INTERNAL FRICTION Where

v = belt velocity (m/s),

lvth = theoretical volume flow (m³/h),

ρ = bulk density of the conveyed material (t/m³),

φSt = coefficient for determination of the volume flow.

More

LENGTH RELATED MASS FLOW (m³/h) Where,

PB0 = braking factor related to the rated torque of all drive motors,

ηges = overall efficiency of all transmission elements between motor and pulley shaft,

PMerf = total capacity of the drive motors required in a steady operating state,

PMinst is the total installed capacity of the drive motors (N).

BRAKING FACTOR Where

g = gravity (9,81 m/s²)

m'Li = mass of the conveyed material, uniformly distributed across a section of the conveyor (kg/m)

m'G = length related mass of the conveyor belt (kg/m)

IRo = idler spacing in top run (m)

hrel = maximum belt sag related to the spacing between the carry idlers (%)

MINIMUM BELT TENSION FOR BELT SAG LIMITATION (top side, loaded) Where

g = gravity (9,81 m/s²)

m'G = length related mass of the conveyor belt (kg/m)

IRu = idler spacing in bottom run (m)

hrel = maximum belt sag related to the spacing between the carry idlers (%)

MINIMUM BELT TENSION FOR BELT SAG LIMITATION (bottom side, unloaded) Where

f = friction factor in top and bottom run

L = conveyor length (m)

g = gravity acceleration (m/s²)

m'R = mass of the idlers (kg/m)

m'G = length related mass of the conveyor belt in both runs (kg/m)

m'L = mass of the conveyor belt with an evenly distributed load (kg/m)

δ = even inclination of the conveyor (°)

PRIMARY RESISTANCES IN AN EVENLY TILTED CONVEYOR The Voigt model consists of a Newtonian damper and Hookean elastic spring connected in parallel.

It is used to explain the creep resp. relaxation behaviour of polymers.

Where

η = dynamic viscosity

τ = total stress

γ = total deformation

D = shear rate

G = shear modulus

VOIGT MODEL Where

C = coefficient (main resistance factor)

f = resistance coefficient

L = belt length (m)

g = acceleration (m/s²)

qRO = mass of the idlers on top side (kg/m)

qRU = mass of the idlers on bottom side (kg/m)

qB = belt mass (kg/m)

qG = mass of the conveyed material (kg/m)

H = lift (m)

FS1 = special main resistances

FS2 = special secondary resistances

PERIPHERAL FORCE MINIMUM TRANSITION LENGTH (m) Where

qG = conveying mass (kg/m)

H = lift (m)

g = acceleration (m/s²)

SLOPE RESISTANCE TRANSITION CURVES (m)

Where

m'G = length related mass of the conveyor belt (kg/m)

g = acceleration (m/s²)

b = width (mm)

δ = troughing angle

l = idler length (mm)

B = belt width (mm)

Tx = drive traction Where

Δle = elastic elongation (mm),

Io = initial length of the test piece(mm).

ELASTIC ELONGATION (ISO 9856) Where

Δ lp = permanent elongation (mm),

Io = initial length of the test piece (mm).

Test

PLASTIC (PERMANENT) ELONGATION (ISO 9856) The E-Modulus (Young's modulus) defines the relationship between stress (force per unit area)

and strain (proportional deformation) in a belt,

where

ΔL = amount by which the length changes (mm)

F = force

Ao = original cross-sectional area

Lo = original length (mm)

ELASTIC MODULUS Where

F = resistances to motion

v = belt speed

DRIVE POWER Where

FH = primary resistances (idlers, belt indentation, etc.)

FN = secondary resistances (feeding, scrapers etc.)

FS = extraordinary resistances

RESISTANCES TO MOTION Where

FGH is the downhill force

FG is the weight force

Gravity acts straight down (= the weight of the conveyor belt) and the support force acts away from the conveyor. Since the conveyor is sloped, there is a net force acting down the slope.

DOWNHILL FORCE The belt friction equation relates the hold-force to the load-force when a belt is is wound around a pulley,

where

e = 2,7183

EYTELWEIN'S EQUATION The RMS is the square root of the arithmetic mean of the squares of the values.     