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),

H lift of the conveyor between discharge and loading area (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

 

Chart

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 specifiying 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

 

 

 

 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

 

 

TRANSITION CURVES (m)

 

 

 

Where

 

Δle = elastic elongation (mm),

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

 

Test

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

FSt = gradient 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.

 

Please also see here.

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