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TECHNICAL / EQUATIONS  

(This chapter is still being improved. Sorry for inconveniences.)

MODULUS OF ELASTICITY

The modulus of elasticity is calculated by dividing the stress by the strain:
Modulus of elasticity
where M is the modulus of elasticity (ISO 9856)
F is the force (N)
εelast is the elastic elongation at the end of the specified number of cycles in N/mm
In other words: The higher the modulus the lower the elastic elongation per unit stress. See definition here

TENSION FORCE

The modulus of elasticity of a material can be used to calculate the tension force it exerts under a specific extension:
Tension force
where T is the tension force
  λ is the modulus of elasticity
  A is the cross-sectional area
  x is the extension
  l is the length (m)

MINIMUM PERIPHERAL FORCE

The minimum belt tensions for transmitting the pulley peripheral forces are calculated as follows:
Minimum Pulley Peripheral Force
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.

TAKE-UP LENGTH

Take-up length
where
SSp is take-up length (m)
  L is centre distance (m)
 

ε is belt elongation, elastic and permanent (%)

As a rough guideline, use 1,5% elongation for textile belts and 0,25% for steel cord conveyor 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.

COEFFICIENT C

Coefficient C

The coefficient C is a function of the length of the installation.
The total resistances without slope and special resistances are divided by the main resistances.

ARRHENIUS EQUATION

Arrhenius equation
where
k is the temperature dependence of the rate constant (of a chemical reaction)
  EA is the activation energy
 

T is the temperature

  R is the gas constant
  A is the prefactor (frequency factor)
The Arrhenius equation describes the quantitative relation between reaction velocity and temperature (as you know, the speed of chemical reactions increase with rising temperature).

STRESS IN RUBBER

Equation for stress in viscoelastic materials
where
σ is the stress
  v is the period of strain oscillation
  δ is the phase lag between stress and strain

STRAIN IN RUBBER

Equation for strain in viscoelastic materials
where
ε is the strain
  ω is the period of strain oscillation
  t is time
STORAGE MODULUS
Storage modulus
where E' is the storage modulus
  σ is the stress
  ε is the strain
  δ is the phase lag between stress and strain

INTERNAL FRICTION

Internal friction of a rubber
where tan δ is the internal friction of a rubber

E' is the storage modulus (N/mm²)
  E'' is the loss modulus (N/mm²)

The tan d is sometimes used to determine the indentation loss of a conveyor belt cover (cf. 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''.

LENGTH RELATED MASS FLOW (m³/h)
Length related mass flow
where v is the belt velocity (m/s),
  lvth is the theoretical volume flow (m³/h),
  ρ is the bulk density of the conveyed material (t/m³),
  φSt is the coefficient for determination of the volume flow.
BRAKING FACTOR
Braking factor
where ΡB0 is the braking factor related to the rated torque of all drive motors,
  ηges is the overall efficiency of all transmission elements between motor and pulley shaft,
  PMerf is the total capacity of the drive motors required in a steady operating state,
  PMinst is the total installed capacity of the drive motors (N).
MINIMUM BELT TENSION FOR BELT SAG LIMITATION (top side, loaded)
Minimum belt tension top run loaded
where g is gravity (9,81 m/s²)
  m'Li is the mass of the conveyed material, uniformly distributed across a section of the conveyor (kg/m)
  m'G is the length related mass of the conveyor belt (kg/m)
  IRo is the idler spacing in top run (m)
  hrel is the maximum belt sag related to the spacing between the carry idlers (%)
MINIMUM BELT TENSION FOR BELT SAG LIMITATION (bottom side, unloaded)
Minimum belt tension of return run
where g is the gravity (9,81 m/s²)
  m'G is the length related mass of the conveyor belt (kg/m)
  IRu is the idler spacing in bottom run (m)
  hrel is the maximum belt sag related to the spcing between the carry idlers (%)
PRIMARY RESISTANCES IN AN EVENLY TILTED CONVEYOR
Primary resistances for an evenly tilted conveyor
where f is the friction factor in top and bottom run
  L is the conveyor length (m)
  g is the gravity acceleration (m/s²)
  m'R is the mass of the idlers (kg/m)
  m'G is the length related mass of the conveyor belt in both runs (kg/m)
  m'L is the mass of the conveyor belt with an evenly distributed load (kg/m)
  δ is the even inclination of the conveyor (°)
MAXWELL MODEL
Maxwell model
where ε is strain
  σ is stress
VOIGT MODEL
Voigt-Kelvin model
where η is dynamic viscosity
  τ is total stress
  γ is total deformation
  D is shear rate
  G is shear modulus
Used to express the relaxation behavior of polymers.
ROLLING RESISTANCE
Rolling resistance
where F is resistance force
  Crr is the dimensionless rolling resistance coefficient
  Nf is the normal force
or  
RRF
where E' is the storage modulus (N/mm²)
  tanδ is the internal friction
MINIMUM TRANSITION LENGTH (m)
Minimum transition length
where B is belt width (mm)
  λ is troughing angle (°)
  S is the safety factor
  KG is the belt parameter
  Kf1 is the troughing parameter
PERIPHERAL FORCE (N)
Peripheral forces
where FH is the main resistance
  FN is the secondary resistance
  FS1 are the special main resistances
  FS2 are the special secondary resistances
  FSt are the resistances due to slope
 
Peripheral force
where PTr is the drive power (pulley)
  v is speed (m/s)
 
Required driving force
where C is the coefficient (main resistance factor)
  f is the resistance coefficient
  L is belt length (m)
  g is acceleration (m/s²)
  qRO is the mass of the idlers on top side (kg/m)
  qRU is the mass of the idlers on bottom side (kg/m)
  qB is the belt mass (kg/m)
  qG is the mass of the conveyed material (kg/m)
  H is the lift (m)
  FS1 are the special main resistances
  FS2 are the special secondary resistances
SLOPE RESISTANCE
Slope Resistance
where qG is the conveying mass (kg/m)
  H is the lift (m)
  g is acceleration (m/s²)
TRANSITION CURVES (m)
Transition curves
where m'G is the length related mass of the conveyor belt (kg/m)
  g is acceleration (m/s²)
  b is width (mm)
  δ is troughing angle
  l is idler length (mm)
  B is belt width (mm)
  Tx is drive traction
ELASTIC ELONGATION (ISO 9856)
Elastic elongation
where Δle is the elastic elongation (mm),
  Io is the initial length of the test piece(mm).
PERMANENT (PLASTIC) ELONGATION (ISO 9856)
Permanent elongation
  Hysteresis
where Δ lp is the permanent elongation (mm),
 

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

For the drawing: FU is 10% force of the belt breaking strength multiplied by the test piece width (N),
  FL is 2% force of the belt breaking strength multiplied by the test piece width (N),
  ΔF is the test force range.
YOUNG'S MODULUS
Young's Modulus
where ΔL is the amount by which the length changes (mm)
  F is the force
  Ao is the original cross-sectional area
  Lo is the original length (mm)
DRIVE POWER
Drive Power
where F are the resistances to motion
  v is belt speed
RESISTANCES TO MOTION
Drive Power-
where FH are the primary resistances (idlers, belt indentation, etc.)
  FN are the secondary resistances (feeding, scrapers etc.)
  FS are extraordinary resistances
  FSt are gradient resistances
DOWNHILL FORCE
Downhill force
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.
See also Clamping Force


EYTELWEIN'S FORMULA
Eytelwein's fomular
where e is 2,7183

 


ROOT MEAN SQUARE
Root Mean Square
   
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