After a mathematical evaluation of how a particle moves through a bends, here a short description of the results.
The formulas are not presented here.
The bend type, going from horizontal to vertical requires special attention, as that bend is crucial for the performance of the whole, properly, designed installation.
In a bend, the product particle hits the outer wall of the bend under an angle whereby not all the kinetic energy is lost, nor the bounce back occurs.
After hitting the wall, the particle will follow the outer bend under the influence of the centrifugal forces acting on the particle.
The wall of the bend exerts the reaction on the particle.
During this sliding along the outer wall of the bend, the particle-velocity will decrease due to friction.
The product-flow in the bend increases in cross-section because of the increasing part of the product-flow hitting the wall and also because of the decreasing particle-velocity.
Complete separation of product and air occurs.
As a result of this complete separation the only existing pressure drop is caused by the air friction-losses in the bend.
Because an 1.5D bend is constructively acceptable to manufacture and wear problems can be easily be coped with by an external box , this type of bend is often applied.
At any moment, the particle-velocity, and the covered distance as well as the covered angle can be calculated as follows (gravity ignored):
When product-particles, coming from a straight pipe enter the bend, these particles will hit the wall and be scattered over the outer wall.
The particles will follow the wall immediately.
The particles, entering the inside of the bend, will hit the already present particles further in the outer bend.
The average hitting angle of a particle can be calculated as the point where the centerline of the incoming pipe intersects the outer wall of the bend.
The perpendicular impact velocity is assumed to be lost in a non-elastic collision.
If the calculated hit-angle > the bend angle, then the hit angle is equal to the bend angle.
The velocity loss in a bend (excluding the gravity component) is the combination of the lost velocity in the impact (perpendicular impact velocity) and the friction velocity along the remaining bend angle:
However, the gravity effect in a bend on the velocity change is depending on the type of the bend:
quadrant 1:from vertical upwards to horizontal
The tangential gravity component decelerates the particle
The radial gravity component decreases the normal force at the outer bend wall and thereby causes less deceleration of the particle
quadrant 2:from horizontal to vertical downwards
The tangential gravity component accelerates the particle
The radial gravity component decreases the normal force at the outer bend wall and thereby causes less deceleration of the particle
quadrant 3:from vertical downwards to horizontal
The tangential gravity component accelerates the particle
The radial gravity component increases the normal force at the outer bend wall and thereby causes increased deceleration of the particle
quadrant 4:from horizontal to vertical upwards
The tangential gravity component decelerates the particle
The radial gravity component increases the normal force at the outer bend wall and thereby causes increased deceleration of the particle
Conclusion:
For small radius bends, the reduction in velocity in a horizontal bend is influenced by the impact angle, which is determined by the bend radius factor. For higher bend radii, the velocity loss becomes almost constant.
The reduction in velocity in a bend in a vertical plane is influenced by the bend radius factor (determining the impact angle) and the influence of the gravity component on the wall friction and tangential acceleration.
The bend in quadrant 2 has the lowest deceleration as the gravity components have both positive values for acceleration. (Lowest for deceleration)
A bend going from horizontal to vertical downwards should be a long radius bend.
The bend in quadrant 4 has the highest deceleration as the gravity components have both negative values for acceleration. (Highest for deceleration)
A bend going from horizontal to vertical upwards should be a short radius bend.
In coal powder injection systems those type of bends are commonly T-bends.
Horizontal bends and bends going from vertical upwards to horizontal and from vertical downwards to horizontal should be short radius bends.
Bend radius ratios smaller than 1.5 must be avoided.
In relation to wear, the impact zone is the most vulnerable spot, which occurs in each bend, short or long.
The angle of impact determines the wearing ratio between impact and sliding.
Mild steel is more wear resistant against impact and hard steel is more resistant against sliding.
This effect requires mild steel bends for short radius bends and hard steel for long radius bends.
Experience in pneumatic cement conveying shows that 1.5D bends show almost equal wear based on conveyed tonnage as long radius bends.
Therefore, 1.5D bends in pneumatic conveying installations are widely used.
This theory does not apply for very long radius bends, where multiple impact zones can exist.
In that case, a long radius bend consists of a series of smaller angle bends.
Pre-assumption: In a bend only velocity losses occur and 100% separation of conveying gas and material.
The minimal product velocity occurs when a column of material of the bulk density is filling the complete pipe cross-sectional area at the given conveying rate.
To guarantee this minimum product velocity at the bend outlet, the product at the bend inlet must have sufficient velocity (impulse) to comply with:
To reach a minimum product velocity at the bend outlet, a minimum product inlet velocity is required.
This implies that the minimum gas flow, which is required for sufficient material impulse to
reach the end of a bend with enough velocity to fill the pipe cross section increases with the local gas pressure.
This also indicates that the chosen gas velocities of a pneumatic conveying system must be based on this minimum required gas flow for bends.
In a bend, full separation is reached, whereby the material velocity decreases because of friction and gravity for bends where the vertical material velocity component is upwards.
The bend cross sectional area is considered “filled”, when an equivalent of 0.9*D (0.81*Area) is reached for the material flow, while the gas flow and the material flow are still separated.
When this degree of filling is reached, the flowing gas as assumed to interact again with the material.
The material is then re-accelerated and decelerated by the wall friction of the bend, also causing additional pressure drop.
At lower velocities, due to deceleration in the bend, the filled area increases
At higher material mass flows, the filled area increases.
Material mass flow and velocity are coupled through the gas pressure.
At higher material flow, the conveying pressure is also higher, causing lower velocities.
Both effects lead to higher filling ratios and the maximum filled condition will be reached at a smaller angle in the bend ■
Bend theory
After a mathematical evaluation of how a particle moves through a bends, here a short description of the results.
The formulas are not presented here.
The bend type, going from horizontal to vertical requires special attention, as that bend is crucial for the performance of the whole, properly, designed installation.
In a bend, the product particle hits the outer wall of the bend under an angle whereby not all the kinetic energy is lost, nor the bounce back occurs.
After hitting the wall, the particle will follow the outer bend under the influence of the centrifugal forces acting on the particle.
The wall of the bend exerts the reaction on the particle.
During this sliding along the outer wall of the bend, the particle-velocity will decrease due to friction.
The product-flow in the bend increases in cross-section because of the increasing part of the product-flow hitting the wall and also because of the decreasing particle-velocity.
Complete separation of product and air occurs.
As a result of this complete separation the only existing pressure drop is caused by the air friction-losses in the bend.
Because an 1.5D bend is constructively acceptable to manufacture and wear problems can be easily be coped with by an external box , this type of bend is often applied.
At any moment, the particle-velocity, and the covered distance as well as the covered angle can be calculated as follows (gravity ignored):
When product-particles, coming from a straight pipe enter the bend, these particles will hit the wall and be scattered over the outer wall.
The particles will follow the wall immediately.
The particles, entering the inside of the bend, will hit the already present particles further in the outer bend.
The average hitting angle of a particle can be calculated as the point where the centerline of the incoming pipe intersects the outer wall of the bend.
The perpendicular impact velocity is assumed to be lost in a non-elastic collision.
If the calculated hit-angle > the bend angle, then the hit angle is equal to the bend angle.
The velocity loss in a bend (excluding the gravity component) is the combination of the lost velocity in the impact (perpendicular impact velocity) and the friction velocity along the remaining bend angle:
However, the gravity effect in a bend on the velocity change is depending on the type of the bend:
quadrant 1:from vertical upwards to horizontal
The tangential gravity component decelerates the particle
The radial gravity component decreases the normal force at the outer bend wall and thereby causes less deceleration of the particle
quadrant 2:from horizontal to vertical downwards
The tangential gravity component accelerates the particle
The radial gravity component decreases the normal force at the outer bend wall and thereby causes less deceleration of the particle
quadrant 3:from vertical downwards to horizontal
The tangential gravity component accelerates the particle
The radial gravity component increases the normal force at the outer bend wall and thereby causes increased deceleration of the particle
quadrant 4:from horizontal to vertical upwards
The tangential gravity component decelerates the particle
The radial gravity component increases the normal force at the outer bend wall and thereby causes increased deceleration of the particle
Conclusion:
For small radius bends, the reduction in velocity in a horizontal bend is influenced by the impact angle, which is determined by the bend radius factor. For higher bend radii, the velocity loss becomes almost constant.
The reduction in velocity in a bend in a vertical plane is influenced by the bend radius factor (determining the impact angle) and the influence of the gravity component on the wall friction and tangential acceleration.
The bend in quadrant 2 has the lowest deceleration as the gravity components have both positive values for acceleration. (Lowest for deceleration)
A bend going from horizontal to vertical downwards should be a long radius bend.
The bend in quadrant 4 has the highest deceleration as the gravity components have both negative values for acceleration. (Highest for deceleration)
A bend going from horizontal to vertical upwards should be a short radius bend.
In coal powder injection systems those type of bends are commonly T-bends.
Horizontal bends and bends going from vertical upwards to horizontal and from vertical downwards to horizontal should be short radius bends.
Bend radius ratios smaller than 1.5 must be avoided.
In relation to wear, the impact zone is the most vulnerable spot, which occurs in each bend, short or long.
The angle of impact determines the wearing ratio between impact and sliding.
Mild steel is more wear resistant against impact and hard steel is more resistant against sliding.
This effect requires mild steel bends for short radius bends and hard steel for long radius bends.
Experience in pneumatic cement conveying shows that 1.5D bends show almost equal wear based on conveyed tonnage as long radius bends.
Therefore, 1.5D bends in pneumatic conveying installations are widely used.
This theory does not apply for very long radius bends, where multiple impact zones can exist.
In that case, a long radius bend consists of a series of smaller angle bends.
Pre-assumption: In a bend only velocity losses occur and 100% separation of conveying gas and material.
The minimal product velocity occurs when a column of material of the bulk density is filling the complete pipe cross-sectional area at the given conveying rate.
To guarantee this minimum product velocity at the bend outlet, the product at the bend inlet must have sufficient velocity (impulse) to comply with:
To reach a minimum product velocity at the bend outlet, a minimum product inlet velocity is required.
This implies that the minimum gas flow, which is required for sufficient material impulse to
reach the end of a bend with enough velocity to fill the pipe cross section increases with the local gas pressure.
This also indicates that the chosen gas velocities of a pneumatic conveying system must be based on this minimum required gas flow for bends.
In a bend, full separation is reached, whereby the material velocity decreases because of friction and gravity for bends where the vertical material velocity component is upwards.
The bend cross sectional area is considered “filled”, when an equivalent of 0.9*D (0.81*Area) is reached for the material flow, while the gas flow and the material flow are still separated.
When this degree of filling is reached, the flowing gas as assumed to interact again with the material.
The material is then re-accelerated and decelerated by the wall friction of the bend, also causing additional pressure drop.
At lower velocities, due to deceleration in the bend, the filled area increases
At higher material mass flows, the filled area increases.
Material mass flow and velocity are coupled through the gas pressure.
At higher material flow, the conveying pressure is also higher, causing lower velocities.
Both effects lead to higher filling ratios and the maximum filled condition will be reached at a smaller angle in the bend ■
Teus