6. Material Trajectory – The Starting Point
The starting point is the point at which the material leaves the discharge pulley. It is important here to identify this position as at this point mechanical interaction between material and belt is lost, and the material acts like a projectile with initial velocity subject only to the action of gravity (excluding the effects of air resistance). The material trajectory is fundamental in the design of the chute as it defines the flow of material and the requirements for first impact point and the path followed by the material until it lands. There are numerous examples available in literature defining the methodology to be followed in establishing the material trajectory. The methodology described below is based on the CMA lecture course. There are recent papers presented at Beltcon by D. Hastings which are excellent references to other methodologies and which also give a comparison of the different methods against actual results established by high speed photography. The theory being proved by the practical. The methodology proposed in the CMA Diploma Course is as follows:
- Establish the area of material flowing over the head chute;
- Establish the depth of material flowing over the head chute;
- Establish the centroid of area of the material flowing over the head chute.
The condition of material flowing over the head chute is represented in Fig. 9 for a typical three-roll idler set. Note that the troughed form now flattens out and the area of the trapezium formed when flattened should be equal to that of the troughed configuration.
For a belt loaded to 100% of CEMA and based on normally accepted free-board dimensions, the width of material on the flattened belt may be found from:
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where:
W = belt width [mm]
Wwet= wetted area [m] (Note that units are mixed in order to simplify calculations).
The centroid of area of the trapezium is accepted as 40% of the height yielding
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Setting the area of the troughed belt equal to the area of the trapezium resulting from the flattened belt yields
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where:
A100 = cross sectional area at 100% loading [m2]
λ = angle of repose (typically 34°–37°)
With this, a profile can be defined that the centroid of material would follow around the head pulley and along its trajectory with upper and lower boundaries following this path. Note however, that in the case of the material stream comprising mainly large lumps (greater than say, 150 mm), it is normal to calculate d as a function of the lump size and typically as 60% of the lump size. The point R at which the material stream leaves the belt can now be defined:
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where:
r = pulley radius over lagging [m]
h = belt total thickness [m]
Ca = depth to centre of area of material burden [m]
At the point of separation, the material has the same velocity of the belt V [m/s]. And define factor K:
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where: α = angle of inclination of the belt at the discharge pulley The location of the drop points with their dependency on belt speed is now defined as for K > 1: drop point is at T for K < 1: drop point is at C
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where θ, the angle between the horizontal and the drop point is defined as the release angle.
In the case of slow moving belts where the release angle is calculated as being less than the angle of repose of the material, it is normal to reckon the release angle as being between three and five degrees greater than the angle of repose of the material. Plotting the material trajectory can now be done.
- From the drop point determined above extend a line along the inclination θ determined.
- Decide on set-out spacing.
where: t = time intervals (typically seconds) and mark out
- At each spacing along the line of the release angle drop a vertical of distance
- Join each end point to plot the trajectory of the centre of area of the material.
The upper and lower bounds of the trajectory will follow the upper and lower bounds of the material about the centre of area for a fall of about 2.5 m. Thereafter air-drag and wind may result in deflection of the material stream. Figs. 11 and 12 indicate the difference in material trajectory for fast and slow belt speeds.
Knowing the material trajectory, the flow pattern and the point of first impact can be determined. This is critical in the design of the hood. The hood directs the flow of the material towards the spoon. The material is intercepted at a tangent. The Hood should be designed such that it has the same radius of curvature at the point of impact as the trajectory, i.e. the impact angle should be as small as possible. (Fig. 13)
