presentation

September, 2014.

Introduction

This section presents a technical report of the coal load out bin and other equipment that are presented in figure 1. Among the issues to be addressed in this report are the ideal dimensions of D, H, h1-h3, θ1, V, X,L and x, hc and M. Trigonometry will be vital in meeting this objective (Riley, Hobson, & Bence, 2006).

These dimensions are explained below.

· D the diameter of the load out bin (in m)

· H the total height of the load out bin and frame (in m)

· h1 to h3 the heights of each section of the load out bin (in m)

· θ1 (theta 1) cone and conveyor angle (in degrees)

· V the volume of the load out bin (in m3)

· X, L and x lengths which define the conveyor (in m)

· hC the clearance over the road (in m)

· M the maximum mass of coal in the load out bin (in tonnes)

Figure 1: Proposed coal load out equipment

To begin with, D will be assumed to be at minimal value, which has already been stated as being 8m. The selection of D as 8m is because this falls within the allowable minimum and maximum limits, making it acceptable in terms of size and volume that it will carry. The selection of this dimension will affect the length of x as presented below.

If D is 8m, this will mean that:

· The radius of section B in figure 1 will be , making it

· x =

· = 8 – 4

· = 4 m

Other dimensions for x and D, depending on the selected size of D are presented in the table below

Table 1: Values of x and D

The values above have been established from the expression

In order to obtain the value of H, trigonometric equations will be used to relate these distances (Bolton & Bolton, 2012). Assuming that the load out bin is perfectly vertical and the surface on which it stands is horizontal, the intersection between H and X will be at a right angle. Therefore, the relation between H and X will be as follows:

Where tan 1 =

Therefore, H = X*

This is because the angle ay the top of load out bin is the same as the angle of intersection between the lift conveyor and the horizontal surface.

From the above, L will be expressed as

In order to determine h1-h3, it will be represented as a difference between H and (5m + d). Where d, as mentioned in figure 1, is the chute diameter, which is 1m. This will be represented in equation form as:

= X*

But d + 1m

X*

The cone conveyor angle θ1 (theta 1), as has already been mentioned earlier, will be obtained by the formula below:

Tan 1 =

Therefore, θ1 = tan-1

V, which is the volume of the load out bin, will be calculated by calculating the volumes of sections A, B and C in figure 1 separately.

The volume of A, which will be conical, will be determined by the formula () (Riley, Hobson, & Bence, 2006).

Figure 2: Volume of a conical structure

Source: Riley, Hobson and Bence (2006)

Applying the above formula, the volume of A will be (). This can be presented as:

VA = ()

Section B is cylindrical and thus, its volume will be estimated by the formula

Thus, VB =

Finally, the volume of section C, being conical too, will be estimated by the formula ()

Thus, VC = ()

The total volume of the whole load out bin will therefore be VA + VB + VC

VA + VB + VC = () + + ()

= () + + ())

Thus, V= () + + ())

In order to establish the length of hC as presented in figure 1, trigonometric relations will also be used.

Tan θ1=

=

But tan θ1 =

Therefore,

And hC =

Finally, M which is the maximum mass of the coal in the load out bin will be calculated with the value of D at 10, which results in the minimum value of x. (see table 1).

Therefore, applying the formula of V

V= () + + ())

Replacing D/2 with 5, the volume will be as follows:

V= () + + ())

V=

=

Mass will therefore be V*density, which is 810 kg/m3*

However, at maximum volume, θ1 is at 10◦

Therefore, tan θ1= 0.17633

But h1=

= 5*0.17633 = 0.8816 m

Then, given that θ2= 45◦, h3 = D/2 = 5m

V=

V=154 + 78.54h2

Mass = 810 (154 + 78.54 h2)

= (124740 + 63617.25 h2) kg

= (124.74 + 63.617 h2) tonnes

Conclusion

The expressions that have been presented above provide guidance on how the dimensions of the different elements of section 1 can be obtained. As presented in the equations above, most of the dimensions are affected the value of others. An estimation of the maximum mass that can be supported by the coal bin is essential in guiding the materials that can be used in the provision of support for the entire load.

References

Bolton, W. & Bolton, W., 2012. Mathematics for Engineering. New Jersey: Routledge.

Riley, K.F., Hobson, M.P. & Bence, S.J., 2006. Mathematical Methods for Physics and Engineering: A Comprehensive Guide. Chicago: Cambridge University Press.