for shovel-truck systems with a crusher and conveyors

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Tom 24 2008 Zeszyt 4/3

JACEK M. CZAPLICKI*

The analysis and calculation procedure

for shovel-truck systems with a crusher and conveyors

Introduction

A developed method of comprehensive analysis and calculation of shovel-truck systems was first described by Czaplicki in 2006, and this method was modified and enlarged significantly in his monograph of 2008. This scheme also allows for a new approach towards selected problems of shovel-truck systems considered in Czaplicki’s textbook of 2004;

problems such as application of an inclined hoist in the pit or employment of an in-pit-crusher together with conveyors.

One of the possible modifications in the shovel-truck machinery system is the application of an in-pit movable or mobile crusher together with a certain number of conveyors to take the stream of crushed rock. The main point is that the reason for such an application is reduction of the number of haulers applied in the system, which also means a decrement in the hauling costs, which are otherwise usually very high. However, the costs of crushing and conveying must be included in the whole economic consideration of mine transportation.

If the cost of crushing is low, this kind of solution is advisable.

If economic analysis indicates that application of the crusher should be considered, the main question is how to arrange the system: shovel-truck-crusher-conveyors (for short:

S-T-C-C). Several important questions arise, such as: How many trucks should operate in the pit? How many trucks should be in reserve? What should be the productivity of the crusher?

What should be the transportation capacity of the conveyors? And so on. Generally, we have to know how to analyse and calculate the system, and how to solve its design problems.

* Mining Mechanization Institute, Silesian University of Technology, Gliwice, Poland.

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The purpose of this paper is to present the procedure of analysis and calculation of the S-T-C-C system in the light of the new method described comprehensively in the 2008 monograph.

1. Formulation of the problem

Let us presume that there is an open pit mine in which a machinery system of the shovel-truck type is in operation. Therefore we know the number of working machines and their main operation/exploitation parameters. The mine has decided to change the system of ore hauling, applying an in-pit crusher and a system of belt conveyors to deliver crushed rock to a peripheral dressing plant unit. The problem is how to design this new component of the machinery system and to indicate what kind of changes should be made in the “old” system.

We can describe this problem in mathematical form.

We have the system:

S : < H = H_wÈ H_o: n = n_w+ n_o, A_k;Q: A_w; Q; b = 1, 2, ..., m; r N: k >

for which operation/exploitation parameters are as follows:

P(S,D) : < B_k, Z’, T_t, T_d;t; B_w>

Let us read these notations.

S. The system of power shovels H consists of n machines of the steady-state availability A_k, where n_wunits load waste and n_ounits load ore. The system of dumpersQ of payload Q possesses m trucks fulfilling hauling duties, and r units are in the reserve. The steady-state availability of trucks is A_w. The system of repair stands N consists of k repair units, and we assume that is large enough that the rare event of a queue of failed trucks waiting for repair can be ignored. Thus the problem of k will be non-existent.

P(S,D). The second set of parameters describes the operation/exploitation process of the machinery system that is a function of properties of equipment involved and decisions D made by the truck dispatcher. These parameters are: the accessibility coefficient of shovels B_k, the mean loading time Z’, the mean time of truck travel (haul-dump-return) T_t, the mean time of unloading T_d, the accessibility coefficient of trucks B_wand the mean loading time for spare loaders, which ist times longer than for shovels.

During the system operation – if no priority to the type of material transported is implemented – it makes no difference to which loader a given empty truck is directed and to which dumping point a given full hauler drives. If this new component is an item of the system one part of the truck route will be strictly determined: all trucks loaded by broken ore will always be directed to the crusher. We presume that the system was rationally selected (see for instance chapter 5.4 of monograph 2006 or chapter 8 of monograph 2008), i.e. the system structural parameters < m, r, k > are chosen in the proper way. Installation of a new subsystem makes no difference to the system of loading machines. The system of trucks,

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in turn, is divided into two subsystems but in a stochastic way. It is not strictly ascribed that a given truck is only connected with haulage of a given type of material. Sometimes the truck will transport waste, and sometime ore. Nevertheless, we presume that the system of haulers is divided into two subsystems numerically – a certain number of trucks will on average serve the ore loading machines and a certain number on average will serve machines loading the waste. Because the whole truck system was designed for longer distances and has m units, we in fact have to recalculate this system from the beginning. It will be m_wtrucks hauling the waste and m_ounits hauling the ore to the crusher. We obviously expect that m_w+ m_o< m.

A new size of the reserve of the power r_n= r_w+ r_o, will be forecast: r_n< r. Therefore the repair shop should not be heavily loaded as before.

To design, analyse and calculate the new system we need information on the following six parameters: < T_tc, y, A_cr, B_cr, A_c, u > where: T_tcis the mean time of truck travel serving the crusher, y is the number of trucks that can be unloaded simultaneously to the crusher, Acris the steady-state availability of the crusher, B_cr is the accessibility coefficient, A_c is the steady-state availability of the conveyor and u is the number of conveyors required to connect the outlet crusher with the peripheral dressing plant unit. Notice that the information on the value of y means that y truck payloads can be given in time T_dto the crusher without any choke. Thus, the relationship between the truck payload, dump time and crusher capacity should be rationally matched.

The procedure of analysis and calculation of the system can be made according to the following scheme:

1. Recalculation of the whole system S-T-C-C considering the trucks.

Further part of considerations concerns the ore system alone.

2. Construction of the probability distribution of the number of loading machines able to load with information as to how many power shovels are up.

3. Construction of the probability distribution of the number of trucks in work state for all power shovels up.

4. Construction of the probability distribution of the number of trucks at shovels – assumption: no queue at the crusher.

5. Calculation of conditional system parameters.

6. Calculation of unconditional system parameters.

7. Evaluation of system productivities.

Points 1 – 7 can be calculated using the procedure presented in the 2008 monograph.

8. Calculation of the potential stream of ore that can be delivered to the crusher:

S_pcr= 60yQ/T_d

This output could be obtained if there will always be trucks to dump and the whole continuous subsystem is totally reliable.

9. The steady-state availability A_concalculation of the continuous subsystem: crusher and conveyors.

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10. Calculation of the stream of ore that can be delivered to the crusher taking into account the accessibility of the crusher B_cr(maintenance required)

S_con= A_conB_crS_pcr

11. Construction of the probability distribution of the number of trucks at the crusher.

12. Calculation of the whole system parameters.

13. Conclusions, recommendations and remarks.

2. Case of study

The exploitation process of a shovel-truck system is complicated, and many components must be taken into account if the appropriate level of considerations is to be attained. It has been pointed out (e.g. Czaplicki 2008b) that each exploitation process of this type of machinery system is a unique one, and it is much better to consider each process separately.

Let us consider, for example, the following machinery system:

S : < H = H_wÈ H_o: n_w= 4,n_o= 3, A_k= 0.864, B_k= 0.830; Q = 214t Q : A_w= 0.760;

b = 1, 2, ..., m = 97; r = 20 >

for which

P(S,D) : < B_k= 0.830, Z’ = 1.7 min, T_t= 26.0 min, T_d= 0.9 min; t = 2.9; B_w= 0.880 >

Additional system parameters are:

< T_tc= 9.0 min, y = 2, A_cr= 0.820, B_cr= 0.900, A_c= 0.998, u = 3 >

1. We start to recalculate the system¹.

— Waste subsystem

– The minimum mean number of trucks in work state: h = A_kB_kn_w[(Z' + T_t)/Z'] = 46.7 – The minimum number of trucks needed: h/A_wÞ 62

– Applying the Maryanovitch model and enlarging the result on trucks directed to the pit by 10% we have: < 1.1 m = 55; r = 12 >

— Ore subsystem

– The minimum mean number of trucks in work state:

h=A B n_k _{k o}[(Z¢ +T_tc) /Z¢ =] 13 5.

1 We can make a preliminary assessment of the system. If the power shovels are totally reliable they can load approximately 16 trucks in 9 minutes, whereas the crusher can take 20 payloads of trucks in this time.

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– The minimum number of trucks needed: h/A_w Þ 18

– Applying the Maryanovitch model and enlarging the result by 10% we have:

< 1.1 m = 18; r = 2 >

Conclusion:

After application of the crusher and conveyors the number of trucks required in the system is 30 units less than before.

2. The probability distribution of the number of shovels able to load is described by the binomial function.

If it is presumed that spare loaders are easily accessible (front-end loaders) the reliability of the loading system is determined fully by the reliability of power shovels. The distribution for the case considered is therefore:

P_{k d}⁽₍^zd_{=0 3}⁾ _{. )}= 0.023 P_{k d}⁽₍^zd_{=1 2}⁾ _{. )}= 0.172 P_{k d}⁽₍^zd_{=2 1}⁾ _{. )} = 0.436 P_{k d}⁽₍^zd_{=3 0}⁾ _{. )}= 0.369

where P_{k d i n i}⁽₍^zd₌⁾ _, _-₎ is the probability that i power shovels are able to load and n–i spare loaders, i = 0, 1, …, n.

3. The probability distribution of the number of trucks in work state P_wj^{( )}^p for the ore system obtained from the Maryanovitch model is given in Table 1.

TABLE 1 The probability distribution of the number of trucks in work state for <18, 2>

TABELA 1 Rozk³ad prawdopodobieñstwa liczby ciê¿arówek w stanie gotowoœci do pracy dla <18, 2>

j 9 10 11 12 13 14 15 16 17 18

P_wj^{( )}^p 0.002 0.007 0.022 0.051 0.100 0.159 0.201 0.199 0.148 0.111

4–7. Employing the Sivazlian and Wang model with two randomisations (the number of trucks in work state and the number of power shovels able to load) the probability distribution of the number of haulers at loading machines can be obtained. Based on this characteristic, further interesting probability measures can be determined. The outcomes of these calculations are given in Table 2 with a list of keynotes.

8. The potential stream of ore that can be delivered to the crusher S_pcr = 28533 t/h.

9. The steady-state availability calculation of the continuous subsystem. Knowing the relationship between the steady-state availability and the failure ratek that is k = (1 – A)/A we can calculate the steady-state availability A_conof the continuous subsystem according to the second principle of reduction of series systems of Markov type systems:

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A_con= (1 + uk_c+k_cr)^–1= 0.816

10. The stream of ore that can be delivered to the crusher is:

S_con= A_conB_crS_pcr= 20955 t/h

TABLE 2 Parameters of the ore systemS

TABELA 2 Parametry systemu rud S

Conditional parameters Conditional probabilities

i P_ki⁽^zd⁾ T d_z( , )t Xi Xi/3 p0d p_1d p_2d p_3d p_pd Dd shovels

able to load

min trucks

trucks/lo ading machine

min

3 0.369 1.70 2.6 0.9 0.002 0.098 0.411 0.325 0.164 0.1

2 0.436 2.78 5.7 1.9 0.004 0.049 0.116 0.830 1.0

1 0.172 3.85 8.2 2.7 0.004 0.017 0.979 0.6

0 0.023 4.93 9.7 3.2 0.003 0.997 0.1

Unconditional parameters Unconditional probabilities

TC E_wlk E_wk q Wefk W_efw p₀ p1 p2 p3 pp

min trucks trucks trucks loaded trucks/h

truck work cycles/h

13.41 2.85 15.2 1.7 65.5 68.0 0.000 0.038 0.174 0.174 0.614

Xi– the conditional mean number of trucks at d shovels able to load Xi/3 – the conditional mean number of trucks at 3 machines able to load pbd– the conditional probability that there are b trucks at d loading machines

ppd– the conditional probability that there are more than n trucks at n loading machines Dd– the conditional time loss parameter

TC –the mean truck work cycle including losses Ewlk– the mean number of loaded trucks

Ewk– the mean number of trucks at loading machines q – the mean number of trucks per one loading machine Wefk– the loading machine system effective productivity Wefw– the truck system effective productivity

pb– the probability that there are b trucks at d loading machines; b < n pp– the probability that there are b trucks at d loading machines; b > n

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11. Construction of the probability distribution of the number of trucks at the crusher can be made by applying the Sivazlian and Wang model. Compared to the previous case of application of this model it is significantly simplified, giving following results:

p0= 0.132 p1= 0.701 p2= 0.155 p_p= 0.012

The subscript indicates how many trucks are at the crusher, whereas subscript p means “>2”.

The average number of trucks at the crusherX = 1.04 and the time lost in the truck work cycle due to standing in a queue for loadingD = 0.0 min.

12. Let us now calculate the basic measures of system efficiency. We have three subsystems and we are able to evaluate their productivity.

The shovel productivity is estimated at:

W_k= Q W_efk@ 14017 t/h The truck system productivity:

W_w= Q W_efw@ 14552 t/h The stream of ore flowing through the crusher is:

60 Q X/T_d@ 14837 t/h

Obviously, we intuitively expect that all figures will be approximately the same, taking only into account differences in numbers due to inevitable rounding up during calculations.

This is not right, however. In Sivazlian and Wang’s original paper it is clearly stated that the method of calculation presented only gives approximate solutions. It is also necessary to take into consideration the fact that the heavy traffic condition must be fulfilled (see for instance Czaplicki 2008a). Following short simple calculations it is easy to perceive that in all cases this condition is fulfilled, but with different power. It generates varying accuracy in the outcomes obtained. We may circumspectly assume that the ore system productivity will be approximately 14000 t/h.

13. This productivity result must be compared with the ore productivity of the “old”

system. It is also important to evaluate the size of savings due to the fact that instead of 97 trucks that should fulfil the hauling task and 20 trucks in the reserve we now have 24 trucks less that will be directed to the pit and 6 units less in the reserve. This is quite a high number.

However, the operation costs of the continuous subsystem must be estimated and taken into account in the general economic considerations.

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REFERENCES

[1] C z a p l i c k i J.M., 2008a – Analysis and calculation of shovel-truck systems. Forthcoming.

[2] C z a p l i c k i J.M., 2006 – Modelling of the exploitation process of shovel-truck system. D.Sc. Dissertation.

Papers of Silesian University of Technology, 1740, Gliwice (in Polish).

[3] C z a p l i c k i J.M., 2004 – Elements of theory and practice of cyclic systems for mining and earthmoving.

Silesian University of Technology, Gliwice (in Polish).

[4] C z a p l i c k i J.M., 2008b – A calculation method for a shovel-truck system with an inclined hoist of TruckLift type. (Submitted to the International Journal of Mineral Resources Engineering).

[5] S i v a z l i a n B.D., W a n g K.H., 1989 – System characteristics and economic analysis of the G/G/R machine repair problem with warm standbys using diffusion approximation. Microelectronics & Reliability, 29, 5, 829–848.

PROCEDURA ANALIZY I OBLICZANIA DLA SYSTEMÓW KOPAREK CZERPAKOWYCH Z KRUSZARK¥ I PRZENOŒNIKAMI

S ³ o w a k l u c z o w e

Koparka czerpakowa, mechanizacja, stochastyka, kruszarki, przenoœniki taœmowe

S t r e s z c z e n i e

Niedawno opracowano metodê analizy i obliczania systemów koparek czerpakowych, pozwalaj¹c¹ na wszech- stronne uwzglêdnienie prawie wszystkich g³ównych problemów zwi¹zanych z tego rodzaju systemami maszyn.

Mo¿liw¹ modyfikacj¹ systemu jest zastosowanie mobilnej lub ruchomej kruszarki i przenoœników, która zapewni dalszy przep³yw pokruszonej rudy. Taki system jest po³¹czeniem systemu cyklicznego i ci¹g³ego. Opracowanie przedstawia procedurê modelowania, analizy i obliczania tego rodzaju systemów, wraz z rozleg³¹ orientacj¹ problemów stochastycznych zwi¹zanych z procesem eksploatacji systemu. Bior¹c to pod uwagê, wyci¹gniêto pewne wnioski i poczyniono uwagi o decyduj¹cym znaczeniu.

THE ANALYSIS AND CALCULATION PROCEDURE FOR SHOVEL-TRUCK SYSTEMS WITH A CRUSHER AND CONVEYORS

K e y w o r d s Shovel-truck, mechanization, stochastic, crushers, conveyors

A b s t r a c t

The method of analysis and calculation of shovel-truck systems was recently developed, allowing for comprehensive considerations of almost all major problems connected with this type of machinery system.

One possible modification to the system is the application of an in-pit mobile or movable crusher and conveyors ensuring further broken ore flow. Such a system is a combined cyclic and continuous one. In this paper the procedure of modelling, analysis and calculation of this type of system is presented together with the vast orientation of stochastic problems associated with the system exploitation process. Following the considerations, some conclusions are drawn and some vital remarks made.