1. Introduction
The Brazilian economy is strongly influenced by the behavior of the mining sector. According to the Brazilian Mining Institute [
1,
2], the contribution of the mineral production reached upwards of 4.8% of the Brazilian gross domestic product (GDP) in 2018. The GDP from mining in Brazil increased to Brazilian real-BRL 2717.82 million in the fourth quarter of 2018. Mining production in Brazil averaged 2.82% from 2003 until 2019, reaching an all-time high of 21.4% in January of 2010. The World Trade Statistical Review 2018 ranks Brazil eighth in the world’s iron ore exportation, and third in the last year, behind only India and the Russian Federation.
As result of all mining activities, large quantities of products are generated from beneficiation processes, part of which is destined for trading or later processing, while another part is designated for storage in a tailing dam. For transport, the iron ore is mixed with water to form slurry. Currently, there is a consensus that among the many different types of transportation systems, pipeline transport is the most efficient, mainly for systems with high throughputs [
3,
4,
5,
6,
7].
The design and operation of these slurry transportation systems and, in particular, the performance of the utilized pumps, strongly depend on the properties of the slurry. These properties are a function of the quantity (concentration), size distribution (granulometry), and shape of the ore particles mixed in the water. Unlike the transport of fluids with low viscosity (e.g., water), slurry transportation is much more complex and demands greater power consumption due to the requirement that the fluid velocity be kept greater than the particle deposition velocity.
As a consequence, a hydraulic transport system must be designed and executed carefully so as to achieve the minimum pumping transport cost for specific slurry characteristics. The system costs are related to efficiency in the energy conversion, wherein “efficiency” has several possible interpretations [
8]. These interpretations can be related, for example, to the availability of natural resources, to a better understanding of the processes, or to any other novel approach proposed by considering the importance of using relevant efficiency concepts in calculating the energy cost of pumping liquids.
In this sense, significant research efforts have been given towards establishing the appropriate features to ensure the efficient transport of slurries. The authors of [
9], motivated by a particularly constraining absence of water in Chile, analyzed energy efficiency in long-distance slurry transport by considering the cost of water in addition to energy cost. According to this criterion, when water is considered in the costs of a transportation system, the optimal concentration of the solids to be transported is an increasing function of the throughput. This result improves early findings, in which the weight of transported material relative to the use of water was not correctly defined [
10,
11].
An energy indicator
I, which is based on thermodynamic principles, has been recently used for the assessment of ore slurry transportation systems based on positive displacement pumps [
12] and centrifugal pumps [
13] to characterize minimum power consumption based on the above four defined variables. Two of these variables represent the solid to be transport and are used to define the physical properties of the slurry (the volumetric concentration
Cv and granulometry
D50), while the other two are used for the topological characterization of the pumping system (the pipeline length
L and diameter
D). In the aforementioned studies, the energy indicator was defined as the energy needed to move 1 m
3 of slurry according to specific operating conditions.
The most applicable relationship between these variables could be obtained using the design of experiments (DOE) (see [
14]). When simulations are considered as experiments as in the present study, it is possible to apply the same strategies existing for experiments to the simulations [
15]. The DOE has been used successfully to analyze and optimize different processes through experimental [
16] or computational studies [
17,
18]. This technique of organization and performance the experiments have an advantage over the traditional ones, requires fewer experiments to obtain the same precision in the estimation of the meta-model [
19].
Mining industries are typically located in regions with rugged topography. Consequently, the design of slurry conveyance systems should consider this fact and should be constructed with considerable care and should take into account the different situations in accordance with financial, geographical, and logistical conditions. In this regard, the present study aims to determine the best combination of the four independent variables characterizing the application of relatively short-distance slurry pumping systems to minimize the energy requirement. The methodology developed in [
12,
13] will be applied in three different slurry transportation systems. All variables are significant relative to
I with a similar trend for the three pumping systems. That is, the power requirement is lower when
L,
D50, and
Cv are at their lower levels, and when
D is at its higher level. Regarding second-order interactions, the most significant effect can be observed between
L and
D. Moreover, the surface response model obtained for the three layouts fit the numerical data satisfactorily, with differences of less than 10%. Finally, the optimization of
I showed that the minimum value was obtained for
Cv = 5%,
D50 = 100 µm,
L = 500 m and
D varying between 227 mm and 235 mm as a function of the pumping system.
4. Results and Discussion
A total of 93 simulations were carried out, and the influence of each the four factor on the energy indicator I was analysed. Later, the values of I obtained in the parametric study were used to fit the response surface models, which include the main effects of the independent variables, the effects of the interactions among them and the quadratic terms.
Figure 4 depicts the main effects that the level changes of
L,
D,
D50, and
Cv have on
I for each of the slurry piping system configurations. Points in the graphs correspond to the arithmetic mean of
I in each level of each factor; the average effect of each factor on the response is evidenced. As shown in
Figure 4, the behaviour of the response
I, due to the variation of the factors from one level to another, has the same trend regardless of the slurry piping system. Among the four variables,
L has the greatest amplitude of variation between the highest and the lowest level. The minimum values of
I are reached when
L,
Cv, and
D50 are at their lower levels and when
D is at a higher level.
These results can be best explained from a fluid mechanics point of view. The increase of L, from 500 m to 2000 m, keeping other factors equal, promotes a significant increase of approximately 60% of the indicator I in all cases investigated. By applying an energy balance it is possible to demonstrate that, when the system length increases, there is an increase in distributed head losses and, consequently, the pump must thus provide a higher energy (per unit weight) to transport the slurry to the tailings dam. The factor L has the greatest effect on the energy consumption for all of the analysed systems.
The change in pipe diameter from 200 mm to 250 mm promotes an average reduction of 22.4% of I in all simulations conducted, keeping all other factors constant. It is well-known when the pipe diameter increases, the hydraulic losses, defined by the sum of the distributed head losses and the minor head losses, decreases and, consequently, the required power by the system is lower.
A change from the lower level to the higher level of granulometry, that is, from 100 μm to 300 μm, produces an average increase of I of 24%, keeping the remaining variables constant. By increasing the granulometry, the size, and, consequently, the weight of the solid grains in the slurry also increase. To avoid the sedimentation of particles, a greater energy should be supplied to increase the slurry flow velocity.
The volumetric concentration has the smallest effect on I when compared with other factors, wherein an average increase of I of 9.2% occurs when Cv varies from its lower (5%) to higher level (10%). The influence of Cv on I can be explained by the increase of the solid volume in the slurry.
Apart from these main effects, the interactions among the factors were also investigated and it was determined that the only significant interactions on I are those between D50 and L, as well as between D and L, for all pumping systems examined.
The interaction between these two pairs of variables is shown in
Figure 5. At the top of the figure, it is possible to observe the relationship among
D50,
L and
I for the three system configurations studied. The granulometry exerts almost no influence when
L is at its higher level, but its significance rises noticeably as
L decreases to its lower levels.
At the bottom of the figure, it is possible to observe the relationship among D, L, and I for the three system configurations. The value of I diminishes as D increases and L decreases. The indicator I has a minimum value when D is approximately 225 mm for a lower value of L. The curvature of the response surface approximately for 225 mm for an L value of 500 m confirms this effect.
The prediction models based on the experiments, which respond by the response surfaces shown above, can be used to predict the response I at any point in the space limited by the levels of the factors in the design, provided that the model has an appropriate fit. The adjustment of the models was studied through normal probability graphs and the calculation of two statistical coefficients used in model adjustment, the coefficient of determination , and the standard error of the regression S.
Figure 6 illustrates the normal probability plots of the standardized residuals corresponding to the models for
I. From these graphs, the accuracy of each model can be assessed. Firstly, it should be noted that most of the residuals for all cases are low; only two of 31 samples, for each layout, have an absolute value larger than 2, which is the value typically accepted to identify outliers [
15]. Secondly, the residuals of three models do not depart substantially from a straight line in the normal probability plot, thus confirming that they are very-nearly normally distributed.
On the other hand, the value of for all experiments is higher than 98%, and the value of S is approximately zero. These values meet the appropriate values for both parameters to accept the models. Hence, the quadratic response surface models employed fit the data satisfactorily.
The predicting models are expressed by Equations (9)–(11), for the 1P, PS, and UD cases, respectively:
For Equation (9), = 0.988 and S = 0.028; for Equation (10), = 0.982 and S = 0.026; and for Equation (11), = 0.981 and S = 0.027.
It is apparent that these models have two components: the linear part, which is associated with the main effects, and the non-linear term, which corresponds to the interactions between the parameters.
Figure 7 illustrates the comparison between the simulation results and the results obtained via the reduced prediction models. As can be observed in the right side of
Figure 7, a difference of less than 10% is obtained in all cases, wherein the fitted values are compared with those obtained in the simulations. These results indicate that a good agreement exists between the models and simulations; therefore, an appropriate response for I can be obtained by using the models.
From the obtained models and considering the main objective of the present study, the optimal combination of independent variables required to minimize the indicator
I was determined under the assessed conditions and has been showed in
Table 2.
The standard interval global engine (SIGE) method, as defined by [
30], was applied to obtain the optimal combination.
Table 2 shows the optimal combinations of the independent variables for the pumping systems analysed in the range of investigated values. From these results, it is possible to observe that the optimal values for optimizing (minimizing)
I are as follows:
Cv = 5% (lower level),
D50 = 100 μm (lower level),
L = 500 m (lower level), and
D varying between 227 mm and 235 mm as a function of the pumping system. As is well-known, this value of the diameter is not a normalized value because in the mining industry he commercial pipes has a diameter of 200 mm and 250 mm. Earlier statistical analysis established that 225 mm is the optimal value, but from a practical point-of-view, it is not possible to meet this condition. Therefore, a value of 250 mm was selected. This result is in agreement with the physical foundation of the study phenomena, as previously discussed.
It is important to highlight that the desirability function is an indicator that provides an estimate of the extent to which the solution suggested by SIGE meets the requirements of the response
I. The desirability function ranges from 0 to 1, where values close to 1 correspond to a good fit between the results obtained via models and simulations [
31,
32].
Taking the global results of the optimization into account, it is possible to observe that the indicator I decreases by 11% when the layouts with two pumps (PS and UD) are compared with that of only one pump (1P). It is worth noting that in the 1P configuration, the pump has a higher operating range than the pumps used in layouts PS and UD and thus results in a direct economic impact on operating and maintenance costs. However, the results obtained are unable to explicitly demonstrate whether the PS or UD configuration meets the most effective combination of factors for minimizing I. Choosing the best configuration will likely depend upon other variables not directly associated with the factors studied here. In this case, factors related to logistical conditions in real situations should be analysed.
5. Conclusions
After a careful bibliographical review, and from the results obtained, it is possible to conclude that the slurry transportation system may be characterized by means of four independent variables, two of which are associated with the piping system (length and diameter), while the other two are associated with the physical properties of the slurry (volumetric concentration and granulometry of solid particles). The appropriate relationship amongst them, in order to minimize the power requirement of the conveyor systems, was determined through a numerical study by employing the central composite design technique (combined with the DOE) for optimizing the number of simulations required and analysing the results. The adopted values for the variables correspond to typical values used in the mining industry. To characterize the power requirement, the authors have used an energy indicator I to indicate the energy required to move a unit of slurry within specific operating conditions.
To carry out the analysis, three different slurry transportation systems were assessed. The main difference among them is the quantity of pumps and their location within the systems.
The analysis of the main effects shows that all factors are significant relative to I with a similar trend for the three pumping systems. That is, the power requirement is lower when L, D50, and Cv are at their lower levels and when D is at its higher level. Among the factors, L is the most influential factor when its value changes between its lower and higher levels. From the analysis of the main effects, the minimum values of I are reached when L, Cv, and D50 are at their lower levels and when D is at its higher level. This result has a well-founded theoretical basis and can be explained from a fluid mechanics point-of-view. The interaction among variables, i.e., the second-order interactions, shows that only the relationships between L and D50 and between D and L are significant for I in the three pumping systems investigated. When the effect on I is analysed regarding the interaction between L and D50, it is possible to observe that D50 has different behaviour as a function of the level of L. When L is at its higher levels, the influence of D50 is almost insignificant, but when L trends to lower levels, the significance of D50 is appreciable.
The most significant effect can be observed for factors L and D, resulting in a second-order interaction.
The minimum of I is obtained for D = 235 mm, D = 227 mm, and D = 229 mm for layouts 1P, PS, and UD, respectively. Since no normalized diameter of 225 mm exists, it is suggested to select D = 250 mm for a minimum value of I.
Regression models obtained for the three layouts fit the numerical data satisfactorily, with differences of less than 10%.
An interesting result of this research postulates that between the piping system variables and slurry variables, those related to the piping system are the significant ones.
The results of this work show that through a careful design of the slurry transportation system, numerous benefits can be achieved regarding sustainable development, such as a reduction in water usage and a higher transport of solid throughputs.
Regardless of the reduced interval of values for the variables, the results confirm that the new methodology presented herein, which combines numerical simulation and the design of experiments, has a huge potential to offer relevant information, thereby allowing professionals to establish the appropriate combination of variables to minimize energy consumption in medium-length slurry transportation systems.