Design of high performance thickener hydrocyclones using robust optimization

Journal Pre-proof Design of high performance thickener hydrocyclones using robust optimization Vitor Alves Garcia, Fran Sérgio Lobato, Luiz Gustavo Martins Vieira PII:

S0920-4105(20)30232-1

DOI:

https://doi.org/10.1016/j.petrol.2020.107144

Reference:

PETROL 107144

To appear in:

Journal of Petroleum Science and Engineering

Received Date: 3 December 2019 Accepted Date: 28 February 2020

Please cite this article as: Garcia, V.A., Lobato, Fran.Sé., Vieira, L.G.M., Design of high performance thickener hydrocyclones using robust optimization, Journal of Petroleum Science and Engineering (2020), doi: https://doi.org/10.1016/j.petrol.2020.107144. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.

Design of High Performance Thickener Hydrocyclones using Robust Optimization Vitor Alves Garcia, Fran Sérgio Lobato and Luiz Gustavo Martins Vieira* School of Chemical Engineering, Federal University of Uberlândia, Uberlândia 38408902, Bloco K – Santa Mônica, Uberlândia, MG, Brasil

*Corresponding author: [email protected]

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Abstract: In the literature, a large number of works about the hydrocyclones geometry optimization to obtain high performance separators can be found. However, in these works, during the optimization process, no uncertainties in model, design variables and/or parameters are considered. In these cases, a small variation in the design variable vector can result in a meaningful change on the theoretical optimal design as represented by the minimization of the corresponding vector of objective functions. In this contribution, the Effective Mean Concept is associated with the Multi-Objective Optimization Differential Evolution to obtain solutions less sensitive to perturbations in the design variables during the thickener hydrocyclones design. For this purpose, the proposed multi-objective optimization problem considers the determination of geometric variables to maximize hydrocyclone’s total efficiency (η), to minimize the underflow-to-throughput ratio (RL) and to minimize the Euler number (Eu). The robust results are compared with those obtained considering nominal context (without robustness). The results showed that the tested hydrocyclones present good thickener equipment and that the robust Pareto’s curve presents less diversity in comparison with the nominal solution. Keywords: Hydrocyclone design; Multi-objective optimization; Robust optimization; Differential Evolution; Effective Mean Concept.

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Nomenclature Bδ (p)

δ-neighborhood of point p

Yiexp

Cr

crossover probability

exp Ymax

d63.2

RRB model parameter (m)

Dc Di Do Du Eu

cylindrical section diameter (m) feed inlet diameter (m) overflow tube diameter (m) underflow hole diameter (m) Euler number

Eu exp

Euler number experimentally measured Euler number calculated by the fitted model objective function of models adjustment general objective function mean effective of objective function disturbance factor

Eu mod f fo foeff F h H l L n N Ngen Ns p q r2 RL RL exp RL mod X Y Yical

Y

response experimentally obtained highest experimental response value mean value of the model calculated responses

Greek symbols β model parameters’ matrix δ robustness parameter ∆P pressure drop in hydrocyclone (Pa) εEu relative error in relation to Euler number εRL relative error in relation to underflow-to-throughput ratio εη relative error in relation to total efficiency η total efficiency η exp total efficiency experimentally measured η mod total efficiency calculated by the fitted model θ conical section angle (º)

cylindrical section length (m) conical section length (m) vortex finder length (m) hydrocyclone total length (m) RRB model parameter population size maximum number of generations number of generated neighbors generic point in relation to population generic point in relation to δneighborhood determination coefficient underflow-to-throughput ratio underflow-to-throughput ratio experimentally measured underflow-to-throughput ratio calculated by the fitted model vector of coded geometric variables model’s response response calculated through the model 3

1 Introduction Hydrocyclones are equipment frequently used in industry to accomplish solidliquid and liquid-liquid separation. These equipment basically consist of a cylindrical part coupled to a conical section. In the cylindrical part, there’s a tangential feed duct and, also, a tube at its top, with an extension to the inside part of the equipment, called overflow. The conical section presents a hole at its bottom, called the underflow [1]. These equipment use the centrifugal field, originated by the rotational movement of the suspension, to accomplish the separation between the phases or the classification of solid particles, based on their densities and sizes. The bigger and denser particles are more affected by the centrifugal action, shocking against the separator’s walls and being collected in the underflow, while the smaller and less dense particles are dragged by most of the fluid, which reverses its vertical flow direction and goes out through the overflow pipe. The hydrocyclone present some advantages when compared to other solid-liquid separation equipment, such as filters and centrifuges. Among these benefits, there are the absence of mobile parts, small size, low maintenance cost, high versatility, large operational range, high efficiency, small cut size and low energy consumption [2]. For these reasons, these separators are used in several areas, such as the mineral [3,4], chemical [5,6], petroleum [7,8], food [9–11] and biotechnology industries [12,13]. Specifically, in the drilling of wells in oil and gas industry, a drilling fluid is used to continuously remove the rock fragments produced as the drill penetrates the soil. Due to the expensive cost of this fluid and the environmental regulations, it is necessary to recover the drilling fluid in a way it can be reused in the well drilling process [14,15]. For this purpose, the oil and gas rigs present a solids control system, which consists of a

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set of solid-liquid separation equipment with the goal of recovering the drilling fluid. The hydrocyclones are commonly found among the equipment of the solids control system, and they play an important role in the separation of solids from the drilling fluid. Moreover, the major challenge of the solids control systems is to remove the rock fragments from the drilling fluid with a minimal liquid loss [8]. In this case, the use of thickener hydrocyclones, which present low underflow-to-throughput ratio (RL), can be favorable to minimize the fluid losses. The performance of a hydrocyclone is closely related to proportions between the different parts of its geometry, and, depending on these ratios, these equipment can be adapted to several applications in industry [16,17]. Thus, there are many works with the goal of obtaining enhances in hydrocyclone’s performance through modifications in its geometry [2,3,7,8,16–25]. In design context, various works consist in the application of optimization techniques to obtain the optimal geometric configuration of hydrocyclone that leads to a better equipment performance, for a specific application [2,7,16,21,26]. Among these, it is important to emphasized the good results achieved by Silva et al. [16,21]. In these works, in order to obtain a high performance hydrocyclone, two mono-objective problems were formulated (maximize the total efficiency (η) and minimize the Euler number (Eu), respectively). However, none of these works considered the presence of uncertainties during the optimization process. Frequently, during engineering system design, the model, the design variable vector and the parameter vector are considered free from errors, i.e., they do not contain uncertainties. However, more realistically, small variations in the design variable vector can cause significant variations in the vector of objective functions. As mentioned by

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Ritto et al. [27], the process of modeling engineering systems introduces two types of uncertainties: i) uncertainties related to the parameters of the model, such as geometrical and constitutive parameters (data uncertainties), and ii) uncertainties due to the proposed model. As consequence, the system to be optimized can be very sensitive to small changes in the design variable vector, and thus, small variations in this vector can cause significant changes in the vector of objective functions [28]. In this context, it is important to determine a methodology that produces solutions less sensitive to small variations in the design variables vector. Solutions with this characteristic are called robust solutions and the procedure to find these solutions is named Robust Optimization [29]. During the hydrocyclone operations, it is desirable to achieve high efficiencies, with a low underflow-to-throughput ratio (RL) and a low energy consumption, which is directly related to the Euler number. Thus, the search for hydrocyclone’s geometric configurations which conciliate these three objectives constitutes a multi-objective optimization problem. In this contribution, the design of high performance thickener hydrocyclones considering the influence of uncertainties is studied. For this purpose, the Effective Mean Concept (EMC) [30] is associated with the Multi-objective Optimization Differential Evolution (MODE) algorithm [31], to optimize four test cases (maximize η and minimize RL; maximize η and minimize Eu; minimize RL and minimize Eu; maximize η, minimize RL and minimize Eu, simultaneously). In each test case, the dimensions of the equipment are considered as the design variables. Furthermore, experiments with hydrocyclones similar to those obtained nominally (without robustness) and robustly are carried, so the performance and sensitivity of the equipment obtained by both manners can be observed. The organization of this article is

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as follows. Section 2 introduces briefly the EMC and the MODE algorithm. Sections 3 and 4 present the methodology and results, respectively. Finally, the conclusions are presented in the last section.

2 Basic Concepts 2.1 Effective Mean Concept The influence of robustness during the optimization process can be inserted considering new restrictions and/or new objectives (relationship between the mean and the standard deviation of the vector of objective functions) and probability distribution functions for the design variables and/or objectives. As an alternative to these classical formulations, Deb and Gupta [30] extended the Effective Mean Concept originally proposed for mono-objective problems to the multi-objective context. In this approach, no additional restriction is inserted into the original problem. Thus, the problem is rewritten as a mean vector of original objectives. In this approach, the effective value of an individual p is calculated as:

fo eff ( p, δ ) =

1 Bδ ( p )

∫

q∈Bδ ( p )

fo(q)dq

(1)

In Eq. 1, foeff corresponds to the objective function mean effective of point p. The term Bδ (p) corresponds to the surroundings of point p, whose size has a δ magnitude. The term |Bδ (p)| is the hypervolume of this δ-vicinity. Furthermore, the point q represents a generic neighbor of point p, within the limits of Bδ (p). Finally, fo corresponds to the original objective function of the problem. In addition, the integral of Eq. 1 is calculated using numerical techniques, such as the Monte Carlo method or Latin Hipercube method, through the random generation of a Ns number of neighbors within the limits of Bδ (p).

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In this case, for a multi-objective optimization problem, the procedure involves optimizing all the objective function mean effectives, simultaneously, as shown in Eq. 2.

min

( fo

eff 1

( x, δ ), fo2eff ( x, δ ), ... , fomeff ( x, δ ) )

(2)

Where m is the number of objective functions.

2.2 Multi-objective Optimization Differential Evolution In order to solve the multi-objective problem associated with the hydrocyclone design, the Multi-objective Optimization Differential Evolution (MODE) algorithm [31,32] is considered. In general, this approach consists of the extension of the Differential Evolution (DE) algorithm, proposed by Storn and Price [33], to multiobjective context. The main steps of the MODE algorithm are: 1. An initial population of N solution vectors is generated and the individuals are evaluated according to their objective function values. 2. The crossover operator is applied, generating new individuals through vector operations between the original population individuals, and these operations are weighted by a disturbance factor (F). The new vector values are accepted or rejected depending on a crossover probability (Cr). Then, the population has 2N individuals. 3. The solutions are classified by ranks, in a way that the non-dominated solutions of the entire population are defined as rank 1, the non-dominated solutions among the remaining population (disregarding the solutions already classified in rank 1) are defined as rank 2, and so on until all solutions are classified in a rank. 4. The solutions which are too close to each other are eliminated through the truncation operator, with the objective of preserving the diversity in the population.

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5. The N best ranked solutions are selected to compose the new generation of the population, while the other individuals are discarded. 6. The steps 2 to 5 are repeated until a maximum number of generations (Ngen) is reached. More details about the mathematical development of the MODE algorithm can be found in Lobato [31].

3 Methodology The proposed methodology in this work consists of three steps. The first one corresponds to model estimation of hydrocyclone’s responses as functions of its geometrical variables. The second step consists of nominal and robust optimizations of the hydrocyclone’s dimensions. Finally, the third step considers the experimental verification.

3.1 Models Estimation The models considered for each response (η, RL and Eu) were estimated as functions of the geometrical ratios of hydrocyclone. The considered ratios were Di / Dc, Do / Dc, L / Dc and θ, where Di, Do and Dc correspond, respectively, to the diameters of feed inlet, overflow pipe and hydrocyclone cylindrical part, L corresponds to the total length and θ is the conical section angle. These variables are presented in Fig. 1.

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Fig. 1. Schematic representation of a hydrocyclone.

In Fig. 1, the vortex finder length is indicated by the symbol l, the underflow hole diameter is represented as Du and the lengths of cylindrical and conical parts are designated as h and H, respectively. To estimate the models, experimental data fitting was considered, using the results obtained by Salvador [34], who performed a central composite design (CCD) with 25 experiments and 4 repetitions in the central point. The dimensional ratios of hydrocyclone were encoded (Xi, i=1, …, 4) as shown in Eqs. 3, 4, 5 and 6. X1 =

( Di / Dc ) − 0.21 0.05

(3)

X2 =

( Do / Dc ) − 0.27 0.05

(4)

( L / Dc ) − 5.8 1.1

(5)

X3 =

10

X4 =

θ − 14.5

(6)

3.3

where X is the vector of coded geometrical variables. The fitted models present the form of Eq. 7, where Y corresponds to the response variable (η, RL or Eu) and β is the parameters’ matrix of the model to be fitted. 4

4

4

i =1

i =1

4

Y = β 0 + ∑ βi X i + ∑ βii X i2 + ∑∑ βij X i X j

(7)

i =1 j >i

The parameters’ matrix β for each model was determined by using DE through the minimization of the objective function shown in Eq. 8. This equation corresponds to the sum of quadratic differences between the experimental values and model calculated responses, weighted by the square of the highest experimental value of each response exp ( Ymax ).

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f =∑ i =1

(Yi exp − Yi cal ) 2 exp 2 (Ymax )

(8)

In order to measure the quality of solution, the determination coefficient (r2) is calculated for each model:

∑ (Y 29

r2 = 1−

i =1 29

i

exp

− Yi cal )

∑ (Y − Y ) cal

i =1

2

2

(9)

i

where Yiexp and Yical are the responses obtained experimentally and calculated by the model, respectively, and Y corresponds to mean value of the model calculated responses.

3.2 Robust optimization of hydrocyclone The adjusted models for each response (η, RL and Eu) were considered to formulate four test cases, which are: 11

•

Case 1: Maximize η and minimize RL;

•

Case 2: Maximize η and minimize Eu;

•

Case 3: Minimize RL and minimize Eu;

•

Case 4: Maximize η, minimize RL and minimize Eu.

In this case, the MODE+EMC approach is considered as methodology to obtain the robust solution. For each test case considered in this section, the design variables are the coded geometrical ratios (X1, X2, X3 e X4). The design space for these variables is defined as: 1.66 ≤ Xi ≤ 1.66 (i=1, …, 4). It is important to mention that this domain is set to satisfy the validity range of experiments performed by Salvador [34]. In order to formulate an optimization problem more realistic from the industrial point of view, the following constraints were defined: 70% ≤ η ≤ 100%

(10)

0% ≤ RL ≤ 10%

(11)

0 ≤ Eu ≤ 5000

(12)

Finally, the geometric constraint shown in Eq. 13 was defined to avoid obtaining an impractical solution, i.e., the total length of the hydrocyclone must be greater than the conical part length, which is geometrically related to the θ angle of this part: L>H

(13)

It is important to mention that in the first three test cases, the objective that was not used as objective is used as selection criterion. For example, in the first case study, the Euler number was used as the criterion for choosing a hydrocyclone (postprocessing criterion).

3.3 Experiments

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3.3.1 Tested hydrocyclones After the optimizations, some of the obtained hydrocyclones were selected to perform experimental tests in order to verify their real performance. In cases 1, 2 and 3, the chosen hydrocyclones were the ones in each Pareto’s curve (nominal and robust) with the best value of the non-optimized response. In other words, in case 1, as the objectives were to maximize η and minimize RL, the selected hydrocyclones were the ones with the lowest Eu value in each curve. Similarly, the selected hydrocyclones in case 2 were the ones with the lowest RL in each curve, and, in case 3, the chosen equipment were the ones with the highest η. In the Pareto’s curves of case 4, the chosen equipment were the ones which presented the best value of each response. Hence, three hydrocyclones were selected in each curve, one being that with the highest η of the curve, another being the one with the lowest RL and the last one being the one with the lowest Eu. The hydrocyclones were constructed using the parts available in the Laboratory of Separation and Renewable Energy of the Federal University of Uberlândia. Thus, the equipment considered in each experiment present dimensions close to the optimized hydrocyclones. Furthermore, all the tested hydrocyclones had their cylindrical part diameter (Dc) equal to 30 mm, the underflow hole diameter (Du) equal to 5 mm and the vortex finder length (l) equal to 12 mm.

3.3.2 Particulate Material The particulate material used in this work was quartz. The material density was determined by the technique of helium gas picnometry and its estimated value is equal to 2740 kg/m3.

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The particle size distribution of the particulate material was determined by laser diffraction analysis. The model that best fits the results is the Rosin-Rammler-Bennet (RRB). The determination coefficient obtained in the model fitting has a value equal to 0.98912 and the values of the parameters d63.2 and n are equal to 17.72 μm and 1.09, respectively.

3.3.3 Experimental apparatus and procedure The experimental apparatus is schematically represented in Fig. 2. It consisted of a suspension storage tank of 0.25 m3 capacity, a mechanical stirrer of 1.1 kW of power, a centrifugal pump (3.7 kW), an electromagnetic flow meter Coriolis with a measuring range from 0.0833 to 0.5000 kg/s, a digital pressure gauge (0 to 700 kPa) and, finally, the hydrocyclone. The suspension used in the experiments consists of a quartz and water mixture, with a volumetric concentration equal to 1%. The experimental tests were carried with a pressure drop in the hydrocyclone equal to 147 kPa. The mass flow of the feed stream was obtained by the Coriolis flow meter, while the mass flows of underflow and overflow were determined by gravimetric techniques. Besides that, samples of the feed, underflow and overflow streams were collected in order to determine the mass concentration of these streams, also using gravimetric techniques. The volumetric flow of the streams was calculated using the mass flow value and the density of each stream.

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Fig. 2. Schematic representation of the experimental apparatus: (1) hydrocyclone; (2) storage tank; (3) mechanical stirrer; (4) centrifugal pump; (5) by-pass valve; (6) valve; (7) flow meter; (8) digital pressure gauge.

Furthermore, samples of the feed and underflow streams were collected in order to analyze the size distributions of the particulate material in these streams. These analyses were done through laser diffraction techniques using the equipment Malvern Mastersizer Microplus®. By using the feed stream volumetric flow, it was possible to calculate the Euler number, which is determined as the ratio between the pressure drop in the hydrocyclone and kinetic energy per volume. This response is directly related to energy consumption of the process, thus, it is desirable a minimum value for this response. Another determined response is the underflow-to-throughput ratio, which is calculated as the ratio between the liquid volumetric flow in the underflow and feed streams. This response has an inverse relation with the concentration capacity of the equipment, therefore, it is also desirable a minimum value for this response, so that the concentration capacity of the hydrocyclone is maximized. Finally, the total efficiency is calculated as the ratio between the solid mass flow in the underflow and feed streams.

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This response is an indicator of the separation capacity of the hydrocyclone, hence, it is desirable a maximum value for this response.

4 Results and discussion 4.1 Adjusted Models In order to estimate the design variables in the proposed models, the following design space is considered: -2000 ≤ β ≤ 2000. In addition, the parameters used in the DE algorithm, population size (N), disturbance factor (F), crossover probability (Cr) and number of generations (Ngen) are equal to 50, 0.5, 0.8 and 500, respectively. The obtained models for η, RL and Eu considering the experimental data are shown in Eqs. 14, 15 and 16, respectively.

η = 73.26 + X ' a1 + X ' A 1 X

(14)

 X1   −0.60   0.61 −0.47 0.33 −0.28 X   −5.05    2  e A1 =  −0.47 0.71 0.40 −0.78 . where X =   , a1 =   X3   −0.69   0.33 0.40 1.65 −0.91        −2.35   −0.28 −0.78 −0.91 1.67  X4  RL = 18.43 + X ' a2 + X ' A 2 X

 0.29   −0.10  −10.03  0.06   where a2 = e A2 =   1.17   0.03     −0.90   0.07

(15)

0.07  2.81 0.16 0.29  . 0.16 −0.60 −0.36   0.29 −0.36 −0.19  0.06

0.03

Eu = 1857.71 + X ' a3 + X ' A 3 X

(16)

 −1360.04  662.09 46.2 64.81 36.85  −444.64     e A3 =  46.2 50.90 45.15 12.99  . where a3 =   −346.57   64.81 45.15 153.05 0.06       −153.15   36.85 12.99 0.06 15.27  In these models, it is important to observe that some design variables, for the linear terms coefficients, have a greater individual influence on the responses. Thus, the 16

overflow tube diameter (X2) is the variable which has the greatest effect on the total efficiency and the underflow-to-throughput ratio. On the other hand, the feed inlet diameter (X1) is the variable that most influences the Euler number. The determination coefficients (r2) obtained for the models were equal to 0.9274, 0.9984 and 0.9916 for the total efficiency, underflow-to-throughput ratio and Euler number models, respectively. These values are considered good, i.e., it means that the models represent well the experimental data variability. However, the determination coefficient obtained for the total efficiency model is importantly lower than the coefficients found for the other two models.

4.2 Pareto’s Curves The parameters used in the MODE+EMC algorithm, population size (N), disturbance factor (F), crossover probability (Cr), number of generations (Ngen) and number of neighbors (Ns) are equal to 100, 0.9, 0.9, 500 and 250, respectively. Also, it is important to mention that for each test case, the parameters Dc, Du and l, are considered equals to 30, 5 and 12 mm, respectively.

4.2.1 Case 1: maximize η and minimize RL Fig. 3 shows the nominal and robust Pareto’s curves obtained by using MODE and MODE+EMC for the first test case.

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Fig. 3. Nominal and robust Pareto’s curves for the test case 1.

In Fig. 3, it is evident the conflicting behavior between the objectives (maximize η and minimize RL), i.e., an increase in the efficiency imply an increase in the underflowto-throughput ratio, and a reduction in the underflow-to-throughput ratio imply a reduction in the total efficiency. In addition, the robustness insertion parameter causes a “deterioration” in the robust Pareto’s curve in relation to the nominal case. In this case, it is observed a change of minimum and maximum limits established in each optimization. These two mentioned facts correspond to the cost of inserting robustness in this test case. Still in Fig. 3, some important points of each curve are highlighted. The points 1 and 4 correspond to those of lowest underflow-to-throughput ratio in each curve. The points 3 and 6 are the hydrocyclones with the highest total efficiency of their respective curves. The points 2 and 5 are solutions which presents a compromise between the two objectives. Finally, the points A and B correspond to the hydrocyclones with the lowest values considering the evaluation of the Euler number. These points are represented in Tab. 1.

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Di Do L θ (º) η (%) RL (%) (mm) (mm) (mm) 1 4.56 9.85 120.41 11.95 77.96 5.09 2 4.36 9.65 120.25 19.99 81.64 7.01 3 4.52 8.85 120.22 20.00 84.72 10.00 A 4.36 9.76 120.25 19.98 81.16 6.73 4 4.40 10.57 120.35 19.82 77.46 5.74 5 4.55 9.52 120.28 19.99 80.97 7.55 6 4.65 9.01 120.28 19.98 82.95 9.44 B 4.43 10.40 120.35 19.99 78.39 5.76 Tab. 1. Dimensions and response values for the test case 1.

Point

Nominal

Robust

Eu 4985 5000 4982 4947 4595 4656 4660 4579

In order to establish a standard to the analyzes, the ranges of possible values for all the optimized dimensions are: 3.81 mm ≤ Di ≤ 8.79 mm, 5.61 mm ≤ Do ≤ 10.59 mm, 120.22 mm ≤ L ≤ 227.78 mm and 9.00º ≤ θ ≤ 20.00º. In addition, the variation in the total efficiency between the hydrocyclones of same curve is smaller than the variation in the underflow-to-throughput ratio. This fact indicates that the obtained equipment tend to have total efficiency values closer to each other and underflow-to-throughput ratio with a higher variability among them. The optimized hydrocyclones tend to present, in general, small feed inlet diameter, overflow tube diameter varying from medium to high values, small total length and big conical section angle, except for the hydrocyclone 1. In addition, the dimension which varies most among the equipment of same curve is the overflow tube diameter. Therefore, the separators with the highest total efficiencies (3 and 6) have smaller sizes of Do when compared to the separators with the lowest underflow-tothroughput ratio in their respective curves (1 and 4). These hydrocyclones, in turn, have the highest values of Do. Thus, it is evident that the reduction in the overflow tube diameter causes an increase in the total efficiency of the hydrocyclone, however, with the cost of also raising the underflow-to-throughput ratio.

4.2.2 Case 2: maximize η and minimize Eu

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Fig. 4 shows the nominal and robust Pareto’s curves obtained by using MODE and MODE+EMC for second test case.

Fig. 4. Nominal and robust Pareto’s curves for the test case 2.

In this figure, similar results in relation to first test case can be observed. It is evident the conflicting behavior between the objectives considered (maximize η and minimize Eu). In addition, it is important to mention the linear behavior of each solution. For η less than, approximately, 80%, the Eu is constant due to activation of constraints of the optimization problem. For the robust curve, it is observed a discontinuity close to 82,25%, i.e., there is not any non-dominated solution obtained in this region. Also in Fig. 4, some important equipment of each curve are highlighted. The points 7 and 10 are the hydrocyclones with the lowest Eu for each curve. Meanwhile, the separators 9 and 12 are the ones with the highest η of their respective curves. The points 8 and 11 are hydrocyclones which have a compromise between the two objectives of optimization. Finally, the points C and D were obtained considering the lowest values for RL equation. Also, it is worth mentioning that the hydrocyclone D and

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12 are coincident. Tab. 2 presents the mentioned points of the nominal and robust curves. Di Do L θ (º) η (%) RL (%) (mm) (mm) (mm) 7 7.76 10.58 182.94 9.56 78.31 9.94 8 5.76 10.59 189.00 9.01 82.14 9.99 9 4.11 10.59 197.46 9.01 85.75 10.00 C 7.76 10.59 182.88 9.02 80.11 9.83 10 7.41 10.58 180.32 9.01 79.38 9.60 11 4.96 10.58 186.44 9.02 82.27 9.59 12/D 4.66 9.04 120.25 19.98 82.81 9.22 Tab. 2. Dimensions and response values for the test case 2.

Point

Nominal

Robust

Eu 797 1973 4639 810 858 3106 4616

In this table it is important to observe that the variation in the Euler number among the hydrocyclones is greater than the variation in the total efficiency. This means that, in practice, the obtained hydrocyclones will have total efficiencies closer to each other and Euler numbers more varied among them. In general, the hydrocyclones showed in Tab. 2 present diverse values for the feed inlet diameter, high values for the overflow tube diameter, total length with values varying from medium to high and small conical section angle. The hydrocyclone which stayed out of this trend was the equipment 12/D, which is a separator with the highest total efficiency and lowest underflow-to-throughput ratio of the robust curve. Because the features of this hydrocyclone are different from its equivalent in the nominal curve (hydrocyclone 9), it emphasizes the existing differences between the hydrocyclones obtained through nominal and robust optimization. Therefore, the main differences between this two hydrocyclones are the total length and the angle of the conical part. While L has a high value in the equipment 9, it has a low value in the separator 12. In turn, the value of θ is low in the hydrocyclone 9, and it is high in the hydrocyclone 12. This fact indicates that the hydrocyclone 9 is an equipment with significant sensitivity to perturbations, since in robust optimization there is no separator similar to it.

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Observing the highlighted hydrocyclones of each curve, it is notable that the dimensional variable which varies most among the separators is the feed inlet diameter. While the value of this variable is high in the separators with the lowest Euler numbers, its value is low in the ones with the highest total efficiencies. This fact happens because a bigger feed inlet diameter is associated with a lower suspension average velocity, which reduces the pressure drop inside the hydrocyclone and, consequently, leads to a lower Euler number. On the other hand, a greater average velocity, provided due to a smaller feed inlet diameter, implicates a greater centrifugal action, which favors the recuperation of solids and increase the total efficiency of the process. 4.2.3 Case 3: minimize RL and minimize Eu Fig. 5 presents the nominal and robust Pareto’s curves obtained by the proposed methodology.

Fig. 5. Nominal and robust Pareto’s curves for the test case 3.

As mentioned in earlier test cases, in this figure we can observe the conflicting behavior of the objectives and the “deterioration” and “reduction” of the solutions range of robust solutions in relation to nominal solutions. In addition, the points 13 and 16 represent hydrocyclones with the lowest underflow-to-throughput ratio of their 22

respective curves. The separators 15 and 18 are equipment that present the lowest Euler numbers of each curve. The points 14 and 17 were selected as separators which present a compromise between both objectives. Finally, the points E and F represent hydrocyclones that have the highest values in terms of the total efficiency. Also, it is worth mentioning that the hydrocyclones F and 16 are coincident. These points are presented in Tab. 3. Di Do L θ (º) η (%) RL (%) (mm) (mm) (mm) 5.11 13 4.97 9.89 120.25 12.01 76.85 6.38 14 6.31 10.53 120.31 18.88 70.97 78.17 9.93 15 7.67 10.58 182.81 9.60 79.33 9.70 E 7.72 10.58 180.93 9.16 5.74 16/F 4.44 10.59 120.35 19.64 77.25 6.63 17 6.19 10.58 120.41 19.92 72.02 9.29 18 7.36 9.49 130.26 19.99 72.68 Tab. 3. Dimensions and response values for the test case 3.

Point

Nominal

Robust

Eu 4185 1897 798 808 4510 1955 1349

In this table, it is clear that the variation in the Euler number among the highlighted hydrocyclones is greater than the variation in the underflow-to-throughput ratio among them. This fact indicates that the hydrocyclones obtained in the optimizations will have, in practice, Euler numbers highly varied among them, and underflow-to-throughput ratios also diverse, but not as much as the Euler number. Regarding the dimensions, the selected hydrocyclones present, in general, varied feed inlet diameters, big overflow tube diameters and total length varying from small to medium. The conical section angle presents different behaviors between the nominal and robust Pareto’s curves. The separators selected in the nominal curve have variable θ angle, while the robust curve hydrocyclones present high values for this variable. This fact shows that, in this case, equipment with high conical section angle values tend to be less sensitive to disturbances. In addition, it can be noticed that the hydrocyclones with the lowest RL values, in each curve, are those which have the smallest feed inlet

23

diameters and smallest total lengths, while the separators with the lowest Euler numbers in each curve are the ones with biggest values of Di and L. This behavior occurs because a bigger feed inlet diameter means a lower flow average velocity, and a bigger total length means more intern space in the hydrocyclone. Both mentioned facts result in reduction of pressure drop in the equipment and, consequently, in decrease of the Euler number.

4.2.4 Case 4: maximize η, minimize RL and minimize Eu Fig. 6 present the nominal and robust Pareto’s curves obtained by using MODE and MODE+EMC strategies for the test case 4. In this figure, it can be observed that the nominal Pareto’s curve is more scattered in comparison with the robust solution. This fact indicates that, similar to what was observed in other cases, the insertion of robustness in optimization process concentrates the solutions in a smaller region, i.e., it reduces the diversity of the obtained hydrocyclones.

Fig. 6. Nominal and robust Pareto’s curves for the test case 4.

Also in Fig. 6, some important points of each curve are highlighted. The points 19 and 22 correspond to separators with the highest total efficiencies in the nominal and 24

robust curves, respectively. The hydrocyclones 20 and 23 represent the equipment with the lowest underflow-to-throughput ratio. Finally, the equipment 21 and 24 are the ones with the lowest Euler number in each curve. These points are presented in Tab. 4. Di Do L θ (º) η (%) RL (%) (mm) (mm) (mm) 84.05 9.83 19 4.39 10.58 192.76 9.16 5.26 20 4.82 9.86 120.77 12.14 77.07 78.81 9.77 21 7.81 10.54 180.77 9.32 9.35 22 4.65 9.05 122.00 19.97 82.50 5.77 23 4.46 10.52 120.25 19.65 77.21 8.95 24 7.21 9.57 129.09 19.94 72.55 Tab. 4. Dimensions and response values for the test case 4.

Point Nominal

Robust

Eu 4088 4451 809 4593 4479 1394

In this table, it can be noticed that the Euler number is the response which presents the greatest variation among the selected hydrocyclones. Next, the underflowto-throughput ratio is the second most varying response. Finally, the response which varies less among the hydrocyclones is the total efficiency. Hence, the hydrocyclones obtained during the optimization process tend to present the Euler numbers considerably varied, underflow-to-throughput ratios moderately varied and total efficiencies close to each other. Concerning the dimensions of the highlighted hydrocyclones, they present, in general, varied feed inlet diameter and big overflow tube diameter. Regarding the total length and the conical section angle, the nominal and robust optimizations resulted in equipment with different features for these two dimensions. The hydrocyclones selected from the nominal Pareto’s curve tend to have varied total length, whereas the robust curve hydrocyclones have small values for this variable. This fact indicates that, in this case, the equipment with smaller total lengths tend to be less sensitive to disturbances. In relation to θ angle, the nominal hydrocyclones present varied values for this dimension, whereas the robust separators present high values for it. This fact suggests that the hydrocyclones with big cone angles tend to be less sensitive to disturbances. 25

Furthermore, in Tab. 4, it can be observed that the separators with the highest η have low values of Di. On the other hand, the equipment of minimum RL have the lowest values of L and they also present small Di. Finally, the hydrocyclones of lowest Eu are those with the highest Di. Therefore, it is evident that the feed inlet diameter is a variable with great influence in the hydrocyclone performance when these three objectives are simultaneously optimized.

4.3 Experiments In order to validate the results obtained by optimization, some separators in each Pareto’s curve were chosen. Hydrocyclones “similar” to them were assembled using the parts available in the Laboratory of Separation and Renewable Energy. Tab. 5 shows the selected hydrocyclones of each curve, as well as the respective hydrocyclones assembled from them. Optimized

Di (mm)

Do (mm)

L (mm)

θ (º)

Assembled

Di (mm)

Do (mm)

L (mm)

θ (º)

A

4.36

9.76

120.25

19.98

HOI

4.8

9.5

125.3

19.9

B

4.43

10.40

120.35

19.99

HOI

4.8

9.5

125.3

19.9

C

7.76

10.59

182.88

9.02

HOII

7.6

10.5

180.0

9.0

D

4.66

9.04

120.25

19.98

HOIII

4.8

9.0

125.3

19.9

E

7.72

10.58

180.93

9.16

HOII

7.6

10.5

180.0

9.0

F

4.44

10.59

120.35

19.64

HOI

4.8

9.5

125.3

19.9

19

4.39

10.58

192.76

9.16

HOIV

4.8

10.5

185.0

9.0

20

4.82

9.86

120.77

12.14

HOV

4.8

9.5

130.3

11.3

21

7.81

10.54

180.77

9.32

HOII

7.6

10.5

180.0

9.0

22

4.65

9.05

122.00

19.97

HOIII

4.8

9.0

125.3

19.9

23

4.46

10.52

120.25

19.65

HOI

4.8

9.5

125.3

19.9

24

7.21

9.57

129.09

19.94

HOVI

7.6

9.5

132.3

19.9

Tab. 5. Points selected from the Pareto’s curves and assembled hydrocyclones.

26

In this table, some of the chosen hydrocyclones led to the same constructed separator. This fact is due to proximity in the separators’ dimensions and due to the limitation of available parts to assemble the equipment. For example, the hydrocyclones A and B, selected from case 1, resulted in same constructed hydrocyclone (HOI). Except for the test case 1, in other cases the hydrocyclones selected from the nominal and robust curves originated different assembled separators. In case 2, the nominal hydrocyclone C resulted in the assembly of the separator HOII, whereas the robust equipment D originated the hydrocyclone HOIII. In case 3, the chosen nominal equipment E resulted in the assembly of the hydrocyclone HOII, while the selected robust separator F resulted in the hydrocyclone HOI. Finally, in case 4, in which three hydrocyclones of each curve were chosen, the separators from the nominal curve, 19 (highest η), 20 (lowest RL) and 21 (lowest Eu), originated the hydrocyclones HOIV, HOV and HOII, respectively. The hydrocyclones selected from the robust curve, 22 (highest η), 23 (lowest RL) and 24 (lowest Eu), resulted in the equipment HOIII, HOI and HOVI, respectively. The assembled hydrocyclones were experimentally tested and their results were compared to the response values calculated through the fitted equations for η, RL and Eu, as presented in Tab. 6. Hidrocyclone η mod (%) RL mod (%) Eu mod η exp (%) RL exp (%) Eu exp HOI 77.73 8.36 3980 59.01 7.76 3206 HOII 79.09 9.25 842 68.97 5.67 1123 HOIII 79.92 10.14 4158 64.91 7.61 3202 HOIV 81.76 8,71 3580 58.70 5.65 3512 HOV 76.76 8.39 4314 70.09 7.78 3760 HOVI 71.73 9.43 1261 61.97 9.22 1939 Tab. 6. Experimental and calculated responses of the assembled hydrocyclones. In this table, it can be observed that the total efficiencies of the tested hydrocyclones were, in general, below the values calculated using the model. This fact is justified by the lower determination coefficient obtained for the total efficiency model 27

(r2 = 0.92740), which means that this model has a lower prediction capacity than the other two fitted models. Regarding the underflow-to-throughput ratios and the Euler numbers of the tested hydrocyclones, the obtained values were satisfactory, since, for all the hydrocyclones used in the experiments, the measured responses were within the stablished limits for them. In other words, the underflow-to-throughput ratios were below 10% and the Euler numbers were lower than 5000. In addition, the relative errors of the models were estimated for each tested separator. These results are shown in Tab. 7. Hidrocyclone εη εRL εEu HOI 32% 8% 24% HOII 15% 63% 25% HOIII 23% 33% 30% HOIV 39% 54% 2% HOV 10% 8% 15% HOVI 16% 2% 35% Tab. 7. Relative errors for the constructed hydrocyclones. The data showed in Tab. 7 can be used to perform the comparison between the relative errors considering the nominal and robust assembled hydrocyclones. In the test case 1, as the nominal and robust optimizations resulted in the same constructed hydrocyclone (HOI), the relative errors of all responses are the same for nominal and robust cases. In case 2, the nominal optimization resulted in the hydrocyclone HOII, whereas the robust optimization resulted in the hydrocyclone HOIII. Regarding the relative errors of the responses of these two hydrocyclones, it is noticeable that the hydrocyclone HOIII has relative errors below 40% for all the responses, while the hydrocyclone HOII has the relative error of underflow-to-throughput ratio around 63%, a value considerably high. In the test case 3, the constructed separators obtained from the nominal and robust optimizations were HOII and HOI, respectively. The hydrocyclone HOII has the 28

relative error of the underflow-to-throughput ratio elevated, as it was mentioned. On the other hand, the hydrocyclone HOI has the relative errors of three responses below 40%. Again, the robust optimization resulted in the assembly of a hydrocyclone with relative errors not too high in general, whereas the nominal optimization led to the assembly of a separator with a considerably high relative error of a response. Regarding the test case 4, three hydrocyclones were constructed from each curve, thus the comparison must be made between the equivalent separators of each curve. In other words, the hydrocyclones with the highest total efficiencies of each curve are compared to each other, and, in same way, it is done with the equipment with the lowest underflow-to-throughput ratio and the ones with the lowest Euler number of each curve. Thereby, the hydrocyclones with the highest total efficiencies assembled from the nominal and robust curves were, respectively, HOIV and HOIII. It is noticeable that the nominal hydrocyclone HOIV present a relative error considering the underflow-tothroughput ratio equal to 54%. On the other hand, the robust hydrocyclone HOIII has the relative errors for all responses below the limit of 40%. Regarding the equipment with the lowest underflow-to-throughput ratios, the nominal curve led to the assembly of the hydrocyclone HOV, while the robust curve resulted in the separator HOI. In this case, both equipment have relative errors below 40%. Finally, the hydrocyclones with the lowest Euler numbers assembled from the nominal and robust curves considering the test case 4 were HOII and HOVI, respectively. As it was mentioned in earlier cases, the hydrocyclone HOII present a relative error for the underflow-to-throughput ratio elevated, close to 63%. On the other hand, the robust hydrocyclone HOVI present a relative error for the three responses lesser than 40%.

29

In general, the robust optimization results present relative errors not high in relation to the separators produced by nominal optimization. This fact is due to the lesser sensitivity of the robust hydrocyclones to small uncertainties associated with the experimental process. Therefore, the robust hydrocyclones tend to present experimental results closer to responses calculated through the models, when compared to the nominal separators. Figs. 7 to 9 present the performances of the tested hydrocyclones in comparison with those from the CCD of Salvador [34] in terms of the total efficiency, the underflow-to-throughput ratio and the Euler number, respectively. For this purpose, Tab. 8 presents the configurations considered by Salvador [34]. Hydrocyclone H01 H02 H03 H04 H05 H06 H07 H08 H09 H10 H11 H12 H13 H14 H15 H16 H17 H18 H19 H20 H21 H22 H23 H24 H25

Di (mm) Do (mm) L (mm) 4.8 6.6 141.0 4.8 6.6 141.0 4.8 6.6 207.0 4.8 6.6 207.0 4.8 9.6 141.0 4.8 9.6 141.0 4.8 9.6 207.0 4.8 9.6 207.0 7.8 6.6 141.0 7.8 6.6 141.0 7.8 6.6 207.0 7.8 6.6 207.0 7.8 9.6 141.0 7.8 9.6 141.0 7.8 9.6 207.0 7.8 9.6 207.0 3.9 8.1 174.0 8.7 8.1 174.0 6.3 5.7 174.0 6.3 10.5 174.0 6.3 8.1 117.0 6.3 8.1 228.0 6.3 8.1 174.0 6.3 8.1 174.0 6.3 8.1 174.0 Tab. 8. Hydrocyclones from the CCD of Salvador [34].

30

θ (º)

11.2 17.8 11.2 17.8 11.2 17.8 11.2 17.8 11.2 17.8 11.2 17.8 11.2 17.8 11.2 17.8 14.5 14.5 14.5 14.5 14.5 14.5 9.0 20.0 14.5

In Fig. 7, the total efficiencies of the CCD hydrocyclones (H01 to H25) and the assembled optimal hydrocyclones (HOI to HOVI) are shown.

Fig. 7. Comparison between the results obtained by the proposed methodology and Salvador [34] for the total efficiency.

In Fig. 7, it can be observed that the total efficiencies of the assembled hydrocyclones were, in general, lower than the total efficiencies of the CCD separators. This fact was already expected, since the aim of optimization was to obtain thickener hydrocyclones, which, traditionally, present low total efficiencies. In Fig. 8, it is presented a similar comparison to the one showed in Fig. 7, however the concerned response is the underflow-to-throughput ratio. In the figure, it is noticeable that the assembled hydrocyclones presented lower underflow-to-throughput ratio in relation to the CCD hydrocyclones. This fact confirms that the obtained equipment have, in fact, a high thickener capacity.

31

Fig. 8. Comparison between the results obtained by the proposed methodology and Salvador [34] for the underflow-to-throughput ratio.

In Fig. 9, the hydrocyclone performances are compared regarding the Euler number. It can be observed that the assembled separators present average Euler number values when compared to the CCD hydrocyclones. The exceptions are the equipment HOII an HOVI, which have low values of this response when compared to other hydrocyclones in this figure.

Fig. 9. Comparison between the results obtained by the proposed methodology and Salvador [34] for the Euler number. 32

Among the assembled separators, the hydrocyclone HOII stands out from the others due to its good performances regarding the three considered responses. The mentioned equipment has a total efficiency higher than most of the assembled hydrocyclones, and also an underflow-to-throughput ratio and Euler number among the lowest in Figs 8 and 9. Therefore, the hydrocyclone HOII is the equipment with the greatest performance regarding the three considered responses, among the thickener assembled hydrocyclones.

5 Conclusions In this work, the Effective Mean Concept was associated with the Multi-Objective Optimization Differential Evolution algorithm to determine the hydrocyclone’s dimensions which produce high-performance thickener equipment. For this purpose, four test cases, that consider the combination among the maximization of hydrocyclone total efficiency (η), the minimization of underflow-to-throughput ratio (RL) and the minimization of the Euler number (Eu), were proposed. The results obtained were compared to those obtained nominally (without robustness) and the best solutions were experimentally validated. The obtained results showed that the overflow tube diameter was the variable which most influences, individually, the total efficiency and the underflow-tothroughput ratio. On the other hand, the Euler number is most influenced by the feed inlet diameter. In addition, considering the maximization of the total efficiency and the minimization of the underflow-to-throughput ratio, the overflow tube diameter is the variable with the greatest influence over the responses. Moreover, considering the maximization of the total efficiency and the minimization of the Euler number, the dimension which most influences the hydrocyclone’s responses is the feed inlet

33

diameter. Finally, considering the minimization of the underflow-to-throughput ratio and the minimization of the Euler number, the variables with the greatest effect over the hydrocyclones are the feed inlet diameter and the total length. For the three-objective optimization problem, one can conclude that smaller feed inlet diameters favor the maximization of total efficiency, whereas higher values of this variable benefits the minimization of the Euler number. In addition, smaller total lengths favor a minimal underflow-to-throughput ratio. It is important to mention that the robustness insertion in the optimization process had significant effect over the results. The robust approach implies in removal of the Pareto’s curve in comparison with the nominal solution. In addition, the robustness insertion also causes the reduction of the ranges obtained in relation to nominal solution, i.e., a lesser diversity of obtained hydrocyclones. On the other hand, the experimental results showed the benefits of the robust approach. The robustness insertion in optimization results in hydrocyclones less sensitive to small perturbations on its dimensions, which avoids great differences between the experimental results and the responses calculated by the fitted models. Thus, the behavior of a robustly optimized hydrocyclone is more predictable than that of a separator optimized nominally. Moreover, the assembled thickener hydrocyclones had good performance in the accomplished experiments. Among them, the separator HOII was the one which most stood out, considering the compromise between separation, concentration and energy consumption. Hence, the mentioned equipment can be considered, in terms of these criteria, the best thickener hydrocyclone among the assembled separators of this work. Finally, it is also concluded that the method employed in this work can be used as a design technique to produce hydrocyclones for various industrial applications.

34

Acknowledgments The authors thank the brazilian research funding agencies CAPES, FAPEMIG and CNPq for the financial support provided to this work.

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Highlights •

Robust optimization is configured as a necessary strategy in real process.

•

The optimum values obtained by proposed methodology were experimentally tested.

•

Hydrocyclones with high efficiency, low underflow-to-throughput ratio and low Euler number are obtained.

•

Nominal optimization leads to a bigger diversity of hydrocyclones.

•

Robust optimization results in hydrocyclones less sensitive to perturbations.

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: