# Linear Circuit Analysis: a Tool for Addressing Challenges and Identifying Opportunities in Process Circuit Design

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## Abstract

One of the most important processing engineering tools for designing mineral processing facilities is linear circuit analysis. This technique, originally conceived and developed by Meloy, Williams, and Fuerstenau nearly four decades ago, provides fundamental insights regarding how unit operations interact and respond when arranged in multi-stage processing circuits. Researchers have successfully utilized this tool to improve the operating performance of industrial processing circuits incorporating coal spirals, magnetic separators, mineral spirals, and eddy current separators. More recently, advanced versions of this tool have also been developed to provide a standardized framework for circuit mass balance calculations, to output exact analytical solutions to mass yield and value-based efficiency expressions, and to estimate uncertainty propagation in separation circuits. This article reviews the historical development of linear circuit analysis, describes how the technique has evolved to address more complex circuit design problems, and presents industrial case studies that highlight the importance of this process engineering tool.

## Keywords

Linear circuit analysis Circuit design Circuit optimization Process analysis## 1 Introduction

Mineral processing is the science of particulate separation as it applies to the beneficiation of mining products. Run-of-mine material consists of one or more valuable components often mixed with a significant portion of gangue that must be removed to enhance the marketable value of the raw material. Single-unit separations are typically not able to meet the strict purity requirements for consumer markets; therefore, staged separation circuits are often employed. Since multi-unit separation circuits often represent a large and inflexible capital investment, mine operators are careful to ensure that the circuit is optimally designed to meet the desired operational objectives. The set of interdependent decisions that constitute the circuit design may involve the selection of different unit operations, different equipment models or sizes, different operational parameters, and different unit interconnections.

To assist circuit designers, researchers have attempted to establish standard design methodologies that involve various analytical techniques and tools. These tools are typically guided by some optimization strategy as well as a generic process model applicable for the given separation. Over time, circuit design tools have grown from simple design heuristics (or “rules-of-thumb”) based on operational observations to very-high-level optimization strategies based on sophisticated process models. Recent research trends favor high-level, state-of-the-art numerical optimization algorithms to accommodate the nonlinear design parameters associated with separation circuits. Nevertheless, contemporary industrial practice still favors more heuristic solutions that can incorporate operator experience and common-sense design constraints. Even today, much of preliminary circuit design is still driven by trial and error and accepted industry practices [1, 2, 3].

One of the first comprehensive expositions on the numerical optimization of flotation circuits was provided in 1974 by Mehrotra and Kapur [4]. This work describes the generalized mathematical formulation that was used to determine the cell retention time and circuit structure needed to maximize grade, recovery, or profit in a flotation operation. At the time, the authors used a direct search algorithm to determine the optimal solution, and later updates by the same authors incorporated more advanced search algorithms [5] and more sophisticated flotation models [6]. Over the next 40 years, this mathematical approach to circuit optimization was further developed to include detailed process models, new objective functions, and refined optimization routines. This research and development cycle was extensively reviewed in 1988 [7], with more recent reviews occurring in 2009 [8] and 2018 [9]. Similarly, reviews and descriptions of the various heuristic optimization methods have also been provided in prior literature (e.g., [1, 10, 11, 12, 13, 14, 15]).

While both the heuristic and the numerical approaches provide substantial value to the circuit designer, they fail to provide fundamental insight on why specific circuit designs consistently outperform others. This fundamental understanding can provide substantial value to a circuit designer who is trying to integrate both quantifiable technical factors as well as qualitative intangible factors into an acceptable circuit design. As an alternative to both the empirical heuristic method and the numerical optimization approach, linear circuit analysis is a fundamental tool that provides a convenient way to evaluate competing circuit designs particularly in preliminary design cases where experimental data may be lacking. The essential elements of the approach were developed by Meloy, Williams, and Fuerstenau over a 20-year period in the 1980s and 1990s. More recently, the method has been used to guide circuit design in a number of industrial case studies. Given these theoretical and practice considerations, the objective of this paper is to provide a review of the historical development of linear circuit analysis (LCA) (Section 2), describe several case studies where LCA has been successfully implemented to solve industrial problems (Section 3), and explain recent developments that have expanded the utility of the method (Section 4).

## 2 Historical Development of LCA

### 2.1 LCA Concepts

The concept of LCA was first derived by Meloy in 1983 to provide a method of optimizing multi-unit separation circuit configurations [16]. This original paper eventually developed into a series of publications by Meloy, Williams, and Fuerstenau that examined various theoretical aspects and applications of the methodology. The impact of these papers in the literature spanned nearly two decades with much of the original developments occurring in the 1980s. In the original work [16], a series of circuit design principles were generated from fundamental observations on the algebra concerning binary separation units.

*P*). The mass of material in the concentrate stream is simply the product of the yield and the feed mass (PF), while the recovery to the tailings stream is determined by mass balance (1 −

*P*). By extending this algebra over many units, the recovery for the entire circuit (C/F) may be analytically defined in terms of the recovery in each individual unit (

*P*

_{n}). Figure 1 shows examples of this algebra applied to common circuit configurations. The power of all LCA applications is then derived from this analytical solution.

The LCA methodology is constrained by linearity assumptions. Meloy [16] presents a formal definition of these restrictions, but in summary, linearity states that a unit’s partition curve is not influenced by feed composition or feed rate. While this assumption is not wholly valid for operating units, Meloy [17] states that during the design phase, a larger or smaller unit may be selected to accommodate the required feed rate, and thus, the approach is valid for new circuit designs. Furthermore, the same author has suggested that literature contains support for linearly operated process units and that experimental investigations have confirmed linearity in some cases [18, 19, 20].

*P*= 50% is used as a general indicator of separation capability, and Meloy shows that this slope can be determined for the full circuit by calculating the derivative of the circuit’s analytical recovery with respect to the separation property (

*X*). Figure 2 shows how this derivative applies to a single-unit separator where the separation performance follows that of a typical partition curve. By extending the analysis to multi-unit circuits, the relative separation efficiency of any arbitrary circuit can be compared to that of a single unit to show the relative increase (or decrease) in separation efficiency. Figure 3 shows the result of this calculation for several two and three unit circuits. This data shows that the addition of circulating loads increases separation sharpness; however, staged units may also affect the cut point of partition-based separators even if all units are operating similarly. Meloy’s original paper discusses these outcomes and also presents a means of analyzing unit bypass.

Meloy later expanded upon the analysis procedure to define a methodology for circuit optimization [17]. In this paper, four functions fundamental to separation processes are described mathematically: feed, selectivity, composition, and criteria. The former three functions are defined by three variable types, i.e., particle property, operational, and compositional, though not all functions are defined by all variables. Finally, the criteria function defines the value to be optimized, typically grade or recovery. The optimization then proceeds by (1) defining the criteria function in terms of the three other functions; (2) differentiating with respect to the operational variables; (3) setting the resulting derivative equal to zero; and (4) solving for the operational derivatives. If more than one process variable exists, the procedures may be expanded by taking partial derivatives of the criteria function with respect to each operational variable. This array of equations is then set equal to zero and solved simultaneously. Meloy states that the required data are easily determined by assays or other experimental studies. Furthermore, the process may be applied to various mineral processing unit operations, including flotation, gravity separation, and magnetic and electrostatic circuits. As a final contribution, Meloy notes that the optimum grade and the optimum recovery never occur at the same point.

The principles of LCA were also used to: optimize for grade or recovery [17], analyze dynamic flotation cell models [19], assess multi-feed multi-stage separators [20], and identify the effect of density variations in heavy-media circuits [21]. First, the analytical circuit solutions derived from LCA were integrated with other common process functions, i.e., feed, selectivity, and composition, to show how a circuit can be optimized to a specific criteria [17]. Meloy notes that the optimum grade and the optimum recovery never occur at the same operational point. Next, LCA was synthesized with a dynamic, rate-based lumped parameter flotation model to analyze the dynamic response of flotation circuits to sinusoidal feed variations [19]. The authors compared the dynamic behavior of counter-current and co-current circuits, concluding that co-current circuits are better in all applications. This result was based on the deficiencies of counter-current circuits, including larger required volumes and longer dynamic response times. Finally, co-current flotation banks were shown to be non-oscillatory, while counter-current circuits exhibit osculation frequencies that increase with flotation rate and retention time.

Another paper in the LCA series addresses the optimization of a rougher-scavenger-cleaner dense-media coal-cleaning circuit [21]. Here, the authors seek to address whether the media density in multi-stage coal-cleaning circuits can be optimized to improve overall performance. The authors note that rougher-scavenger-cleaner circuits are not common in coal preparation, especially in gravity separation circuits; nevertheless, the authors conduct the optimization exercise utilizing a standard partition function for the selection function of the dense-media separator. The LCA methodology is used to determine the product function for the entire rougher-scavenger-cleaner circuit, and an incremental approach (by taking the second derivative of the analytical expression) is used to determine the effect of the gravity set point in each unit on the final recovery, grade, concentrate, and circulating load. The results show that the best benefit occurs at relatively low sharpness values. Furthermore, additional benefits can be experienced by increasing the scavenger gravity and decreasing the cleaner gravity. This result is expected, since such modifications will increase the circulating load to the rougher and increased circulating loads are known to enhance separation performance.

### 2.2 Alternative Analytical Approaches

To supplement their core work in LCA, Williams and Meloy later suggested two alternative approaches to circuit configuration design [22, 23]. Both methods were derived from theories similar to LCA; however, the authors sought to reduce the cumbersome mathematics-associated circuit analysis. The first of these methods presents precise definitions for the common colloquial circuit functions: roughers, scavengers, and cleaners [22]. Here, the authors define a rougher as unit whose feed is the circuit feed, a cleaner as unit that is fed by a concentrate stream, and a scavenger as a unit that is fed by a tailings stream. In most processing plants, a single unit will fulfill several of these functions. For example, the rougher in a standard rougher-scavenger-cleaner recycle circuit is actually a rougher, scavenger, and cleaner, since it is processing feed, concentrate, and tailings from various units. The authors argue that a better approach is to design circuits so that the individual unit operations are only fulfilling a single function. This strategy promotes specialized operation for individual cells, since each is pursuing a different process goal. Furthermore, by developing circuits that exploit specialized functions, the feed loading to each unit is substantially reduced. In the paper, the authors use LCA to show four equivalent circuits, each representing a higher degree of specialization. The authors then use the analytical solution to show the degree to which specialization can reduce feed loading, and in many cases, increase metallurgical performance.

The second alternative circuit design approach defined mathematical solutions to three circuit design criteria: (1) the required number of stages, (2) the stage where the feed enters the circuit, (3) the configuration of the product streams [23]. This approach begins by assuming a generic cleaner–type circuit of indeterminate size, with each concentrate advancing serially to the next unit. Tailing streams are recycled to a prior point in the circuit, such that the grade of the recycle stream is greater than or equal to the grade at the point of reentry, a principle originally suggested by Taggart [24]. By establishing this generic superstructure, the three design criteria may be solved algebraically if four desired/operational parameters are specified: (1) the desired global product recovery, (2) the desired global ratio of product to waste, (3) the product-to-waste ratio achievable for each unit, and (4) the feed component ratio. These algebraic functions are intended to guide an initial circuit design, since they will inherently produce non-integer values. By rounding and manipulating different combinations of values, the design criteria that achieve the desired results may be determined. These configurations constitute the “feasible designs” from which a more thorough optimization or design process may originate [23].

A later reaction paper [25] proposed slight alterations to the “feasibility method” employed by [23]. This paper begins by describing potential pitfalls to the original feasibility method, such as the assumption of identical transfer function for each unit, the conversion of recycle streams to waste streams when the recycle parameter was ambiguous, and the lack of a standard methodology when non-integer values were calculated. Rather than first generically defining the number of stages for the entire plant, the author assumes that each circuit will have one rougher stage, and an indeterminate number of scavenger and cleaner stages [25]. The number of units in each stage is calculated independently using equations that relate the waste specification to the number of scavenger stages and the concentrate specification to the number of cleaner stages. Next, the reentry point of the concentrate waste streams is determined by implementing the same recycle principle proposed by [24] and employed by [23]: namely, the waste stream must be recycled back such that it enters a stream with a lower or equal grade. An analogous approach is taken for the reentry of the scavenger concentrate products. After the calculation of these four parameters, the author proposed three rules to guide selection when non-integer values are calculated: (1) the number of recycle stages must be greater than or equal to one, (2) all recycle streams must be recycled into the circuit (i.e., no open circuits), and (3) values for the number of cleaner and scavenger units should be rounded up, unless they are extremely close to the floor value. The final rule provides added conservatism since the initial calculations do not consider the influence of recycle streams. Even after these rules are applied, several feasible solutions may persist. In these cases, an economic analysis or a decision based on the separation factor, the beneficiation ratio, or the valuable component recovery is suggested [25].

Noting the utility of LCA and the analytical solution, Williams, Meloy, and Fuerstenau derived a methodology to rapidly produce analytical solutions to separation circuits [26]. In this paper, the authors note the drawbacks to traditional circuit analysis, namely the cumbersome mathematics, as well as the deficiencies of numerical optimization approaches, such as the inability to introduce common-sense principles from the designer. This approach, tailored from the principles of graph theory, provides a technique of relating the recovery of individual units to the full circuit recovery. In their nomenclature, separation units are designated as modules that are connected by branches by identifying loops in the circuit configuration; the overall circuit recovery may be calculated by a standard approach. The authors present an example from the literature that contains five units and required the simultaneous solution of 12 equations [27]. Williams, Meloy, and Fuerstenau suggest that, when mastered, the graph theory approach should take ten minutes for a similarly sized problem [26].

### 2.3 Other Adaptations and Implementations

Other authors have also highlighted the need for robust mathematical optimization in the circuit design problem, despite the nonlinear objective functions and discrete selection variables that complicate the underlying mathematics [28, 29, 30]. Yingling’s first paper introduces a novel approach to the mathematical representation of the circuit configuration based on the theory of steady-state evolution in Markov chains [28]. This formalistic approach to probabilistic separation was formed as an extension of LCA. Yingling notes the desire for an analytical circuit solution (especially in optimization problems), but discredits the case-by-case algebraic approach taken by Meloy [16]. Instead, Yingling proposes a flowgraph reduction strategy based on elementary reduction rules [31]. With the Markov assumption, the separation state of a given unit is not dependent on the prior states of the process. Combining this approach with potential theory of Markov chains, Yingling is able to produce a more efficient, but mathematically equivalent, solution for the steady-state behavior of the circuit. This approach incorporates the circuit superstructure with flow distribution nodes. The state of this superstructure along with the operational parameters is defined as the circuit control policy that is varied to optimize an economically driven reward function. The optimization algorithm proposed by the author is based on stochastic dynamic programming with extended techniques to account for the multiple particle classes present in flotation systems. This optimization relies on discrete layout alternatives, defined by the circuit designer; however, Yingling is regarded as one of the first authors to provide a formalistic approach to the circuit superstructure concept and an economic objective function [28].

Yingling’s later two-part series reviewed prior work in circuit optimization and extended the original work in Markov chains [29, 30]. Yingling’s review categorized prior work into two classifications: (1) those that use direct search techniques to optimize the operational parameters and the circuit layout simultaneously and (2) those that use a two-stage optimization to first establish the configuration before solving the parameters. The author notes that many of the authors in the first group produce solutions that contain too many flow streams, as the optimization algorithms blindly attempt to expand the circuit optimization problem. The second group of authors rarely consider the impact of stream flows in the circuit configuration step and generally ignore economic considerations. Yingling concludes that neither approach is ultimately sufficient for the circuit design problem. In response, the final paper [30] extends the procedures developed in the original. Most notably, a new optimization routine was developed that allows for both discrete and continuous stream splitting nodes. This algorithm is stated to be more efficient and actually more robust than many direct search methods that cannot determine the appropriate number of cells within a flotation bank. A similar ambiguous, though economically based, objective function is used. Examples of the solution robustness are presented.

A recent adaptation of LCA is sensitivity analysis (SA) [3]. The authors present SA as an ideal trade-off between empirical and heuristic insight and numerical optimization strategies. Since global optimization through experiments is nearly impossible, SA is used to determine the nodes in the circuit that produce the greatest impact. Subsequently, empirical insight and experiments can be used to optimize or improve performance at those nodes. In SA, each unit is examined individually and the final results are compared to determine the most influential unit. As in LCA, the first required step is to determine an analytical expression for the circuit yield in terms of each unit operation’s independent recovery function. In defining this expression, terms referring to units not under scrutiny are lumped into a single, constant parameter. By mathematically manipulating this global recovery function, an expression can be determined that indicates if a species is being diluted or concentrated, depending on the value of the lumped parameter. Next, the partial derivative of the global recovery function is determined with respect to the recovery of the unit under scrutiny. The magnitude of this partial derivative is then determined and plotted for various expected values of the individual recovery functions. Local minima and maxima in the plots are noted. This process is then repeated by taking the partial derivative with respect to each unit, the behavior of the plots are identified, and the overall magnitude of each partial derivative is compared to determine the unit with the greatest influence on the circuit. Unfortunately, the behavior of the sensitivity graph changes, depending on the performance of other units in the circuit. However, if the general behavior of an operating circuit is known, SA may be used to determine the unit that merits the most attention. Once the operation of this unit is altered, the circuit will produce a new high-sensitivity unit and the process may be repeated. The authors conclude the paper by demonstrating the method on a hypothetical flotation circuit using a standard perfectly mixed reactor model. Since that original SA paper, the authors have developed the method to include global SA considerations [32, 33, 34].

### 2.4 Summary and Key Takeaways

- 1.
Only circuit configurations that involve recycle to prior units are capable of increasing the relative separation efficiency.

- 2.
For a cleaner circuit without recycle, additional stages of processing will increase the purity of the concentrate; however, the relative separation efficiency will decline if three or more cleaning units are used.

- 3.
For a scavenger circuit without recycle, additional stages of processing will increase the recovery to the concentrate; however, the relative separation efficiency will decline if three or more scavenging units are used.

- 4.
The largest improvements in relative efficiency are obtained using counter-current cleaner/scavenger circuits where downstream products are recycled back to the preceding operation.

- 5.
Products generated after the first separator should not cross between the scavenger and cleaner branches of the circuit without first being recycled through the initial separator.

- 6.
Units positioned off of the main scavenger and cleaner legs do not increase separation sharpness.

- 7.
The relative efficiency for a circuit increases in direct proportion to the number of cleaning and scavenging stages in each branch of the circuit. Perfect separation is obtainable as the number of units down the scavenger and cleaner branch approach infinity.

- 8.
The grade of a particular stream within the circuit should not be used as a basis for determining where that particular stream should be directed within the circuit.

- 9.
The most efficient separator should be placed as the head of the circuit, as this unit has the greatest influence on overall separation efficiency.

While these rules will generally result in the most efficient circuit, Noble and Luttrell [37] later show at least one exception to rules 1 and 5. A circuit in which the cleaner tailings are directed to the scavenger feed, while the scavenger concentrate is directed to the cleaner feed, violates rules 1 and 5, yet produces a relative separation efficiency higher than that of a single unit. Nevertheless, this circuit is still trumped by the counter-current balanced circuit where secondary products are recycled back to the head of the circuit.

## 3 Review of LCA Case Studies

### 3.1 Improved Spiral Separation

While LCA is a fundamental concept based on theoretical constructs, it has been applied in many cases to solve industrial process engineering challenges. In one of the first industrial case studies using LCA, Luttrell et al. demonstrate how the fundamental information gained from LCA can be used to improve the performance of the spiral circuit in a coal preparation plants [38]. Despite several competing technologies, spirals have become the preferred alternative for fine coal separations (0.15 × 1.0 mm), particularly in the USA [39]. While spirals tend to have low maintenance and operating costs, they do not always produce a sharp separation. The orientation of circulation channels in the flowing film tends to trap high-density rock in the clean coal product and produce a middlings stream of misplaced material. Both of these inefficiencies can be offset by staged separation, and the compound spiral eventually grew out of this need.

After the Luttrell et al. publication, similar studies confirmed the performance gains of the proposed circuit configuration [40], while other authors have proposed even more sophisticated circuits with four or five spiral units [41, 42].

### 3.2 Improved Magnetic Separation

### 3.3 Heavy Mineral Sands Plant

The “wet” portion of heavy mineral sands processing plants often use gravity separation technologies (e.g., spirals) to separate light sand minerals, such as quartz, away from the more valuable heavy minerals, including zircon, rutile, ilmenite, and others. In an effort to meet strict quality standards, these circuits can become exceedingly complex, sometimes using more than 10 stages of separation. These complex circuits thus produce numerous internal recirculation streams that must be carefully optimized to ensure suitable performance.

In a landmark case study, McKeon and Luttrell show how LCA was used to reconfigure and optimize a large-scale spiral circuit processing heavy mineral sands [35, 47]. The original plant flowsheet utilized 686 spirals organized in 14 individual separation stages. The circuit produced an ultimate heavy mineral sands recovery of 93%; however, material was reprocessed through the circuit seven times to meet the grade threshold. In the series of publications, McKeon and Luttrell show how the existing circuit violated many of the LCA circuit design rules and could thus be improved by careful modification. After modifying the circuit to meet LCA guidelines, the plant was able to obtain a 94.7% recovery at the desired grade in only a single pass. The authors were also able to reduce the number of spirals to 542 and the number of separation stages to 11. This work is particularly noteworthy as the authors did not attempt to directly calculate the LCA separation efficiency parameters (the algebraic solution for 14 separation units is exceedingly difficult to obtain). Rather, the significant performance improvements were driven simply by LCA design rules.

### 3.4 Eddy Current Separations

The applications of LCA are not solely limited to mineral processing problems. Since the LCA framework is built upon a generic partition function, the method can be readily applied in any field that relies on separation of particulate material. In a recent paper, Shuttleworth et al. [48] showed how LCA can be used to improve processing efficiency in the scrap-recycling industry. Many scrap-recycling facilities (particularly those specializing in automotive scrap) use eddy current separators to sort valuable non-ferrous “zorba” metals from non-metallic waste. Like many separation technologies, eddy current separators are subject to natural inefficiencies that limit the overall separation performance and lead to contamination in the final product. The issue with contamination has been compounded in recent years, as major consumers of scrap metal have instituted strict purity limits on scrap imports. For example, China’s 2013 “Green Fence” initiate mandated that imported scrap contain less than 1.5% impurity [49, 50, 51].

For eddy current machines, this misplacement is commonly caused by improper orientation of objects entering the eddy current field, interferences created by solid-solid contacts, and fouling of splitters by material buildup. All of these inefficiencies are caused by random misplacement and thus can be reduced by reprocessing. However, these recycle streams must be carefully configured to ensure optimal performance is obtained. Shuttleworth et al. show how LCA can be used to optimize multi-stage eddy current separation to achieve both high grade and high recovery simultaneously.

## 4 Advanced Techniques

### 4.1 Automated LCA Mass Balance Calculations

The traditional LCA approach utilizes an algebraic solution to the circuit recovery function to evaluate the sharpness of separation for middling material. While the adequacy of this approach is evident in the numerous case studies and industrial applications, it is constrained by several limitations. First, the traditional approach requires manual and cumbersome algebra to derive the algebraic circuit solution. In many of the case studies presented above, either the LCA solution was not solved directly or the total number of units was limited to just two or three. As the number of processing units increases beyond three, the manual algebra becomes exceedingly complex, and the benefits of the method are easily negated.

As early as 1992, Williams, Fuerstenau, and Meloy attempted to identify an alternative to the manual algebraic approach to circuit solutions [26]. This proposed solution utilized graph theory. While the method was mathematically valid, it did require a working knowledge of graph theory concepts, which likely dissuaded many practicing engineers. As an alternative approach, Noble and Luttrell proposed a matrix reduction approach that utilized simple connection matrices in conjunction with a symbolic algebraic computational routine [37]. The result is a consistent and computationally efficient algorithm that can be used to produce algebraic solutions to the circuit recovery function. This algorithm can thus be used to incorporate LCA into larger circuit design problems, such as industrial flotation circuits. To demonstrate the utility of the algorithm, the authors use the matrix algorithm to solve more than 30 common circuit configurations (two and three units). Moreover, to demonstrate the limits of the algorithm, the authors solve the 14-stage original spiral circuit as well as the 11-stage modified circuit that was described by McKeon and Luttrell [35, 47]. The algorithm confirmed the improvement to the separation sharpness that was observed empirically in the earlier study.

### 4.2 Novel Objective Functions

A second criticism of traditional circuit analysis is that the objective function (i.e., separation sharpness at the midpoint) does not adequately articulate circuit performance in all cases. In recent years, several authors have attempted to supplant traditional technical metrics of separation performance (e.g., grade, recovery, separation efficiency) with direct or indirect economic indicators. For example, Cisternas et al. [52] show how the specific economic objective function influences the optimal circuit selection, while also showing how the optimal circuit is very sensitive to the product price. Numerous studies and fundamental considerations have shown that the misplacement of pure particles, rather than the misplacement of middling particles, is a primary driver of economic circuit performance. In well-liberated systems, the inventory of pure particles grossly overwhelms the inventory of pure particles, and the “micro-pricing” principle shows that these pure particles (both ore and gangue) tend to have a larger unit value [53, 54, 55, 56, 57].

Noting these combinatory issues associated with pure particle misplacement, Noble and Luttrell [58] proposed an alternative approach to circuit analysis that utilizes the “moment of inertia” rather than the separation sharpness as the indicative circuit characteristic produced from the analysis. In the paper, the authors provide the conceptual and mathematical framework for the moment of inertia. Rather than the simple derivative at the midpoint (i.e., the sharpness value), the moment of inertia is calculated by first conceptually generating a thin plate with a two-dimensional shape equal to the difference between the real partition curve and the ideal partition curve. If this thin plate is rotated around the desired cut point, classical mechanics dictates that the resistance to that rotation is given by the moment of inertia. The moment of inertia is larger when more mass resides away from the rotational access. For the case of the partition curve, this mathematical feature of the moment of inertia implies that misplacement further away from the cut point will be more strongly penalized than misplacement closer to the cut point.

Using this theoretical construct, Noble and Luttrell show the mathematical derivation of how the moment of inertia can be calculated from the LCA circuit solution. They demonstrate this calculation for more than 30 circuits and show how the value relates to the traditional separation sharpness parameter. In addition, Noble and Luttrell also show how the LCA circuit solution can be used to determine circuit yield. They derive a quantitative “yield score” parameter that indicates whether a specific circuit has a tendency to increase or decrease yield relative to any other circuit. In closing, they show how these two metrics (yield score and moment of inertia) can be integrated into a more comprehensive circuit design methodology using a Pareto efficiency approach.

### 4.3 Circuit Uncertainty Evaluation

Most traditional circuit optimization approaches, including the LCA approach, solely focus on deterministic evaluations. Unfortunately, the circuit design process is subject to several sources of uncertainty. For example, replicate batch flotation testing and modeling may show that a particular industrial unit has an expected recovery of 90%; however, the variation in the raw data and the modeling approach may suggest that the expected value follows some distribution from 80 to 95%. Questions then arise on how this uncertainty will be propagated throughout the circuit and influence the final circuit recovery.

Until the last decade, very few studies considered the influence of uncertain processing parameters on overall circuit performance. Some exceptions to this trend include Xiao and Vien [59] who used Monte Carlo simulation to evaluate flow rates in flotation circuits; Lucay et al. [3], Sepúlveda et al. [32], and Lucay et al. [34] who used local and global sensitivity analyses to evaluate the influence of individual units on final circuit performance; Sepulveda et al. [33] and Montenegro et al. [60] who investigated the influence of statistical distribution types, Jamett et al. [61] who used stochastic optimization to optimize flotation circuit performance, and Amini and Noble [62] who use Taguchi’s method as an alternative to Monte Carlo simulation. All of these approaches show how uncertainty influences circuit performance, and most provide tools for analyzing and potentially mitigating that uncertainty.

However, as was the case for the deterministic approaches, very few of these mathematical approaches provide any fundamental knowledge on why particular circuits suppress or induce uncertainty. To solve this problem, Amini and Noble [63] proposed an approach to uncertainty analysis based on LCA. They use the LCA algebraic circuit solution in conjunction with the law of the propagation of errors to determine the overall uncertainty imparted to the circuit by individual units. This formulation provides an efficient approach to quickly assess whether a specific circuit design will have the tendency to increase or decrease the uncertainty in both grade and recovery predictions. They show that some circuits (such as high-recovery scavenging circuits) can reduce the uncertainty in circuit recovery below that of a single unit, while others, such as high-recovery cleaner circuits, increase the potential range of possible circuit recovery values. The nature of this change (either inhibiting or propagating uncertainty) is dependent both on the circuit design and on the value of the unit recoveries. LCA provides a mechanism to predict these changes, and the authors validate this approach using independent projections derived from Monte Carlo simulation.

## 5 Summary and Conclusions

- 1.
Despite its limitations, LCA can be used to impart substantive improvements to operating circuits. Case studies have shown to demonstrate this utility for coal spirals [38], magnetic separators [36], heavy mineral sands separation [35], and eddy current separators [48].

- 2.
Recent advances in the method of formulating LCA solutions [37] have extended the applicability of the method to include techno-economic objective functions [58] and circuit uncertainty analysis [63].

- 3.
LCA can provide insight on how a particular change can influence circuit performance, but it cannot precisely define the level of performance change. As such, LCA is best used in conjunction with other process design tools, including modeling and simulation, numeric optimization, and the circuit designer’s experience. An example of a comprehensive circuit design approach using LCA is given by [58].

## Notes

### Compliance with Ethical Standards

### Conflict of Interest

The authors declare that they have no conflict of interest.

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