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Virginia Tech | CHAPTER 5. AN ALGORITHM FOR ANALYTICAL SOLUTIONS AND ANALYSIS
OF MINERAL PROCESSING CIRCUITS
100
90
80
70
60
50
40
30
20
10
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
)%(
yrevoceR
High Bypass
Cut Point
Sharpness
Low Bypass
Actual Partition
Ideal Partition
Dimensionless Property
Figure5.1: Typicalpartitionfunctionwiththeprimaryparameterslabeled: cut-point,sharp-
ness, high bypass, and low bypass.
10,000 tons of sand with a defined particle size distribution). The separator then distributes
that feed to one of two products in a proportion dependent on the individual particle’s mag-
nitude of the property. This distribution of the feed to the two products may be interpreted
as a probability (e.g. a 1,000 micron particle has a 100% chance of going to screen overflow;
a 150 micron particle has a 20% chance of going to screen overflow).
The function defining these probabilities may be visualized as a smooth S-shaped curve,
as shown in Figure 5.1. The horizontal axis may be manifested as any property, either
tangible or abstract, which a separation process can be based on (e.g., size, density, magnetic
susceptibility, conductivity, Stokes diameter, floatability, color, boiling temperature, shape,
hardness, etc.). Furthermore, the vertical axis may represent the recovery to either product,
depending on the system.
While several researchers have proposed mathematical fits to the partition function,
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OF MINERAL PROCESSING CIRCUITS
all essentially rely on four parameters: high bypass, low bypass, cut-point, and separation
sharpness (King, 2001). High bypass is the percentage of high property material that reports
to the desired product. Graphically, this is the location maximum asymptote of the partition
function. Similarly, thelowbypassistheamountoflowpropertythatreportstotheincorrect
product or they low asymptote of the partition function. The cut-point is the property
value at which a particle has a 50% chance of reporting to either product. Finally, the
separation sharpness is the slope of the curve at this 50% point. A higher slope indicates
that the separator can better distinguish middling particles. Intuitively, each parameter of
the partition function reveals some information on the separation capability of a single unit.
One mathematical form of the partition function is given by:
1
P = (θ −θ ) +θ (5.1)
H L L
1+exp(α(1−Z))
where P is the partition probability, θ is the high bypass, θ is the low bypass, α is the
H L
separation sharpness, and Z is the property normalized by the cut-point (X/X ) (King,
50
2001). In a perfect separator, the partition curve is a step function, i.e., particles lower than
the desired separation point will have a zero probability of reporting to the product, while
particles with a higher property will have a 100% probability of reporting to product.
5.2.2 Circuit Analysis
Linear circuit analysis is an approach originally described by Meloy (1983). In the
past, this tool has been used to evaluate heavy mineral circuits (McKeon & Luttrell, 2005),
coal spiral circuits (Luttrell, Kohmuench, Stanley, & Trump, 1998), and magnetic separators
(Luttrell, Forrest, & Mankosa, 2002). This procedure provides a systemic approach to the
evaluation of processing circuits and the ability of these circuits to overcome partition imper-
fections. In order to evaluate the strength of a circuit, an analytical solution for the circuit
recovery is first required. Fundamentally, recovery is simply defined as the proportion of
mass that reports to the concentrate relative to the original feed mass (C/F). This propor-
tion can be defined for the total mass of the stream (i.e., yield) or of a particular component
(i.e., recovery). To determine this analytical expression, the recovery of each unit is defined
by a probability value, P. This P value may be considered a deterministic, single value for
a given property class or a functional value, such as a partition function. For a single unit,
the solution is trivial (Figure 5.2). The concentrate mass (C) is the P value multiplied by
the feed mass (F):
C = PF.
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OF MINERAL PROCESSING CIRCUITS
Figure 5.2: Circuit analysis solution for a single cell.
Following similar logic, the tailings mass (T) may be calculated by:
T = (1−P)F.
Extending this procedure, the circuit concentrate and tailings streams can be calculated
for any circuit configuration. As a simple example, consider a rougher-cleaner circuit, with
no recycle (Figure 5.3). The first unit produces the same solutions presented in the trivial
case. If the first unit’s concentrate (FP ) is then introduced as feed to the second unit, the
1
final concentrate and final tails can be calculated by:
C = FP P
1 2
T = F(1−P )+FP (1−P ).
1 1 2
If a recycle stream is introduced to the rougher-cleaner circuit, an additional equation
must be written to account for the additional unknown. In this example, the first unit’s feed
is not explicitly known, so an arbitrary F(cid:48) variable is assigned. However, by adding another
equation to account for the initial node, F(cid:48) can be solved in terms of F, and the appropriate
substitutions can be made to solve for C/F. These procedures are shown in Figure 5.4.
Onceananalyticalsolutiontotherecoveryhasbeenobtained, variousparametersofthe
circuit strength may be evaluated. One of the most useful parameters presented by Meloy
is relative separation sharpness. By re-examining the partition function (Figure 5.1), the α
valueisdefinedastheslopeofthepartitionfunctionatP = 0.50. Extendingtheconcept, the
partition function may be applicable for a single unit, or for an entire circuit. If the circuit’s
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Virginia Tech | CHAPTER 5. AN ALGORITHM FOR ANALYTICAL SOLUTIONS AND ANALYSIS
OF MINERAL PROCESSING CIRCUITS
do not affect the probability that a particle will report to a given stream, (ii) the fraction
of particles of a given property in each stream remains unchanged as feed rate increases,
and (iii) the partition curve is feed independent. Simply, these assumptions state that the
partition function for a given unit is fixed, with respect to the feed rate and composition.
While real processing circuits do not necessarily adhere to the linearity assumption, Meloy
argues that for the design-case, all circuits are linear, since the size or number of the units
has not yet been determined. Consequently, the equipment may be sized to accommodate
any feed rate such that linear behavior is achieved.
5.3 Software Development
5.3.1 Matrix Reduction Analytical Solution Algorithm
The value of circuit analysis lies in its ability to provide fundamental evaluation without
extensive data or computational resources. Unfortunately, when the circuit size extends
beyond three or four units, the math becomes overly cumbersome, and the value of the final
solution is diluted by the effort needed to achieve it. In order to overcome this deterrent, a
graphically-basedsoftwareprogramwasdevelopedtocomputecircuitanalysissolutions. The
resulting platform integrates a Microsoft Excel graphical user interface and a new matrix
reduction solution algorithm to determine analytical stream solutions for circuits of any
magnitude and any potential configuration.
To apply the matrix reduction algorithm, three circuit-descriptive arrays must first be
constructed: the feed matrix (F), the products matrix (P), and the initial condition vector
(cid:126)
(C). The two matrices have dimensions of M×N where M is the number of streams, and N
is the number of units. The initial condition vector has dimensions of M×1. In constructing
the feed vector, each element is given a value of 1 or 0. A value of F = 1 indicates that
m,n
stream m feeds unit n, while a value of 0 indicates that stream m does not feed unit n. The
mathematical representation is given by:
Unit Unit ··· Unit
1 2 N
Stream F F ··· F
1 1,1 1,2 1,N
Stream F F ··· F
F m,n = . .
.
2 2 . . .,1 2 . . .,2 ... 2 . . .,N .
Stream F F ··· F
M M,1 M,2 M,N
The products matrix follows a similar construction. Here the matrix element value
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OF MINERAL PROCESSING CIRCUITS
represents the analytical or numeric transfer function which relates stream m to unit n. For
separation units, a value of P or (1 − P ) is used to describe the concentrate or tailings
n n
products, respectively. For junction units, the appropriate matrix element is given a value
of 1 to indicate that all of the product is produced in the single stream. Finally, products of
a non-selective splitter unit may be described by the numeric fraction indicating the split.
For example, 0.5 indicates one product of a 50-50 split. Formally, P is given by:
Unit Unit ··· Unit
1 2 N
Stream P P ··· P
1 1,1 1,2 1,N
Stream P P ··· P
P m,n = . .
.
2 2 . . .,1 2 . . .,2 ... 2 . . .,N .
Stream P P ··· P
M M,1 M,2 M,N
The initial condition vector simply indicates the value of the circuit feed. If the mth
stream is a circuit feed stream, a value of 1 is assigned; otherwise, each element is given a
value of 0. The vector is formally defined:
Stream C
1 1,1
Stream C
C(cid:126) m,1 = . . 2 . .2,1 .
. .
Stream C
M M,1
Onceallthreecircuit-descriptivearrayshavebeendefined, theanalyticalcircuitsolution
may be solved by formulating the following linear system:
(I−P×F(cid:48))A(cid:126)
=
C(cid:126)
(5.2)
(cid:126)
where I is the M ×M identity matrix, and A is the circuit analytical solution (M ×1) for
each stream.
To illustrate the utilization of this matrix reduction methodology, consider the rougher-
cleaner recycle circuit given in Figure 5.4. First, the feed and products matrices are con-
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Virginia Tech | CHAPTER 5. AN ALGORITHM FOR ANALYTICAL SOLUTIONS AND ANALYSIS
OF MINERAL PROCESSING CIRCUITS
The linear system may now be formulated according to Equation 5.2:
(I−P×F(cid:48))A(cid:126)
=
C(cid:126)
1 0 0 0 0 P −1 A 0
1 1
0 1 0 0 0 −P A 0
1 2
0 P 2 −1 1 0 0 0 A 3 = 0 .
0 −P 0 1 0 0 A 0
2 4
0 0 0 0 1 0 A 1
5
0 0 −1 0 −1 1 A 0
6
Solution of the linear system yields:
Cell Tail −(P −1)/(P P −P +1)
1 1 1 2 1
Cell Con P /(P P −P +1)
1 1 1 2 1
A(cid:126) =
Cell 2Tail −(P 1(P
2
−1))/(P 1P
2
−P
1
+1)
.
6,1
Cell Con (P P )/(P P −P +1)
2 1 2 1 2 1
Feed 1
F(cid:48) 1/(P P −P +1)
1 2 1
This matrix reduction methodology produces an analytical solution to every circuit
stream in terms of individual unit transfer functions. The expression for the final circuit
product (Stream 4: Cell Con) is mathematically equivalent to the expression formed alge-
2
braically in Figure 5.4. Finally, the procedure is infinitively scalable and can accommodate
circuits with any number of streams and units.
5.3.2 Software Interface
The matrix reduction analytical solution algorithm is incorporated into a Excel-based
software package which provides graphically-based circuit drawing tools. The software uses
a custom ribbon tab on the Microsoft Excel program to access the drawing and other cal-
culation tools. The available drawing components are limited to nodes, simple separators,
and streams. By using Excel’s drawing tools, the user can input the flowsheet, identify the
feed and concentrate streams, and calculate the circuit analysis solution. Figure 5.5 shows
a screenshot of the program’s interface (a), and the custom ribbon tab highlighted (b).
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Virginia Tech | CHAPTER 5. AN ALGORITHM FOR ANALYTICAL SOLUTIONS AND ANALYSIS
OF MINERAL PROCESSING CIRCUITS
(a) Microsoft Excel platform
(b) Custom ribbon tab.
Figure 5.5: Circuit analysis software interface.
Once prompted, the software calculates several values indicative of a circuit’s strength
andpresentstheseinadialogwindow. Theuseristhenfreetoaltertheflowsheet,recalculate,
or start over.
5.3.3 Analysis Features
The most significant output of the software is the analytical solution to the C/F value.
The software next uses a numerical technique to compute the derivative of the C/F function
at P = 0.5, assuming all P values are equal. The result is Meloy’s relative separation
sharpness value which is a good single indicator of circuit performance.
Further study beyond the separation sharpness has shown that the other parameters of
the circuit’s partition function (bypass values and relative cut-point) can also be determined
once the analytical circuit solution is known. The low bypass (θ ) defines a particular
L
probability for the smallest property class. If given a circuit’s recovery (C/F) function, the
circuit’s bypass may be calculated by substituting in each P value which corresponds to θ .
L
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OF MINERAL PROCESSING CIRCUITS
Mathematically, this calculation is defined as:
θ = C/F(P = θ )
L,Circuit L,Unit
For example, consider the rougher-cleaner, no recycle circuit (C/F = P2). If both units
have a low bypass of 10%, the circuit’s low bypass may be calculated as P2 = (10%)2 = 1%.
Likewise, thehighbypassmaybecalculatedinasimilarfashion, substitutingtheappropriate
value for P.
Inordertopredictthecircuit’scut-point,afunctionalformmustbegiventotheP value.
For the circuit analysis software, Equation 5.1 is used. The choice of this function is based
on calculation simplicity and is rather arbitrary. Most known partition functions predict
similar behavior around the cut-point, so significant deviation is not expected regardless of
thespecificfunctionchosen(King, 2001). Byincorporatingthepartitionfunctionalform, the
analytical circuit solution, and proprietary solution algorithms, the circuit analysis software
is able to analytically predict the circuit’s relative change in cut-point as a function of each
unit’s α value
5.4 Applications
5.4.1 Simple Examples
To demonstrate the power of the software, the results of several simple two- and three-
unit circuits are presented. Figure 5.6 shows the two-unit circuit configurations, Figure 5.7
shows the three-unit circuit configurations, and Table 5.2 summarizes the analytical circuit
solutions and the calculated parameters of the circuits’ partition functions.
Since the relative cut-point is dependent on an inputted α value for each unit, an
arbitrary value of 4 was chosen for this analysis, in order to show the relative change between
the circuit configurations. A higher α value would result in a smaller change in the cut-point,
while a lower α value will result in a greater change. Furthermore, the high and low bypass
values were calculated by assuming a low bypass of 10% for each unit and a high bypass of
90% for each unit.
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5.4.2 Discussion
Theanalysisofthesimplecircuitsleadstoseveralsuggestiveresults. First,byexamining
the separation sharpness values for the two-unit circuits, the immediate conclusion is that
enhanced sharpness can only result from recycle streams. This conclusion was presented in
the original circuit analysis paper (Meloy, 1983).
The relative cut-point and bypass values follow an expected pattern. For the two-
unit circuits, the greatest change in cut-point is realized by the non-recycling circuits. The
cleaner circuits, which reprocess high property material, tend to increase the cut-point while
scavenger circuits decrease the cut-point in equal portion. Since the three-unit circuits are
both symmetric, the cut-point is not changed.
Additionally, a trade-off exists for the bypass values of the two-unit circuits. The
rougher-cleaner, no recycle circuit substantially reduces the low bypass, at the expense of
misplacing more high-property material, while the rougher-scavenger circuit reduces high
bypass at the expense of misplaced low-property material.
Interestingly, Circuit 5 shows no improvement in any parameter from that of a single
unit, despite the increased resources. This result implies that in more complex circuits,
inappropriately implemented separation units may be completely inert.
5.4.3 Industrial Application
In a prior communication, McKeon and Luttrell (2005) used the fundamentals of circuit
analysistomodifyaheavymineralsandswetplant. Theoriginalplantconsistedof686spiral
units with 14 stages of upgrading. The modified plant greatly reduced this requirement, as
it consisted of 542 units with 11 stages of upgrading. Given the magnitude and complexity
of these plants an analytical circuit solution and relative separation sharpness were not
presented. Rather, the authors reported actual performance gains which are summarized in
Table 5.3. While the modified circuit did not substantially enhance the ultimate recovery
(94.7% vs. 93.0%), it did significantly reduce the number of passes needed to achieve this
recovery (7 to 1).
To verify the suggestions and industrial applicability of the circuit analysis software,
these two circuits were analyzed to determine the analytical circuit solution and the relative
separation sharpness. For the analysis, all units were assumed to have the same P value.
Furthermore, since these spirals produced three products (concentrate, middlings, and tail-
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Virginia Tech | CHAPTER 5. AN ALGORITHM FOR ANALYTICAL SOLUTIONS AND ANALYSIS
OF MINERAL PROCESSING CIRCUITS
Table 5.3: Summarized data for heavy mineral sands plants as presented by (McKeon &
Luttrell, 2005).
Parameter Original Circuit Modified Circuit
Ultimate Recovery 93.0% 94.7%
Number of Required Passes 7 1
Number of Stages 14 11
Number of Spirals 686 542
ings), each unit was modeled as two units in series. Figure 5.8 shows a screenshot from the
software after calculating the analytical solution (C/F) andthe relative separationsharpness
(SE) for the original circuit. Figure 5.9 shows a similar screenshot for the modified circuit.
The software confirms the original authors’ claims. The second circuit showed a greater
separation sharpness (4.29 compared to 2.36) which was anticipated by the plant modifica-
tions and verified by the enhanced plant performance.
5.5 Summary and Conclusions
AsoftwarepackagehasbeendevelopedwhichintegratesMeloy’scircuitanalysisconcept
with solution algorithms that minimize the cumbersome algebra. The result is a tool which
can provide efficient and quick insight on mineral processing circuits without the need for
extensive data sets and computing resources. The implications of this tool are to provide
insight and guidance. The circuit analysis software will never completely offset the need
for simulation or empirical experience; however the solution algorithm may offset iterative
calculation for process simulations. Predictions of grade, recovery, and other performance
indicators will still require numeric simulation when a deterministic result is expected, and
empiricalexperienceisstillneededtoensurethatcommon-senselimitationsarenotexceeded.
Nevertheless, this tool is intended to limit the required number of simulations. Rather than
designing a circuit by performing multiple simulations on random and experience-driven
circuit configurations, the principles can guide the design and limit the field of proposed
designs. Such guidance can hasten the design process while providing truly engineered
circuit solutions.
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DESIGN
the valuable and invaluable components of the run-of-mine material. In solid-solid mineral
systems, these constituents are often labeled ore and gangue. Nevertheless, a more general
interpretation can be applied to consider the separation of any component which ultimately
increases profitability from any component which ultimately decreases profitability. For
example, coal drying can be viewed as a beneficiation process which separates the dry coal
material from the water to a point which sufficiently justifies the drying cost.
Given various physical limitations, the efficiency of separation in standalone units is
severely limited. To overcome this deficiency, mineral processing plants often employ staged
separation which can incorporate numerous separation devices of different sizes and opera-
tional characteristics. The unit operations and the interconnection between individual units
are collectively described as the separation circuit.
The design of separation circuits is an open-ended, ill-defined engineering problem
mostly experienced in two contexts: plant modifications and greenfield designs. The plant
modification problem involves adding resources to an existing plant to pursue the enhance-
ment of one or two process objectives (e.g. add a unit to increase overall recovery of fine
material). Plant modification problems often entail a constrained solution, and comparisons
between the initial and modified plant determine the real performance gains. The modifica-
tions may be limited to a specific section of the plant, as the costs, benefits, and risks of the
modifications are balanced.
Alternatively, the greenfield problem usually entails more creativity and enhanced risk.
The greenfield site does not have an existing plant to establish or compare site-specific
process objectives, so historic indicators and experimental data must be used to establish
benchmarks. Often, the circuit designer leverages prior experience to define a starting point,
whilelaboratoryandpilot-scaleanalysesareusedtorefinethespecificoperationalparameters
to produce in the final solution.
In both the greenfield designs and plant modifications, the final solution must answer
four questions:
1. Which separation processes are to be utilized?
2. What is the total size and number of units needed?
3. What are the standard operational parameters for each unit?
4. How should the interconnection between the units be configured?
Circuit designers often approach the four circuit design problems independently while
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DESIGN
incorporating knowledge from three paradigms: (1) empirical observation/traditional knowl-
edge; (2) simulation and phenomenological predictions; (3) circuit analysis and optimization.
6.1.2 Separation Circuit Design Techniques
Today, many state-of-the-art solutions largely utilize process models. Over the last
40 years, the fidelity of these models has increased from low-level empirical curve fits, to
phenomenological and other physics-based predictive models. Many mineral processing unit
operations have mature phenomenological models which require experimental testing to tune
the parameters. In general, the reliability of the simulation is principally dependent on
the size and accuracy of the data set. As a result, substantial experimental work must
be performed to ensure validity and repeatability. Several commercial software packages
incorporate these contemporary models and data analysis software. The most widely used
examples include Limn (Nageswararao, Wiseman, & Napier-Munn, 2004; Hand & Wiseman,
2010),JKSimMet(Cameron&Morrison,1991; Richardson,2002),andModsim(King,2001).
At a higher level of fidelity, purely theoretical models are in development for some unit
operations, but are still largely considered immature (Do, 2010; Kelley, Noble, Luttrell, &
Yoon, 2012). Furthermore, computation fluid dynamics (CFD) and discrete element method
(DEM) simulations are becoming increasingly popular as an alternative to experimental
testing.
Despite the sophistication of current process modeling and simulation, the circuit de-
signer’sexperiencecannotbeunderstatedordenigratedasadesigntool. Thecircuitdesigner
can consider pragmatic factors not inherent to standard process models (e.g. brand loyalty,
maintenance familiarity, process control complexity, etc.).
Lastly, circuit analysis and optimization is a broad category of design tools which in-
corporate fundamental analytics and numeric optimization. Some of these tools, such as
Meloy’s linear circuit analysis (Meloy, 1983), predate contemporary simulation since they
rely on simple algebra, rather than computationally intensive mathematics. Other examples
of analytical tools include procedures for determining the overall circuit size (M. Williams &
Meloy, 1991; Galvez, 1998), the relative size between parts of the circuit (Sutherland, 1981),
the amount of material recirculated (Loveday & Marchant, 1972; Lauder & McKee, 1986;
Loveday & Brouckaert, 1995), the point of reentry for recirculating loads (M. Williams &
Meloy, 1991; Galvez, 1998), and the unit which most greatly influences the overall circuit
performance (Lucay, Mellado, Cisternas, & Galvez, 2012).
Most recently, researches have attempted to synthesize the analytical techniques, the
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DESIGN
modelingandsimulationtools, andtoalimitedextent, theempiricalinsight, inordertoform
robust circuit optimizers. An extensive review of circuit optimization strategies has been
given by (Mendez, Galvez, & Cisternas, 2009). While older optimizers solved the problems
of stream interconnection and unit parameters separately, more modern tools resolve both
questions simultaneously. Often, the optimizers impose a generic circuit superstructure
which contains all of the possible solutions originally established by the circuit designer.
The final circuit is generated from this superstructure. Researchers have utilized numerous
optimization algorithms, including: genetic search algorithms (Guria, Verma, Gupta, &
Mehrotra, 2005; Guria, Varma, Mehrotra, & Gupta, n.d.) mixed-integer linear programming
(Cisternas, Mndez, Glvez, & Jorquera, 2006), integer programming, as well as other various
search strategies (e.g. Schena, Villeneuve,& Nol, 1996; Schena, Zanin, & Chiarandini, 1997).
Unfortunately, many of these methods have been criticized for the complexity of the final
circuit solutions, and the inclusion of impractical circuit elements, such as stream splitting
nodes.
Despite the veracity and plurality of circuit design tools, many remain underutilized,
ignored, or poorly understood with respect to industrial integration. Much of circuit design
(especially Question 4) is driven by trial-and-error, either in simulation or laboratory testing.
Realisticsimulationrequiresextensivelaboratorydatawhichmaybeunavailableoruntimely,
especially during the preliminary design phases. Analytical tools, such as linear circuit anal-
ysis usually involve cumbersome mathematics which often outweighs the perceived benefits
of the techniques. As a result, separation circuit design remains a labor-intensive and costly
process.
This paper presents a software package which may be used to assist the circuit designer
in both greenfield and plant modification problems, especially in the absence of detailed
feed and separation data. The Circuit Analysis Reduction Tool (CART TM) provides rapid
analytical solutions to user-defined circuit layouts. These analytical solutions may then be
used in numeric simulation, sensitivity analysis, or other various circuit design tools. In
particular, the integration of linear circuit analysis provides one basis for fundamental cir-
cuit comparison. This paper provides a detailed review of prior work which has utilized
analytical circuit solutions, namely Meloy’s linear circuit analysis (1983) and Lucay’s Sensi-
tivity Analysis (2012). Next, the CARTTMsoftware is described and presented in context of
these various analytical solution uses. An optimization approach to the circuit modification
problem is presented, and finally, the utility of the software is validated by an application
example. Opportunities for refinement and further study are described in the conclusions.
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6.2 Utilization of Analytical Circuit Solutions
Of the analytical circuit design tools, Meloy’s linear circuit analysis (LCA) and Lucay’s
sensitivity analysis (SA) are the most generally applicable for all separation systems, since
they are based on modular separation fundamentals rather than unique model-dependent
heuristics. Both methods of circuit analysis require the final circuit concentrate to be pre-
sented as an analytical function of the generic transfer functions for each unit.
6.2.1 Linear Circuit Analysis
Meloy first presented a method of determining the analytical circuit solution via simple
algebra. Several authors have described and utilized Meloys algebraic technique (Luttrell,
Kohmuench, Stanley, & Trump, 1998; Luttrell, Forrest, & Mankosa, 2002; McKeon & Lut-
trell, 2005, 2012). In summary, each unit is assumed to be a binary separator with a generic
transfer function, P . As a result, each unit produces a concentrate product, FP +n, and
n
a tailings product, F(1−P ). The algebra is extended by determining the downstream unit
n
feed values in terms of other transition functions. The final concentrate and overall circuit
transition function (C/F) are solved in terms of the various P values. Figure 6.1 shows a
n
simple example of this algebra.
Meloy’s LCA uses this analytical solution to determine the relative separation efficiency
between the circuit and a single unit. The slope of a single unit’s partition curve at 50%
recovery is a reasonable estimation of the unit’s efficiency. For many separators, this value
entails identifiable meaning as the imperfection or Ep value. By evaluating the derivative
of the analytical function at the 50% recovery point, the overall separation sharpness of the
circuit is identified. This value roughly corresponds to the relative increase in separation
sharpness from an individual unit to the overall circuit (in this paper, the value is termed
“Meloy’s circuit strength parameter”). A more rigorous description of the mathematics is
presented by Meloy (1983).
6.2.2 Sensitivity Analysis
Lucay’s SA is a more recent utilization of analytical circuit solutions. In SA, each unit
in the circuit is evaluated to determine its influence on the overall circuit. By identifying
the most influential unit, experimental and optimization efforts can be directed to specific
areas of the circuit which will return the greatest benefits. In the procedure, the analytical
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F’P (1-P )
1 2
F
P P C
1 2
F’ F’P
1
T
C (cid:3) F’P P
1 2
P P
1 2
T (cid:3) F’(cid:10)1-P ) C/F (cid:3)
1
1-P (cid:10)1-P )
1 2
F’ (cid:3) F(cid:13)F’P (cid:10)1-P )
1 2
F’ (cid:3) F/(cid:15)1-P (cid:10)1-P )]
1 2
Figure 6.1: Example of circuit analysis algebra.
expression for the circuit is first determined, and terms referring to units not under scrutiny
are lumped into a single constant 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 plotted for various expected values of the individual
recovery functions.
Local minima and maxima in the plots indicate the relative sensitivity of the unit under
scrutiny. This process is then repeated by taking the partial derivative with respect to each
unit, and the overall magnitude of each partial derivative is compared to determine most
influential unit in the circuit.
6.3 Analytical Solution Algorithm
After identifying the utility of the analytical circuit solution in fundamental separation
analysis, several authors have described procedures for simplifying cumbersome mathematics
associated with circuit reduction (Yingling, 1988; M. C. Williams, Fuerstenau, & Meloy,
1992). Unfortunately, these procedures require understanding of flowgraph reduction and
graph theory concepts. Even if those concepts are mastered, the algorithms still require
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manual calculation and somewhat obtrusive time requirements. M. C. Williams et al. (1992)
claims that an experienced practitioner can use the graph theory approach to solve a five
unit problem in ten minutes.
Despite these efforts, analytical circuit solutions remain underutilized. To reach their
potential value, the derived analytical expressions must meet two criteria: (1) undoubted
accuracy and (2) efficient determination. The first criterion refers only to the equation;
simply, does the derived equation match the true analytical solution of the circuit. If the
circuit designer has any cause to doubt the validity of the analytical expression, the utility
of any subsequent method is negated. When the algebraic or flowgraph reduction techniques
are followed steadfastly, this criterion will be met. The second criterion reflects the time and
effort required to achieve the solution. For many circuits, this value increases exponentially
as additional units and complex recycle patterns are introduced to the circuit. For any
manual technique, these two objectives cannot be consistently met simultaneously. If more
care is placed in ensuring an accurate solution, the process will be inefficient. Alternatively,
if the solution technique is hastened, accuracy cannot be undoubtedly confirmed.
This paradox has been resolved through the development of a computational-efficient
software-based analytical solution algorithm. This algorithm has been incorporated into
a simple user interface based within the Microsoft Excel platform. The resulting software
package, the Circuit Analysis Reduction Tool (CARTTM), allows users to construct a separa-
tion circuit using standard flowsheet drawing tools including separators, nodes, and streams.
Once the flowsheet is complete, the calculations are initiated, and the analytical circuit solu-
tion, aswellasMeloy’scircuitstrengthparameterandothercircuitparametersareproduced.
A prior paper has introduced this tool and shown its applicability in a heavy mineral sands
wet plant (Noble, Luttrell, & Silva, 2012).
Since the original presentation, the algorithm has been updated to accommodate differ-
ent recovery functions for the different units. Formerly, the algorithm produced an analytical
solution which assumed the transfer function to be identical for all units. While this assump-
tion allows the calculation of Meloy’s circuit strength parameter, it limits the application
of the analytical solution to real circuits, since circuit units rarely have identical recoveries.
Additionally, the multi-unit analytical solution provides other uses, via streamlined simu-
lation, simple sensitivity analysis, and operational parameter optimization. An example of
the software’s output is shown in Figure 6.2. This circuit is identical to the one solved
algebraically in Figure 6.1.
The analytical solutions produced by the CARTTMsoftware meets the two aforemen-
tioned criteria. The accuracy of the solutions has been validated by comparing the resulting
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DESIGN
solutions to those found from careful manual calculations. Approximately, 25 to 50 simple
circuits have been verified in this manner, including circuit sizes ranging from two to seven
units. Furthermore, the theory behind the algorithm ensures absolute scalability. The solu-
tion algorithm effectively does not distinguish between a small circuit and a large circuit in
any respect other than computational time.
With respect to computational efficiency, the algorithm far exceeds manual computa-
tion. The actual calculation time for most standard circuits (four to eight units) ranges up
to 6 seconds on standard laptop and desktop PC’s (Intel Core i7 2.70 GHz processor). If
all units are assumed to be identical, the calculation is somewhat faster. As an extreme
example, the highly-complex 19 unit circuit presented by McKeon and Luttrell (2012) was
solved considering unique unit transition functions. The algorithm produced the solution in
78 seconds.
6.4 Calculation Approaches for Circuit Simulation
Beyondfundamentalcircuitanalysis, analyticalsolutionsprovideseveralsupplementary
benefits to traditional circuit simulation, especially with regard to the calculation method-
ology for circuits that contain recirculating loads. In linear circuits (unit performance is
independent of unit feed), an analytical solution completely eliminates the need for an iter-
ative or computational solution. The circuit performance may be solved directly by simple
algebra. For nonlinear circuits (unit performance is dependent on unit feed), an analyti-
cal solution does not completely eliminate the need for iteration, but it does substantially
streamline and simplify the simulation procedure.
6.4.1 Iterative Approach
Many traditional circuit simulators employ an iterative technique to accommodate re-
circulating loads and solve for the final circuit products. In this approach, an initial guess is
chosen for the recirculating load, and the recovery calculations are repeated until the values
stabilize within some predetermined tolerance or until a specified number of iterations are
performed. This approach inherently introduces error unless a large number of iterations
are performed. The number of required iterations is arbitrarily defined, since the iterative
solution error depends upon the circuit complexity as well as the actual recovery values for
individual units.
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For nonlinear circuits, two nested iterative loops are required. As in linear circuits, the
iterationmuststabilizethevalueofthecirculatingloads, andthecalculationmustiteratively
solve for each unit’s actual recovery value since the recovery is dependent upon the unit feed
rate. While the computational power of most PC’s is sufficient to handle simple systems,
this approach becomes cumbersome for larger problems. Simulations involving double or
triple distributed parameters (e.g. flotation plants simulated with size, liberation, and rate-
constant classes) can be especially problematic.
6.4.2 Analytical Approach
The CARTTMsoftware provides a simplified calculation strategy for circuit simulation.
The CARTTMalgorithm produces an analytical solution to the final circuit concentrate as a
function of each unit’s individual recovery. For linear circuits, intermediate and circulating
streams do not need to be calculated, since the final concentrate value is given analytically.
The only information required for simulation is the individual recovery values for each unit.
With an analytical solution in hand, these simulations can easily be conducted by hand or
with simple spreadsheet software.
Alternatively, non-linear circuits still require an iterative or numeric solution technique.
While the analytical solution negates the need for recirculating load calculations, the in-
dividual unit recoveries must be determined numerically. One approach to solving these
problems is to write a system of equations consisting of the analytical circuit solution and
the nonlinear transfer functions for each unit. This system can then be solved by a standard
nonlinear numeric technique.
6.5 Optimization Algorithm
In addition to the analytical circuit algorithm, the CARTTMsoftware includes an opti-
mization algorithm, based on Meloy’s LCA. This optimization algorithm has been developed
to specifically address the circuit modification problem. Given an existing circuit, the algo-
rithm determines the best location within the circuit to add another separation unit without
altering the structure of the original circuit. The objective function, defining best location,
is given by Meloy’s circuit strength parameter.
The search algorithm first identifies the feasible and infeasible solution space. Various
practical constraints limit the potential solutions and enhance the efficiency of the optimiza-
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DESIGN
tion algorithm. Overall, the approach of defining the feasible space is tailored to the circuit
modification problem. The additional unit may be fed by any current circuit product or
placed between existing units such that the initial downstream unit is fed by the new unit.
Otherwise, the products of the new unit may be reintroduced at any point in the circuit.
Finally, after the additions, any product stream in the circuit may be recognized as the new
circuit concentrate.
The optimization algorithm utilizes a direct search technique which attempts every cir-
cuit combination that adheres to the stated constraints and falls within the feasible solution
space. The analytical solution is produced and the derivative is calculated for each design al-
ternative. Given the calculation efficiency of the CARTTMsoftware, several thousand circuit
calculations can be performed within a reasonable time frame. The optimization of adding
a fourth unit to an existing three unit circuit performed 350 circuit analysis calculations and
took approximately 30 seconds on a standard laptop PC with an Intel Core i7 2.70 GHz
processor.
6.6 Application Example
The CARTTMsoftware and the optimization algorithm were applied to a hypothetical
galena flotation plant, using the FLoatSim circuit simulator to produce quantitative results.
The plant has an existing configuration and performance capacity. In this example, the
circuit is to be modified by adding an additional unit which will enhance recovery and/or
grade without reducing either. Consequently, the addition must fundamentally enhance the
separation potential of the circuit rather than just shifting the circuit to a new operating
point on the same grade-recovery curve.
The plant consists of two primary circuit legs: a rougher bank and a cleaner bank. High
grade concentrate from the first rougher cell is passed directly to the final concentrate, while
concentrate from the remaining rougher cells is directed to the cleaning bank. The cleaner
cell is supported by a cleaner scavenger. Altogether, the circuit is represented schematically
by four binary separators (Figure 6.3a).
Initial simulations show that the circuit produced a lead recovery of 95.7% at a grade
of 57.22%. CARTTMsoftware indicated that Meloy’s circuit strength for this configuration
is 1.1901 (Figure 6.3b).
Inefficient trial-and-error simulations were conducted attempting various configurations
which added scavenging units to support the rougher section of the plant. None of these sim-
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DESIGN
(a) Circuit A - original configuration. (b) Analysis Summary for Circuit A.
Figure 6.3: Circuit A Analysis
(a) Circuit B - scavenger addition. (b) Analysis Summary for Circuit B.
Figure 6.4: Circuit B Analysis
ulations met the modification objectives, and none were able to increase the circuit strength
parameter. The best of these results produced a recovery of 96.3% at a grade of 56.65%
(Figure 6.4).
After several preliminary simulation attempts, the CARTTMoptimization algorithm was
invoked. The results showed that the circuit strength parameter could be increased to a
value of 2.00 by adding a cleaning unit after the initial rougher concentrate (Figure 6.5).
The additional cleaning stage naturally produced a higher grade product, but the additional
support also permitted the initial rougher cell to run at higher recovery value (via enhanced
air flow rate) without compromising the final product. As a result, more total material
was recovered in the rougher bank, while the overall circuit showed significantly enhanced
lead grade (62.70%) and enhanced recovery (96.11%). Table 6.1 summarizes these three
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DESIGN
are addressed by trial-and-error, either through data-intensive simulation or labor-intensive
laboratory experimentation.
ThispaperhasdescribedtheCircuitAnalysisReductionTool(CARTTM)andpresented
its utility as a fundamental tool for separation circuit design. The CARTTMuses proprietary
algorithms to produce an analytical circuit solutions which subsequently provides several
means of analysis and utilization. The derived analytical solution may be used in simple
simulations to estimate circuit recoveries and perform simple sensitivity analyses. Addition-
ally, the software automatically calculates Meloy’s circuit strength parameter and includes
an optimization algorithm which determines the next best place to add a unit to the circuit.
The utility of the software has been shown through an application example.
From the information presented in this presented in this paper, four conclusions are
derived:
1. The analytical solution produced by the CARTTMsoftware meets the two criteria for
utility: undoubted accuracy and efficient determination.
2. The analytical solution negates the need for iterative calculation in linear circuits.
Appropriate usage can also significantly streamline the simulation of nonlinear circuits.
3. The CARTTMsoftware has substantial value in preliminary circuit evaluation. The tool
can efficiently guide designers to preferred solutions and substantially reduce the need
for trial-and-error.
4. The CARTTMsoftware cannot replace model-based simulation. The final solution will
require fine parameter tuning and an estimate of expected performance measurements.
These tasks are best accomplished through experiments and simulation.
Despite the current power of the CARTTMsoftware, additional refinements will enhance
the applicability of the analyses to real circuit designs. Most significantly, the objective
function to the optimization algorithm must be adapted to reflect more relevant information.
Currently, the objective function is Meloy’s circuit strength parameter, which only considers
the relative efficiency of middling separation. In many separation systems, pure particles
represent a significantly higher value per unit weight and a higher portion of the circuit feed
(if well liberated). The efficiency of separation units which poorly distinguish these pure
particles (i.e. entrainment in flotation, bypass in cyclones) is not reflected in Meloy’s circuit
strength parameter. Consequently, an optimization routine based on middling separation
cannot produce valid results in these systems. Rather, a new objective function must be
devised which considers pure particle as well as middling separation.
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Virginia Tech | Chapter 7
The Partition Moment of Inertia as a
Technical-Economic Separation
Performance Measure
(ABSTRACT)
The partition curve is widely used in particulate separation to diagnose and compare
separation behavior between different operating conditions, feed characteristics, and unit
operations. Several traditional surrogate parameters have been defined in the literature and
used to consolidate information from the entire curve into a single value. Many of these pa-
rameters are based on the curve’s slope through the middling transition region, and common
formulations include the separation sharpness, Ecart probable, and imperfection. Unfortu-
nately, these surrogate measures fail to fully describe the process economics since the area
of interest is isolated to the middling particles. This flaw is further compounded consid-
ering the disproportionately high influence pure particles have on final process economics,
via increased incremental value and increased feed percentage in well-liberated systems. To
account for these common biases, a new performance measure, the partition moment of iner-
tia, is proposed. This paper describes the derivation of the partition moment of inertia and
demonstratesthecalculationforsingle-unitpartitionsaswellassimplecircuitconfigurations.
Finally, the veracity of the value is demonstrated in a coal separation case study.
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TECHNICAL-ECONOMIC SEPARATION PERFORMANCE MEASURE
7.1 Introduction
7.1.1 Partition Curves
Partition curves are commonly used in the mineral processing discipline to characterize
the separation efficiency of binary separators. First introduced by Tromp (1937), the par-
tition curve graphically shows how individual particles are distributed between concentrate
and tailings streams in a binary separator. Figure 7.1 shows a generic partition curve, high-
lighting the difference between an ideal and a real separation. The abscissa of a partition
curve is a continuous scale reflecting the magnitude of the separation property of interest (of-
ten denoted Z), while the ordinate defines the probability that a particle of a given property
will report to the concentrate product (often denoted P). Given the ease of two-dimensional
graphical analysis, partition curve curves are common in separations that exploit a single,
easily measured physical property.
Partition curves serve as a suitable measure of process performance as long as the
separation property corresponds to the process objective. For example, the performance
of classification cyclones is often characterized by partition curves showing recovery to un-
derflow as a function of particle size. In this case, the partition curve permits detailed
process evaluation since the predominant or sole separation property (particle size in this
case) matches the process objective (size classification). Conversely, traditional partition
curve analyses are largely absent in more complex processes where the separation property
is ill-defined or difficult to measure, such as in froth flotation. Nevertheless, the underlying
principles are still present, provided that the separation property corresponds to the process
objective (i.e. floatability corresponds to composition).
Several mathematical functions which define the partition curve have been proposed.
King (2001) has provided a detailed investigation of 9 different mathematical functions,
whileStratfordandNapier-Munn(1986)haveprovidedgeneralrecommendationsforsuitable
partition function formulations. By mathematically fitting experimental separation data to
a known partition function, the parameters of the model may be used for diagnostic or
comparative analyses (Armstrong & Whitmore, 1982; Rong & Lyman, 1985; Jowett, 1986;
Tamilmani & Kapur, 1986). These comparisons are common when comparing different
separation technologies which exploit the same separation property (jigs, vessels, and spirals,
for example). One common partition function is given by the Whiten model:
1
P(Z) = (θ −θ ) +θ . (7.1)
High Low 1+eα(1−Z) Low
where P is the partition probability, θ and θ are the high and low bypass values, α is
High Low
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a fitting parameter related to the partition slope, and Z is the normalized cut-point.
7.1.2 Traditional Partition Performance Measures
The overall geometric and algebraic characteristics of the partition curve provide infor-
mation on the type and quantity of misplacement occurring within the particulate separator.
For example, the location of the final asymptotes (θ and θ ) indicate the separator’s
High Low
susceptibility to bypass of pure particles. If the low property portion of the curve does
not close at 0%, some portion of the low quality product will always report to concentrate.
The amount of this misplaced portion is quantified by the location of the asymptote. A
second geometric feature used to characterize the separation performance is the slope of the
partition curve through the transition region. A steeper slope indicates that the separator
is increasingly selective, and an infinitively large slope indicates a perfect separation. This
value is often cited as the sole indication of performance in a partition separator in diag-
nostic and comparative studies. Two common formulations of this slope are given by the
Ecart probable (E ) and the imperfection (I) (Leonard, 1991; King, 2001). Many of the
P
mathematical partition functions also include a fitting parameter (often denoted α) which
directly corresponds to this slope. The Ecart probable is calculated by:
|d −d |
75 25
E =
p
2
whered andd refertothex-axispropertyvalueswhichrefertothe75%and25%partition
75 25
probabilities. The imperfection provides a way of normalizing the Ecart probable by the 50%
partition property value (d ):
50
E
p
I = .
(d −1)
50
The Ecart probable and the imperfection are commonly used in raw data analysis
since the values can be quickly obtained from a simple graphical analysis. A more precise
definition of the instantaneous partition slope (or separation sharpness, SS) can be derived
by differentiating Equation 7.1 at the cut-point. This reduction yields:
α(θ −θ )
High Low
SS = .
4
Unfortunately, when the partition slope is used as the sole characteristic performance
measure, information on the high and low bypass values is lost. A process with a high sepa-
ration slope could in fact be producing an inadequate product if the bypass is considerably
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high. The error area is one measure meant to overcome this limitation, while consolidating
information from the entire partition curve. Mathematically, the error area is the differ-
ence between the ideal separation and the true separation defined by the partition curve.
A graphical example of the error area is presented in Figure 7.1. If the partition function
(P(Z)) as well as the minimum, maximum, and desired Z values (Z , Z , and Z )
Min Max Desire
are known, the error area (EA) may be calculated by:
(cid:90) (cid:90) ZDesire (cid:90) ZMax
EA = dA = P(Z)dZ + (1−P(Z))dZ. (7.2)
A ZMin ZDesire
While the partition curve analysis is most readily applied to single separation units,
its principles are also applicable to full separation circuits. Linear circuit analysis provides
one methodology of extending the mathematics from a single unit to a full circuit (Meloy,
1983a). Here, simplealgebraisfirstusedtoderiveananalyticalcircuitrecoveryexpressionas
a function of individual unit recoveries. The derivative of this analytical expression is used to
determine the slope of the circuit partition curve through the transition zone. Circuit slope
valuesgreaterthanoneindicatethatthecircuitcandistinguishmiddlingparticlesbetterthan
a single unit, while values less than one indicate that the circuit reduces middling separation
performance. Similarly, the circuit analytical expression can be used to determine how other
partition factors (cut-point and bypass) are influenced by the configuration.
In this paper, the separation sharpness, imperfection, and error are are collectively iden-
tified as the traditional partition performance indicators. These values are commonly used
in industrial and academic settings as surrogate measures for real separation performance.
They conveniently reduce the full partition data to a single parameter that may be used
for comparative or diagnostic studies. Unfortunately, this consolidation results in a loss of
information on the real performance of the separation.
7.1.3 Micro-Pricing and Incremental Quality
Smelter and utility contracts often include premium and penalty clauses to incentivize
delivery of raw materials that meet a certain quality standard. These price adjustments
are applied to a base cost when the delivered quality deviates from a standard quality (i.e.,
“± $1.00 per ton for each 100 Btu/lb over/under 13,000 Btu/lb”). The “micro-pricing”
principle dictates that individual particles contribute independently to the final recovered
revenue (Luttrell, Honaker, & Yoon, 2004; Luttrell & Honaker, 2005; Luttrell, Keles, &
Honaker, 2009). Accordingly, the “incremental value” of each particle (or class of particles
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with similar composition) is determined by the contract penalty and premium values as
applied to that individual particle.
As an example, in a coal-ash system, contract specifications typically apply premiums
and penalties for heat content and ash values which deviate from a standard value. By
applying these price adjustments, each washability class can be defined by the incremental
value the recovered material in that class will contribute to the final revenue. Low ash,
high Btu classes will contribute an incremental value substantially above the base price,
while pure rock material will contribute a large negative return. Often this principle is
extended to show that the ideal cut-point occurs at the washability class that contributes
zero incremental value. Furthermore, optimal blending can be achieved when all parallel
circuits are operated at the same incremental quality (Salama, 1989; Lyman, 1993; Luttrell,
Catarious, Miller, & Stanley, 2000; Luttrell, Barbee, & Stanley, 2003; Luttrell et al., 2009;
Mohanta, Chakraborty, & Meikap, 2011).
Unfortunately, the traditional, derivative-based separation indicators fail to reflect the
economic incentives of particular separation processes and circuit configurations. Since the
partition slope only accounts for middling separation, performance measures derived from
this value cannot account for deviant behavior at the end of the curves. According to the
micro-pricing principle, particle classes at the ends of the curves have a much stronger influ-
ence on process economics than the middling material. Extreme Z values often correspond
to classes which invoke high premiums or penalties, provided that the separation property
corresponds to the process objective. Additionally, in a well liberated system, these pure
particles represent a higher portion of the feed material. When compared to the derivative-
based measures, the error area shows marked improvement since it considers misplacement
along the entire property axis. However, this value does not inherently give additional weight
to the pure particles which have a greater influence on final revenue. Consequently, all of
the performance measures fundamentally lack the ability to diagnose separation performance
from an economic perspective.
7.1.4 Overview
This paper proposes a new separation performance measure, the partition moment of
inertia (MOI). This parameter may be derived from standard partition analysis and is
analogous to the mass moment of inertia for rigid-body rotational dynamics. Calculation of
the parameter may be conducted for single units or extended to circuit configurations via
linear circuit analysis. From the physics perspective, the mass moment of inertia indicates
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TECHNICAL-ECONOMIC SEPARATION PERFORMANCE MEASURE
a body’s resistance to rotational motion about a given axis. Mass which is located further
away from the rotation axis is more heavily weighted in the calculation. Similarly, the
partition moment of inertia evaluates the partition error’s ability to “rotate” about a given
axis, namely the partition cut-point. In this calculation, misplacement further way from
the cut-point (i.e. pure particles) is more heavily weighted than misplacement of middling
material. This paper derives this formula using the mass moment of inertia as a blueprint for
the methodology. A precise mathematical formulation of the parameter is provided along
with sample calculations for single units and simple circuits. Finally, the veracity of the
performance indicator is investigated for a coal preparation case study.
7.2 The Moment of Inertia
7.2.1 Mechanical Background
In rotational mechanics, the mass moment of inertia (I) is a parameter which defines a
body’s resistance to rotation about a given axis. From the perspective of rigid-body kinetics,
the moment of inertia is the rotational equivalent of mass. By definition, mass signifies a
body’sabilitytoresistlinearacceleration; whereas, themomentofinertiasignifiesthebody’s
ability to resist angular acceleration. Mathematically, a three-dimensional body’s moment
of inertia is given by:
(cid:90)
I = ρ(r)r2dV (7.3)
V
where dV is an incremental volume element, r is the perpendicular radius from the axis of
rotation to the volume element, and ρ(r) is the mass density function. For uniform-density
bodies, the density function is constant and can be factored out of the integral. These
physical dimensions are shown for an arbitrary body in Figure 7.2.
The value of the mass moment of inertia may be increased by increasing the mass of
material located away from the axis of rotation. Pragmatically, this increase is most readily
achieved by either increasing the radius of the body or by increasing the peripheral density.
7.2.2 Applications to Single Separators
The incremental quality concept asserts that particles with extreme property values
have a larger incremental influence on the final recovered revenue. For example, in a simple
coal-ash system, pure coal and pure rock particles carry the largest incremental value when
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compared to intermediate middlings. Furthermore, since most mineral systems are well
liberated, the initial mass fraction of pure particles is much greater than the fraction of
middlings. Unfortunately, all traditional performance measures fail to explicitly account for
this natural economic bias. Conversely, a more descriptive separation performance measure
shouldplacealargeremphasisonpureparticlemisplacementsincethismaterialsubstantially
influences the final economic result.
The moment of inertia calculation provides a basis to apply differential weighting based
on the property class and its distance from the cut-point. The mechanical analogy is formed
by assuming the partition error area to be a thin plate with the thickness going into the
paper. The rotational axis is set parallel to the y-axis at the normalized cut-point, and
the density of the body is related to the incremental cost function or may be assumed to
be uniform if no costing data is available. The rotation of the partition error area plate
around the cut-point axis is governed by the moment of inertia. Larger bypass values result
in more peripheral mass, leading to a larger moment of inertia. Conversely, a lower partition
slope would lead to more overall mass and an increased moment of inertia. However, since
this mass is centrally located, it would not be weighted as heavily as the peripheral mass
influenced by the unit bypass. A visual representation of this principle is provided in Figure
7.3.
Mathematically, the partition moment of inertia (MOI) may be derived by applying
the geometry of Figure 7.3 to the generic mass moment of inertia calculation (Equation 7.3),
while assuming that the incremental cost function corresponds to the plate density. The
MOI derivation follows a similar pattern as the partition error area calculation (Equation
7.2) since the rotational geometry of interest is identical to the error area. The final value is
dependent on the specific partition function (P(Z)), the optional cost function (C(Z)), as
well as the minimum, maximum, and desired normalized property values (Z , Z , and
Min Max
Z ). The generic derivation is given by:
Desire
(cid:90) ZDesire (cid:90) ZMax
MOI = C(Z)P(Z)(Z −Z)2dZ + C(Z)(1−P(Z))(Z −Z )2dZ.
Desire Desire
ZMin ZDesire
(7.4)
As it is defined, the partition moment of inertia always places a higher weight on pure
particles. If incremental costing data is available, these values may be included to account
forthe“density”differenceoftherotatedarea. Ifnocostinformationisavailable, a“uniform
density” may be assumed. In this case, the unweighted MOI value still indirectly accounts
forthethepureparticles(bythesquareofthemomentarmlength). Ingeneral, theapproach
supports both methodologies, depending on the availability of data.
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To show how the derived MOI parameter relates to standard partition function pa-
rameters (α, high bypass, and low bypass), a simple single unit simulation was conducted.
The Whiten partition model (Equation 7.1) was used to produce a partition curve through
a factorial sweep of α (1 to 30) and low bypass (0 to 30%) values. Equation 7.4 was used to
calculate the partition moment of inertia, assuming a constant cost function (C(Z)) and Z
values of 0.5, 1.0, and 1.5 for the minimum, desired, and maximum, respectively. Figure 7.4
shows the results with MOI plotted as a function of α and low bypass.
This simulation indicates the theoretical relationship between MOI and α. While the
MOI integrals were calculated over a range of 0.5 to 1.0, the relative behavior between the
parameters will not change, provided that the cut-point is the average of the two extremes.
At low α values, small increases in α correspond to large decreases in MOI. At larger α
values (> 15 in this case) incremental changes in the partition slope have very little influence
on the partition MOI. Simply, these α values are elevated beyond the point of diminishing
returns. While these diminishing returns are intuitively well known, their impact on process
economics are ill-defined.
From a technical-economic perspective, the MOI-α-bypass relationship suggests that
notallgainsinseparationsharpnessproduceproportionalincreasesineconomicperformance.
Despite the perception of diminishing returns, proportional increases in α yield contrasting
results. For example, consider two separators operating with 0% bypass and respective α
values of 5 and 15. For the first separator (original α of 5), a twofold increase in α produces
a 74% reduction in MOI (MOI reduction from 1.19 to 0.31). Alternatively, a two fold
increase in the second separator (original α of 10) yields an 87% reduction in MOI (MOI
reduction from 0.10 to 0.013).
Alternatively, forafixedα value, alinearrelationshipexistsbetweenthelowbypassand
theMOI parameter. Asanticipated, absolutereductioninbypassdoesnotshowdiminishing
returns. Each unit of properly placed pure material corresponds directly to an increase in
the technical-economic separation performance.
7.2.3 Applications to Separation Circuits
Linearcircuitanalysisdescribeshowtheseparationsharpnessindicatorcanbeextended
from a single unit to a circuit of units by the analytical circuit solution (Meloy, 1983a,
1983b). In an analogous manner, the analytical circuit solution can be used to extend the
moment of inertia calculation from a single unit to a full circuit. Many of these steps become
increasingly computationally intensive as the circuit complexity increases. Numeric routines
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for the integral calculation are recommended in order to preserve the efficiency and integrity
of the performance indicator.
First, algebra or the CARTTMalgorithm (see Chapter 5) is used to derive an analytical
circuit solution in terms of unit P values. Next, all P values are assumed equal to produce a
simplified analytical solution. This analytical solution may be plotted against unit P values
from 0 to 1 to produce the circuit partition function. The moment of inertia calculation is
then applied to this circuit partition function by using Equation 7.4. The circuit recovery
expression (C/F) is substituted for P(Z), the x-axis value (P) is substituted for Z, and 0,
0.5, and 1 are assumed as the integration limits. The cost function may be implemented
as a function of unit P, or a uniform cost may be assumed if no data is available. The
final calculation yields the circuit MOI. Since many of these values are low in absolute
magnitude and difficult to compare, the values reported in this paper are always expressed
as a percent of the circuit MOI of a single unit (a value of approximately 0.01042). With
this convention, circuit MOI values above 100% correspond to configurations worse than a
single unit, while values less than 100% indicate improved separation.
To illustrate the calculation methodology, a sample calculation for the circuit MOI of
a single unit (C/F = P) is shown:
(cid:90) ZDesire (cid:90) ZMax
MOI = (C/F)(Z −P)2dP + (1−(C/F))(P −Z )2dP
Desire Desire
ZMin ZDesire
(cid:90) 0.5 (cid:90) 1
= (P)(0.5−P)2dP + (1−P)(P −0.5)2dP
0 0.5
(cid:90) 0.5 (cid:90) 1
= [0.25P −P2 +P3]dP + [(P2 −P +0.25)−(P3 −P2 +0.25P)]dP
0 0.5
(cid:90) 0.5 (cid:90) 1
= [0.25P −P2 +P3]dP + [0.25−1.25P +2P2 −P3]dP
0 0.5
(cid:20)
0.25 1 1
(cid:21)0.5 (cid:20)
1.25 2 1
(cid:21)1
= P2 − P3 + P4 + 0.25P − P2 + P3 − P4
2 3 4 2 3 4
0 0.5
= 0.005280+0.005208
= 0.01042
As more units are included in the circuit configuration, the complexity of the analytical
solution increases rapidly, thus reducing the likelihood of a simple integration. As more
terms are introduced into the equation, numerical integration becomes a much more useful
approach. For example, consider the circuit MOI calculation for a two-unit rougher-cleaner
with recycle to the head (C/F = P2/(1−P +P2)). A numeric routine is used to solve the
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integral after the second step:
(cid:90) ZDesire (cid:90) ZMax
MOI = (C/F)(Z −P)2dP + (1−(C/F))(P −Z )2dP
Absolute Desire Desire
ZMin ZDesire
(cid:90) 0.5 (cid:90) 1
= (P2/(1−P +P2))(0.5−P)2dP + (1−(P2/(1−P +P2)))(P −0.5)2dP
0 0.5
= 0.00743
MOI = 0.00743/0.01042 = 71.3%
Consequently, the addition of the cleaner with recycle produces a 71.3% reduction in the
MOI from that of a single unit.
To analyze and compare the utility of the circuit MOI parameter, the integral cal-
culation as well as the sharpness indicator (SE) have been evaluated for various two and
three-unit circuit configurations. Figure 7.5 shows the configurations that were considered
in this demonstration. All circuits were evaluated for both a standard operating condition
as well as several “bypass” conditions. In these conditions (as shown in C1 of Figure 7.5),
a defined portion of the feed bypasses the separation unit and reports directly to the unit
concentrate. This model corresponds to real processes which undergo a selective and a
non-selective recovery mechanism, such as flotation entrainment or screen blinding.
Figure 7.6 shows the circuit partition curves for the six simple configurations included
in this analysis. These plots depict the circuit recovery (C/F) as a function of individual
unit recovery, assuming each unit has the same P value. These partition functions indicate
whether a circuit is predominantly cleaning (circuit recovery is generally lower than unit re-
covery) or scavenging (circuit recovery is generally higher than unit recovery). For extremely
complex circuits, the predominant function can be difficult to identify by merely studying
the configuration. However, when the circuit partition function is plotted with a normalizing
line ((0,0) to (1,1)), zones of enriched or reduced recovery are easily denoted.
Table 7.1 summarizes the analytical solution, circuit SE, and circuit MOI values for
each of the six circuits under the standard, no bypass conditions. Table 7.2 extends these
results for 10%, 20%, and 30% bypass levels.
The aggregate circuit analysis data for the six circuits is summarized in Figure 7.7. In
this plot, circuit MOI is plotted against circuit SE for each unique circuit configuration
and unit bypass level. The circuit configurations are grouped explicitly by the series color
and marker style, while the unit bypass values are denoted only for the C6 data series.
Nevertheless, the unit bypass values for the other data series may be interpreted implicitly
by applying the same positional pattern. For each data series, the rightmost point (the
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F
F
C
F F T
10 90 F C C
C
T T F
T
C
T
C1: Single Unit C2: Scavenger with Recycle C3: Cleaner with Recycle
F F
F C C
C T T
T F F
C C
F T T
C F
T C
T
C6: Rougher-Scavenger-Cleaner
C4: Cleaner Open C5: Scavenger Open Circuit Full Recycle
Figure 7.5: Basic circuit configurations used in circuit MOI calculation comparison. Note
that from a circuit analysis perspective, each unit cell in the multi-unit circuits (C2 - C6)
actually corresponds to a feed splitter followed by a unit cell as depicted in C1. Splitters
were omitted from the latter drawings to conserve space.
Table 7.1: Circuit Analysis Comparison for Basic Circuits.
Circuit Circuit Analytical Circuit Circuit
ID Description Solution SE MOI
C1 Single Unit P 1.00 100.0
C2 R-S (r) P/(P2 −P +1) 1.33 71.3
C3 R-C (r) P2/(1−P +P2) 1.33 71.3
C4 R-C (o) P2 1.00 100.0
C5 R-S (o) 2P −P2 1.00 100.0
C6 R-S-C (r) P2/(2P2 −2P +1) 2.00 31.8
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highest SE value or the lowest MOI value corresponds to the 0% bypass condition. Each
successive data point (moving left and up) corresponds to increasing increments of unit
bypass.
The aggregated circuit MOI−SE trend mirrors the behavior exemplified in the single-
unit partition function analysis (Figure 7.4). At elevated SE values, incremental SE gains
yield diminished reductions in MOI, when compared to similar gains at reduced SE values.
Furthermore, thisgeneraltrenddoesnotproduceaone-to-onecomparisonforalldatapoints,
implying that the MOI performance measure will not always produce the same rankings as
the SE parameter. For example, consider the comparison between the C6, 20% bypass point
and the C3, 0% bypass point. The SE ranking shows preference to the C6 configuration
(SE value of 1.42 compared to 1.33 for C3), while the MOI ranking shows preference to the
C3 configuration (MOI of 71.31 for C3 compared to 77.91 for C6). This result is anticipated
since the MOI parameter places a larger penalty on bypassed pure particles.
To further illustrate the discrepancies between the two circuit analysis measures, Figure
7.7 includes quadrant designations centered around the single unit circuit (1,100). These
four quadrants indicate differences in comparative behavior. Quadrants I and III indicate
divergent conclusions, while Quadrants II and IV show similar conclusions; although, the
magnitude of the improvement may not correlate with the magnitude of the value. For
example, in Quadrant I, SE shows separation improvement while MOI shows separation
deterioration compared to a single unit. Alternatively, in Quadrant IV, both parameters
show circuit improvement. In principle, these quadrant axes may be centered around any
point on the plot to illustrate differences in comparisons at that point.
7.3 Application Example: Coal Separation Economics
7.3.1 Methodology
To analyze the veracity of the partition moment of inertia and other separation perfor-
mance indicators, a hypothetical coal cleaning case is presented. The last section, comparing
circuit MOI and circuit SE, indicated that the two parameters occasionally produce diver-
gent results in ranking circuit configurations. Furthermore, the two performance measures
show a nonlinear relationship, which indicates that both performance measures cannot di-
rectly correlate to economic performance simultaneously. These seeming contradictions indi-
cate that one performance measure likely provides a better indication of technical-economic
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performance.
This coal economics case study uses washability and sales contract data to calculate the
realrecoveredrevenuefromahypotheticaldensity-basedseparator. Theseparatorismodeled
underanumberofdifferentoperatingconditions,usingagenericpartitionfunction(Equation
7.1)todefinetheseparation. Fromthecalculatedpartitiondata,varioustraditionaltechnical
separation performance indicators (separation sharpness, error area, and imperfection) are
calculated along with the partition moment of inertia. All of these surrogate measures
are then compared to the economic results (recovered revenue and misplaced revenue) to
determine which performance indicator produces results which most closely correspond to
the economic output.
The coal contract data used in this example is shown in Table 7.3. While these val-
ues do not necessarily represent any real contract, they do generally correspond to typical
values found in steam coal contracts in the year 2013. These values, including the Btu pre-
mium/penalty, ash premium/penalty, and sales penalty are used to calculate the incremental
revenue by density class. In this example, impurity clauses, sulfur penalties, and moisture
requirements are ignored for the sake of simplicity. Sulfur standards will slightly increase
the incremental value if the cleaning unit selectively rejects sulfur, but the general conclu-
sions will remain. Since size and differential moisture reduction are not considered in this
example, the addition of a moisture clause in this contract will affect the incremental value
consistently for all density fractions. Nevertheless, all cleaned products are assumed to have
8.00% moisture which influences the heat content of the final product.
Despite the assumed simplifications, the contract does include premium and penalty
clauses for ash and heat content as well as a sales cost penalty. The washability data (Table
7.4)wasused alongwiththecontractto createacost function formulated solelyasa function
of particle density. This washability data represents a fairly well liberated, easy to clean coal,
with 63.2% of the mass in the two extreme density classes. A plot of 1/SG versus ash (Figure
7.8) shows a near linear relationship between the two washability parameters, supporting the
validity of the washability analysis.
Table 7.5 extends the washability data to demonstrate the incremental cost calculation.
The base price is constant for all density classes, regardless of the quality. The ash penalty is
applied per density class using the individual ash assay for the class as the basis for penalty.
The Btu penalty/premium is determined by first calculating the heat content incrementally
for each density class. This value is defined as the dry ash free heat (daf, 15,000 Btu/lb in
this case) less the portion of in-class ash and moisture. As mentioned above, 8% moisture is
assumed for all classes. With the heat content calculated for each class, the Btu adjustment
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100
80
60
40
20
0
0 0.2 0.4 0.6 0.8 1
1/SG
hsA
laudividnI
Figure 7.8: Individual ash versus 1/specific gravity from coal washability data.
is applied. The remaining positive revenue in each class (if any is remaining after the heat
and ash adjustments) is subjected to a 10% sales cost. A constant $2.25 sales cost is applied
to all classes, regardless of net revenue status. The final summation of the base cost and all
adjustment yields the net value as a function of SG.
Using the contract and washability data, a cut SG of 1.60 was selected. This value
corresponds to the ash class which produces nearly zero incremental revenue. Cut-points
greater than this value recover particles which result in a net penalty, while cut-points lower
than this value reject particles which result in a net premium. The micro-pricing principle
dictates that the optimal revenue is produced from this SG cut-point.
With the cut-point set, a two-dimensional parameter sweep simulation was conducted
using a factorial combination of α and low bypass values in the partition function. The low
bypass was simulated from 0 to 30% in increments of 1%, while α was simulated from 10
to 30 in increments of 0.5. In total, this matrix produced 1271 independent simulations.
For each simulation, the partition function was used to determine the recovery by density
class. The contract and washability data was then used to determine the product quality
and incremental contract value of each density class. The recovery and incremental value
were multiplied and summed to produce the final recovered product value.
In addition to the economic indicators, several performance measures were calculated
for each simulation run. The traditional performance indicators included the separation
sharpness, the imperfection, and the error area. The moment of inertia was calculated using
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both the unweighted (no costing data) and weighted methods. In the weighted case, the
absolute value of the incremental cost by density class was used as the density function.
7.3.2 Results and Analysis
In total, 1,271 parameter sweep simulations were conducted to compare recovered rev-
enue against several technical and technical-economic performance measures. The graphical
results of these simulations are shown in Figures 7.9 and 7.10. In these plots, the raw eco-
nomic performance is plotted as a function of the α value and the low bypass used in the
simulation. These plots are shown as smooth surfaces with a corresponding color contour on
the x-y plane. The misplaced revenue plot (Figure 7.10) was formulated by subtracting the
maximum revenue ($3,570) from the recovered revenue. While these parameters are simple
inverses of each other, the availability of both curves will allow convenient comparison to
inverse performance measures. For example, increased separation sharpness yields increased
revenue. Consequently, the separation sharpness parameter can be easily compared to the
recovered revenue curve since one should drive the other. Alternatively, increases in the
moment of inertia, error area, and imperfection correspond to reductions in separation effi-
ciency. Consequently, these parameters will be compared to the misplaced revenue in order
to form a one-to-one correspondence.
Figure 7.11 shows the recovered revenue as a function of the five separation performance
indicators as individual plots. The data included on each individual plot corresponds to the
1,271 parameter sweep simulations. Data lines indicate simulations conducted at constant
bypassbutvaryingαvalues. Ineachcase,thelowerlinesindicatethehigherbypassvalues. A
legend is omitted from this plot since too many data series are included, and since the intent
is not to compare how bypass and α influence the various parameters. Instead, the intent is
to analyze the spread of real performance values (in this case, the recovered revenue) that are
represented by a single surrogate performance indicator value. From a metrics standpoint, a
moreusefulsurrogateperformancemeasurewouldrepresentasmallrangeofrealperformance
values by a single value, with the ideal surrogate measure forming a one-to-one relationship
between the derived and real measures. Simply, the different drivers of performance (α and
low bypass, in this case) need to be buried into the surrogate performance measure.
Figure 7.11 indicates that both moment of inertia values (and especially the cost-
weighted moment of inertia) are much better surrogate measures than the traditional per-
formance indicators, based on the spread of revenue values that a single surrogate value can
produce. For example, depending on the specific mix of α and low bypass values, a sepa-
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ration sharpness of 4 can correspond to a recovered revenue of $2,400 to $3,300, a range of
$900. Over the full span of the data, this range does not decrease substantially. On the other
hand, the widest range of the weighted MOI occurs at a value of approximately 1.5. Here,
the recovered revenue only spans a range of $300 ($2,500 to $2,800). Furthermore, as the
values extend to the periphery, the range of recovered revenue values reduces substantially.
Graphically, these range comparisons show that the moment of inertia parameters produce
more aggregate linear behavior, which implies that the “hidden” drivers of recovered revenue
are also proportionally driving the moment of inertia.
A second way of making the same comparison is by looking at side-by-side surface
contours of the real and the surrogate measurements as a function of α and low bypass.
Theseplotswilldepictthespecificregionswheretheeconomicperformancebehavessimilarly
to the derived measure. A more ideal surrogate measure would produce similar, though
proportionally adjusted, behavior across the entire region. Figures 7.12 and 7.13 show this
data for the traditional and moment of inertia performance indicators, respectively. The
left column of plots show the derived performance measures, while the right column of plots
shows the corresponding real economic measure. In this case, all parameters (with the
exception of separation sharpness) were compared side-to-side with misplaced revenue, since
higher values in these measurements correspond to decreased performance. Alternatively,
separationsharpness was compareddirectlywithrecoveredrevenuetomatchthe appropriate
comparison. The missing regions of the imperfection contour indicate that the value is non-
defined for those parameter values. This result was often due to the partition curve never
crossing 25% which is required for the imperfection calculation.
Visual comparisons between the surface contours indicate that the moment of inertia
parameters show close agreement with the financial indicators across the full range of param-
eter values. Figure 7.12 shows that the separation sharpness and error area values are too
heavily influence by the α value when compared to their corresponding economic indicators.
The slope of the contour lines in the separation sharpness plot does not change as a function
of bypass value. On the other hand, the slope of the moment of inertia parameters does
reflect the changes in low bypass value. This visualization can be quantified by counting
the number of contour lines that a particular indicator crosses at a fixed y-axis (α) value.
At α = 20, separation sharpness crosses 1 contour line, error area crosses 2, imperfection
crosses4, unweightedMOI crosses5, andtheweightedMOI crosses5. Forthecomparisons,
the recovered revenue crosses 6, and the misplaced revenue crosses 5. While the graphical
rendering can be manipulated to force these results, the failure of traditional performance
measures to account for unit bypass is evident in the geometry of the contours.
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7.4 Summary & Conclusions
This paper has introduced the partition moment of inertia (MOI) as a technical-
economic separation performance indicator. This value is derived from the mechanical def-
inition of the mass moment of inertia and used to account for the disproportionately high
influence pure particles have on the final process economics. In most systems, these pure
particles represent a higher portion of the initial feed weight (via liberation) as well a higher
unit economic value (via incremental costing). A robust technical-economic performance
indicator must account for this consistent bias.
From a mechanical perspective, the mass moment of inertia penalizes mass in propor-
tion to its distance from the rotational axis. Mass farther away from the axis of rotation
contributes more heavily to the final moment of inertia value. In an analogous fashion, a
traditional partition curve places the valuable pure particles at the periphery of the x-axis,
most distant from the central cut-point. If the error area of the partition curve is modeled
as a thin plate rotating about the cut-point, the mass moment of inertia of that plate would
disproportionately penalize error area at the periphery. Furthermore, an incremental costing
function may be applied to represent the “density” of the thin plate. Thus the final model
produces a derived performance indicator, the partition moment of inertia (MOI), which
successfully accounts for the value of misplaced pure particles.
Formally, the partition moment of inertia is defined by:
(cid:90) ZDesire (cid:90) ZMax
MOI = C(Z)P(Z)(Z −Z)2dZ + C(Z)(1−P(Z))(Z −Z )2dZ
Desire Desire
ZMin ZDesire
where C(Z) is the cost or density function, P(Z) is the partition function, and Z , Z ,
Min Max
and Z are the the minimum, maximum, and desired normalized property values, re-
Desire
spectively. This calculation can be extended to circuit configurations by substituting the
analytical circuit solution as (P(Z)) and the unit transfer function as Z. The integration
limits in the circuit case are defined as 0, 0.5, and 1.
Sample calculations for six basic circuit configurations show that at fixed unit bypass
levels, the circuit separation sharpness (SE) produces similar rankings as the partition mo-
ment of inertia (MOI). However, as bypass is factored into the system, the two factors
often produce conflicting circuit rankings. To defend the partition moment of inertia as the
superior measure, a parameter sweep simulation was conducted using coal separation and
economic data. The results compared the real economic performance of the separator, in
terms of recovered revenue, to several derived performance indicators (separation sharpness,
error area, imperfection, unweighted MOI, and cost-weighted MOI). Contour plots and
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holistic comparisons show that the MOI behavior correlates very well to the pure economic
measures across a full range of separation operating points.
From this work, three final conclusions are formed:
1. Traditional performance indicators do not deliberately or inherently account for the
disproportionate value of pure particles. Most are based solely or largely on the sepa-
ration of middling material which, while difficult to separate, do not account for large
incremental values or a large portion of the feed material.
2. In comparing circuit configurations, the circuit MOI and circuit SE produce similar
but occasionally slightly divergent trends. With the addition of a costing function this
discrepancy is further intensified.
3. The aggregated, near-linear trend between the moment of inertia and the pure eco-
nomic indicators (Figure 7.11) indicate that the MOI parameter robustly accounts for
the various driving parameters of the economic gain. The lack of any aggregate trend
among the traditional performance indicators shows their inability to account for syn-
ergistic gains and losses by multiple driving parameters. As a result, the traditional
performance indicators should be restricted to comparisons along a single dimension
(such as only changes in α at a constant bypass value).
7.5 Bibliography
Armstrong, M., & Whitmore, R. (1982). The mathematical modeling of coal washability,
1st australia coal preparation conf.
Jowett, A. (1986). Anappraisalofpartitioncurvesforcoal-cleaningprocesses. International
Journal of Mineral Processing, 16(1), 75–95.
King, R. (2001). Modeling and simulation of mineral processing systems. Elsevier.
Leonard, J. (1991). Coal preparation, 5th ed. SME.
Luttrell, G., Barbee, C., & Stanley, F. (2003). Optimum cutpoints for heavy medium
separations. Advances in Gravity Concentration, 81.
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Experimental Validation of Analytical
Circuit Design Methodologies
(ABSTRACT)
A virtual experimental study was conducted using the Working Model 2D program to
compare various circuit configurations. This dynamic discrete element modeling program
provides an environment to construct physical systems and observe how these systems will
react to various input conditions. The electrostatic physics model was used to construct a
particulate electrostatic separator within the virtual environment. This separator was then
arranged into 17 circuit configurations and tested under 3 levels of forced unit bypass. A feed
charge consisting of 55 particles of varying electrostatic charges was implemented into the
separation circuits and monitored to determine the selectivity of the various configurations.
This experimental data was compared to the circuit analysis separation sharpness (SE),
moment of inertia (MOI), and yield score (YS) parameters. The results indicate that the
circuit analysis methodology provides exceptional capacity to rank circuits on the basis of
mass yield and selectivity. Isolated comparisons and holistic results are presented.
8.1 Introduction
8.1.1 Background
Over the past 30 years, numerous scientific and industrial studies have proposed various
mineral processing circuit design methodologies which attempt to optimize the allocation of
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separation resources within the processing plant. Over this time period, the methodologies
have undergone various transitions, including the increased inclusion of process models and
computeralgorithmsaswellasthegeneralshiftfromapurelytechnicalsolution(e.g., Lauder
& McKee, 1986) to partially economic solutions (Abu-Ali & Sabour, 2003). This author
has extensively reviewed the circuit design literature elsewhere (see Chapter 2), but some
noteworthy concepts in the development of the discipline include Sutherland’s (1981) use
of process models to distribute residence time in a flotation circuit, Williams’ and Meloy’s
decade-long development of “circuit analysis” as a means of generically evaluating separation
circuits (for example, Meloy, 1983a; Williams, Fuerstenau, & Meloy, 1992), and Yingling’s
(1990) use of a superstructure and optimization algorithm.
Despite these scientific developments, very few studies include a comprehensive empir-
ical validation of separation circuit design methodologies in a controlled experiment. For
example, Sutherland (1981) compares 26 flotation circuit configuration designs, with each
design distributing the total plant residence time differently between the rougher, cleaner,
and scavenger banks. While this work did lead to several generally-accepted principles (such
as the benefits of a balanced configuration), the author only considered two different stream
configurations: the rougher-cleaner-recleaner and the rougher-scavenger-cleaner. Further-
more, the data was generated from purely mathematical (and purposefully simple) flotation
models, rather than real separation devices.
Other circuit design methodologies, such as linear circuit analysis, were originally pre-
sented as purely analytical thought exercises with no external validation (Meloy, 1983a,
1983b; Meloy, Clark, & Glista, 1986). The lack of original experimentation likely led to the
exclusion of this principle in real industrial circuit design problems for nearly 20 years. Nev-
ertheless, other authors noted the legitimacy of Meloy’s mathematical approach (Yingling,
1990; Luttrell, Kohmuench, Stanley, & Trump, 1998), and the the holistic methodology was
eventually used to redesign a heavy mineral sands plant, leading to drastic performance im-
provements(McKeon&Luttrell, 2005, 2012). Whilethisresultlooselyverifiestheprinciple’s
applicability, the study only included a binary comparison of two circuit configurations, and
the final circuit analysis was not extended to completion, given the overbearing complexity
of the circuits. As a result, the outcomes of the design methodology were not quantified,
leaving the potential for alternate solutions. Other studies, such as the column flotation
circuit comparison presented by Tao, Luttrell, and Yoon (2000) extend the methodology to
more circuit designs (six in this case) but still fail to quantify the outcomes in a controlled
setting.
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8.1.2 Review of Analytical Methods
Traditionally, the circuit analysis methodology encompasses two primary stages. First,
the analytical circuit solution is produced via algebraic manipulation. A generic transfer
function (P ) is defined for each binary separation unit, resulting in simple mathematical
i
functions for the concentrate (C = PF) and the tailings (T = (1−P)F). These principles
are extended to account for multi-unit separations, including open circuits and recirculating
loads. An extensive review of this technique is provided in Chapter 5. Second, the analytical
circuit solution is used to define a separation sharpness indicator (SE). This value is mathe-
matically defined as the derivative of the analytical circuit solution at P = 0.5. For the sake
of simplicity in calculating thederivative, allP valuesare assumed to be equal (P = P). The
i
resulting SE value defines the relative sharpness of the circuit’s partition curve to the sharp-
ness of a single unit. Pragmatically, this value reveals the circuit’s capacity to distinguish
middling or otherwise indiscriminate particles.
A new software package, the Circuit Analysis Reduction Tool (CART), has streamlined
both of these circuit analysis steps. This program provides a graphical interface for user-
definedflowsheetinputandusesproprietaryalgorithmstosolvetheanalyticalcircuitsolution
and the SE parameter. The utilization and application of this software is presented in
Chapter 6.
Various real separation processes include some type of non-selective recovery mecha-
nisms, generally labeled unit bypass. Some examples include entrainment in flotation and
hydrocyclones, blinding or plugging in industrial screens, and entrapment in spiral sep-
arators). The circuit analysis algebra provides a basis to include these phenomenon via
non-selective splitting. In the circuit analysis model, a defined portion of the feed (equal
to the bypass) is redirected around the separation, automatically reporting to the appro-
priate product. Unfortunately, while the algebra provides the potential for inclusion, the
SE performance measure cannot entirely account for the performance losses. Typically,
the addition of unit bypass does prompt a reduction in the SE parameter; however, since
the value only inherently considers middling separation, the influence of unit bypass tends
to be understated. This limitation is magnified considering the high incremental value of
non-middling particles and the typically high percentage of these particles in well liberated
feeds.
To overcome these limitations, this author has proposed a secondary performance mea-
sure which is derived from the analytical circuit solution. The partition moment of inertia
(MOI) represents the physical resistance of the partition area against rotation about the
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cut-point. This derived parameter is analogous to a mechanical moment of inertia which
represents a rigid body’s resistance to rotation about an axis. In the mechanical moment
of inertia, mass further away from the rotational axis contributes exponentially more to the
MOI value since that mass is more difficult to rotate. Similarly, partition error further away
from the cut-point contributes exponentially more to the partition MOI since these non-
middling particles contribute substantially more to lost revenue. An extensive mathematical
treatment of the partition MOI as well as its calculation in single units and circuits is given
in Chapter 7.
Along with these measures of circuit selectivity, the analytical solution may also be
used as an indicator of the circuit mass yield. When considering two partition separators,
the area between the partition curves designates the incremental difference in circuit yield.
The same principle applies to circuit partition curves. By integrating the difference between
two circuit analytical functions, the differential yield between the two circuits is obtained.
Mathematically, this integral is given by:
(cid:90) 1
Y = W(Z)[(C/F) −(C/F) ] dP
i 2 1
0
where Y is the incremental yield, W(Z) is the mass distribution function, and (C/F)
i i
are the circuit analytical functions. As it is formulated, a positive Y value indicates that
i
circuit 2 produces a greater yield, while a negative value indicates that circuit 1 produces
a greater yield. The mass distribution function is included to account for non-uniform feed
distributions. If this function is not known, a constant value (i.e. 1) may be substituted and
factored out of the integral.
A generic formulation of this incremental yield calculation is given by the yield score
(YS). This value is defined as the yield differential between a given circuit and the single-
unit circuit (C/F = P). The yield score indicates the yield gain or loss compared to a single
unit and may be used as a holistic ranking parameter used in circuit analysis. Since circuit
analysis generally relies on limited data, the weighting function is commonly disregarded in
calculating the yield score. Formally, the yield score is given by:
(cid:90) 1
YS = [(C/F)−P] dP. (8.1)
0
The yield score inherently carries no information on circuit selectivity. However, the
yield score does indicate a circuit’s ability to recover pure mass of feed material. While
yield is never a sole process objective, the yield score is still useful in intermediate or incre-
mental optimization which targets circuit production since recovered tonnage often strongly
influences the circuit’s revenue.
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8.1.3 Working Model Program
This study evaluates various circuit designs using a virtual experimental analysis con-
ducted with the Working Model 2D software package (Copyright (cid:13)c 2005-2013 Design Sim-
ulation Technologies, Inc.). This commercial computer aided engineering program provides
an environment for dynamic discrete element modeling, based on various Newtonian physics
principles. The Working Model 2D development environment provides an interface to con-
struct simple geometries (circles, arcs, polygons), which model real physical entities in a
virtual two-dimensional environment. The intensive physical properties of the user-defined
geometries(includingfrictioncoefficients, theelasticcoefficient, andtheparticlecharge)may
be adjusted independently or selected from a default menu which includes common materi-
als (plastic, rubber, steel, rock, etc.) Various physical motion constraints, including locking
joints and pin joints, may be applied to the geometries in order to construct simple machines.
Finally, the user may apply several constant or potential force sources (point loads, torques,
springs, rotational motors, gravity fields) to the original geometries. The Working Model
program then uses dynamic Newtonian physics models to predict how the geometries will
react to the environmental conditions as a function of time. Physical outputs of the system,
body particle position, velocity, and acceleration may be logged during the simulation.
The physical models in the Working Model 2D program include functions for simple
collision models as well as various force fields, including gravity, wind, magnetics, and elec-
trostatics. The application of these models has been incorporated in various other scientific
studies covering a wide range of disciplines, including prosthetics (Dechev, Cleghorn, &
Naumann, 2001), biomechanics (Linnell, Wu, Baudin, & Gervais, 2007; Delattre & Moretto,
2008), energy harvesting (Wang, Chen, & Sung, 2010), and robot design (Thueer & Sieg-
wart, 2010). A cursory review of the literature in the mineral processing field yielded no
published studies which have used Working Model. Nevertheless, Working Model was used
in this study to analyze a virtual electrostatic separation device arranged in various circuit
configurations. The multi-body, dynamic environment provided the means to create discrete
particles as well as the actual separator. The simulation then uses the physical models to
depict how particles are separated as they flow through the system.
8.1.4 Overview
This study empirically evaluates the circuit analysis design methodologies via controlled
and comprehensive experimentation. The Working Model 2D program is used to generate 17
circuit configurations which are tested under various levels of force bypass. This approach
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models the behavior of real physical systems while providing the context for unlimited circuit
variations and cost-effective, yet rigorous, data analysis. In addition to the experimental
separation data, the CARTTMprogram is used to define the SE value and the MOI value
for each of the 17 circuit configurations at varying degrees of anticipated bypass.
The remainder of this paper describes the experimental details and the specific bulk
experimental results. The virtual experimental setup which utilizes the Working Model 2D
program is first presented. The specific geometries of the virtual separator as well as the feed
characteristics and data post-processing algorithms are described. Next, comparative results
show the correspondence between the analytical circuit evaluations (SE and MOI) and the
real separation performance measured in the virtual experiments. Finally, opportunities for
further study are described in the conclusions.
8.2 Experimental
8.2.1 Experimental Setup
The Working Model 2D program was used to create a hypothetical electrostatic sepa-
rator. This device uses a strong positively charged electrostatic plate which causes falling
material toeither be pulledor throwndepending upon thegiven chargeof the individual par-
ticle. Two product bins are arranged to collect the processed material, given the trajectory
after being influenced by the electrostatic plate. A range of particle charges was included in
the feed to model both “liberated” and middling constituents.
The specific geometry of the standard single-unit separator (Figure 8.1) includes a
hopper with a feed distributer, an electrostatic plate, a separation divider, and two product
bins. For all simulations, the electrostatic plate was set to a constant charge value, 6.0×10−5
Coulombs. This value was selected from initial shakedown testing to produce a moderately
inefficient separator. Higher charge values produce better separation; however, high single-
unit separation efficiency tends to mask the gains produced by the circuit configuration. In
the simulated environment, this electrostatic plate measures 1.1 by 0.3 meters.
The location of the separation divider and the shape of the feed hopper define the cut-
point for the separator. These geometries were set so that the standard single unit cut-point
would be at the zero-charge particle. The separation divider has a height of 1.8 meters, and
the center point is 2.2 meters from the left edge of the bottom plate. The bottom plate has
a total length of 4.76 meters. The entire device, including the hopper, is 8.5 meters tall, and
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The basic geometry of the electrostatic separator does not represent or correspond to
any “real” process. Conversely, the Working Model device was designed solely to produce
a naturally inefficient separation within a virtual environment. The single-unit separator
was then duplicated and reconfigured to produce numerous simple circuit configurations
so that the final results of this virtual experiment diagnose efficiency gains and losses due
to circuit configuration. An example of a rougher-scavenger-cleaner circuit in the working
model environment is shown in Figure 8.3. The intent of this study is not to investigate
the realism of the modeled separation process; rather, the intent is to quantitatively study
the influence of circuit design on separation performance in an inherently stochastic process.
Other merits and criticism of this approach are thoroughly discussed in Section 8.4.1.
The feed charge includes 11 different particle types with charges varying from -1.0×10−7
to 1.0×10−7 Coulombs in increments of 2.0×10−8 Coulombs. Each distinct particle charge
corresponds to a specific color, and each measures 10 cm in diameter. Five particles of
each charge are included, resulting in a total feed of 55 particles. Table 8.1 summarizes the
standard feed charge for the electrostatic separator.
8.2.2 Circuits
During the experiments, 17 unique circuit configurations were tested at three levels
of forced bypass. Figure 8.4 shows a simplified circuit analysis schematic for each circuit
configuration. Various code letters ranging from C1 to C31 are used to designate the circuits.
C1 is the standard single-unit cell, and the remaining configurations fall into one of three
generic design approaches:
1. C2 - C5: Two unit open and recycle circuits. Configurations include rougher-cleaner
and rougher-scavenger arrangements.
2. C12 - C17: Various rougher-cleaner-recleaner circuits. Configurations include all po-
tentially beneficial recycle patterns.
3. C26 - C31 Variousbalancedrougher-scavenger-cleanercircuits. Configurationsinclude
all potentially beneficial recycle patterns.
Deleterious configurations were purposely omitted from the analysis. For example,
rougher-cleaner-recleaner circuits which re-direct tailings streams forward in the circuit were
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not included. Nevertheless, the remaining configurations are only considered potentially
beneficial as circuit analysis and the expermental results may show that some configurations
waste resources and inhibit separation performance.
These particular circuits were selected in order to model common decisions posed to
circuit designers. For example, the choice of a recleaner cell or a scavenger cell in the three
unit case is common for many flotation circuits. The direction of the force bypass in these
experiments disqualified the consideration of rougher-scavenger-scavenger circuits. Since
material is short circuited to the concentrate rather than the tailings in these experiments,
the additional recleaning circuits were prioritized. Had the bypass forced material to the
tailings, the decision would be reversed.
8.2.3 Procedures
The virtual electrostatic separation experiments were conducted within the Working
Model 2D program. Upon launching the program, the appropriate single-unit geometry
was constructed. This cell was then copied to create the desired circuit superstructure.
The 55 particle feed charge was placed in the first hopper using a semi-randomized initial
configuration. With all of the geometries, physical properties, and initial conditions of
the experiment set, the Working Model virtual test was initiated. While the program was
running, the particles were observed as they passed throughout the entire circuit. Once all of
the particles came to rest in a final product bin, the total recoveries were tallied by counting
the particle colors in each bin.
Recycle streams were implemented manually. For these circuit configurations, specific
product bins were modeled to represent recycle streams. After the initial test run was com-
pleted, particles which were recovered in these bins were manually moved to the appropriate
circuit point and a supplemental run was conducted. During these supplemental runs, all of
the original feed charge was maintained in the test; however, the appearance of the original
feed was turned off, focusing the visual attention on the recirculating particles. As a result,
the original feed particles still influenced the separation behavior; however, the supplemental
test runs only showed the recirculated particles of interest. Additional supplemental runs
werecompleteduntiltheentirerecirculatedloadeventuallycametorestinafinalproductbin
rather than a recirculation bin. For some circuit configurations as many as 10 supplemental
runs were required to finalize the circuit.
Each circuit configuration was tested for five independent experimental runs in order
to appraise the reproducibility of the results. The additional test runs were reconfigured
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F
10 90 F
C
T
C1: Single Unit
F F
C C
F F T T F F
C C C C
T T F F T T
C C
T T
C2: Scavenger with Recycle C3: Cleaner with Recycle C4: Cleaner Open C5: Scavenger Open Circuit
F F F F
C C C C
T T T T
F F F F
C C C C
T T T T
F F F F
C C C C
T T T T
C13: Rougher-Cleaner-Recleaner C14: Rougher-Cleaner-Recleaner C15: Rougher-Cleaner-Recleaner Cleaner
C12: Rougher-Cleaner-Recleaner Open Recleaner Back 1 Recleaner Back 2 Back 1
F F
C C F F F F
T T C C C C
T T T T
F F
C C
T T
F F
F F C C
C C T T
T T
C16: Rougher-Cleaner-Recleaner C17: Rougher-Cleaner-Recleaner Recycle C26: Rougher-Scavenger-Cleaner C27: Rougher-Scavenger-Cleaner
Countercurrent All to Head Scavenger Con to Cleaner Cleaner Tail to Scavenger
F F F F F F F F
C C C C C C C C
T T T T T T T T
F F F F
C C C C
T T T T
C28: Rougher-Scavenger-Cleaner C29: Rougher-Scavenger-Cleaner C30: Rougher-Scavenger-Cleaner C31: Rougher-Scavenger-Cleaner
Double Cross Cleaner Tails Recycle Full Recycle Scavenger Con to Feed
Figure 8.4: Circuit Configurations used in Working Model Simulations. Note that from a
circuit analysis perspective, each unit cell in the circuits (C2 - C31) actually corresponds
to a feed splitter followed by a unit cell as depicted in C1. Splitters were omitted from the
latter drawings to conserve space.
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by randomizing the initial feed condition. Various particles were randomly selected and
moved to other positions in the feed hopper, and the size and location of the feed distributer
were adjusted. This randomization produced slightly varying results, as the specific particle-
particleinteractionschangedwiththealteredfeedconditions. Therecoveryresultspresented
in this study predominantly reflect the simple average of the five test runs, while the error
bars correspond to one standard deviation of the five test population.
In total, 17 circuit configurations were tested at three levels of forced bypass. Each
condition was repeated five times, leading to 255 independent virtual experiments.
8.2.4 Experimental Post-Processing
The results of each experimental run were enumerated using a custom MATLAB image
analysis script (Copyright (cid:13)c 1994-2013 The MathWorks, Inc.). A screenshot from one of
the product bins is input into the routine along with the number of pixels in a single ball.
This value is dependent upon the screen resolution and the zoom level in the Working Model
program; however, a quick analysis of a single ball easily yields this calibration. The script
file then decodes the image using the CIE L*a*b* color space, counting the total number
of image pixels with a designated color. Different colors in the L*a*b* color space are
designated by two-dimensional coordinates in a 200 x 200 matrix. The Euclidean distance
between two points in this plane corresponds directly to color distinction. The actual value
in a specific matrix element represents the number of pixels with that color coordinate. The
colors selected for the electrostatic particles were sufficiently distanced in the color plane to
allow definitive distinction. Figure 8.5 shows a sample 2D histogram depicting the initial
ball charge. The white intensity of a given point indicates the number of counted pixels at
that point.
With the 2D pixel histogram constructed, the total pixels of each particle color were
summed. These totals were divided by the pixels per ball calibration, and the final results
were automatically exported into Excel. Figure 8.6 shows the screen output from the MAT-
LAB script for a sample product. This process was repeated for each product bin in a given
simulation.
Numerous mathematical and analytical methods were used to consolidate the results.
In lieu of artificial pricing or contract data, the true separation capacity of the circuit was
determined by the total recovered charge parameter. This value was determined by summing
the total charge present in the final concentrate product bin. The number of particles of a
given color was multiplied by the charge of that color, and this product was summed for all
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a*
*b
−100
−80 Blue Purple
−60
−40
Light Blue
−20
0 Cyan
Grey Pink
20
40
Maroon
60 Yellow
Dark Green Red
80
Orange
Green
100
−100 −50 0 50 100
Figure 8.6: Sample image analysis collage used in data post-processing. Collage includes
original image, L*a*b* two-dimensional histogram, and final ball count.
particle colors. Mathematically, this values is given by:
N
(cid:88)
TC = N C
i i
i=1
where TC is the total recovered charge, N is the number of recovered particles in color class
i, and C is the charge of particles in color class i. The TC value applies a natural pre-
mium/penalty mechanism since valuable particles contribute an increasingly positive value,
while invaluable particles contribute an increasingly negative value. The net sum is subject
to both increases and decreases from individual particle classes.
Along with the mass yield and total recovered charge statistics, partition curve fitting
was also used to consolidate the data. For this analysis, the parameters of the Whiten par-
tition function, including separation sharpness (α), normalized cut-point (Z), high bypass
(θ ), and low bypass (θ ), were determined for each averaged experimental run by minimiz-
H L
ing the sum-of-the-squared error between the predicted and experimental data points. The
Whiten partition function defines the particle recovery (P) by:
(cid:20) (cid:21)
1
P = (θ −θ ) 1− +θ . (8.2)
H L 1+eα(Z−1) L
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For most analyses, the raw experimental data is preferred over the smoothed partition fit;
however, these parameters are still useful in identifying how the circuit configuration influ-
ences key parameters, including cut-point and bypass.
Several analytical circuit measures were determined for the various experimental circuit
configurations. The CARTTMprogram was first used to determine the simplified circuit
analytical function, which was then utilized in downstream analysis. The circuit SE and
circuit MOI were used as indicators of circuit selectivity, while the yield score was used
as the indicator of mass yield. These values were determined for each circuit configuration
at various unit bypass levels. These analytically-derived values were then compared to the
experimental values to evaluate the correlations.
8.3 Results
8.3.1 Working Model Simulations
The Working Model 2D program was used to simulate 255 circuit runs. This section
presents the consolidated data for these experiments as well as detailed data for specific
experimental points. The bulk experimental data, including recovery values for each test
run is included in the Appendix to this dissertation.
First, the experimental data was used to create partition curves showing recovery to
concentrate (right bin) as a function of individual particle charge. Figure 8.7 shows partition
curves for the single-unit circuit (C1) at the three levels of forced bypass. The data point
represents the average recovery for the five experimental runs, while the error bar represents
the standard deviation of the experimental values. The line was found by fitting the Whiten
partition model (Equation 8.2) to the data by minimization of the weighted sum-of-the-
squared error.
This analysis is extended for the other 16 multi-unit circuit configurations. Figures 8.8,
8.9, and 8.10 show the experimental partition curves for the no, medium, and high bypass
conditions, respectively. Collectively, these plots illustrate the circuit’s ability to influence
the partition cut-point, separation sharpness, unit bypass. Additionally, Table 8.2, shows
the fitting parameters used to determine the Whiten partition model for each data set.
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Finally, two bulk measures of circuit performance were calculated directly for each
circuit configuration: the mass yield and recovered charge. These values were calculated
directly from the experimental data and are completely independent from the partition
analysis. These values correlate to traditional particulate processing performance indicators
and collectively describe the separator’s efficiency and selectivity. Tabular data for the yield
and recovered charge are presented for each experimental run in Table 8.3.
8.3.2 Circuit Analysis
In addition to the Working Model Simulations, the CARTTM(see Chapter 6) program
was used to conduct circuit analyses on each of the 17 circuit configurations at various levels
of bypass ranging from 0 to 35%. Figure 8.11 shows the circuit partition functions for each
of the multi-unit circuits. These graphs present the simplified analytical circuit solution,
assuming the recovery in each unit cell is equal and no unit bypass is present. The addition
of individual unit bypass changes the circuit analytical function, typically increasing the
complexity. The graphs show the final circuit recovery as a function of the individual unit
cell recoveries. These partition functions are used to determine the circuit SE and circuit
MOI values. Once again, these graphs are only valid in the 0% bypass case.
Tabular data for the circuit analysis parameters are shown in Tables 8.4 and 8.5 for the
circuit SE and circuit MOI parameters, respectively. This data is presented as a function
of unit bypass, which is identical for all units throughout the circuit. These data indicate
that as unit bypass increases, the quality of the separation deteriorates; however, the specific
circuit configuration defines the degree of deterioration. Simply, the performance of some
circuits is less susceptible to large bypass in individual units. Graphical presentations of this
data for the 16 multi-unit circuits is presented in Figures 8.12 and 8.13.
8.4 Discussion
8.4.1 Justification for Experimental Methodology
In this work, the Working Model 2D software package was used to create a virtual
separation device based on dynamic discreet element physics-based modeling. While the
model does not correspond to or represent any “real” physical process, the virtual device
still performs a “real” particulate separation, which is inherently inefficient and probabilistic.
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Unlike a pure numeric model, the Working Model experiment provides a distribution of
results. Theseparationispredominantlyinfluencedbytheintensivephysicalpropertiesofthe
feed material; however, the specific loading of the particles into the separator creates natural
particle-particle interactions which also ultimately influence the separation performance.
Changes in the specific particle arrangement generate slightly different results; however, the
bulk distributions show that the device produces a meaningful separation.
Inprinciple, circuitanalysisonlydescribestheroleofunitinterconnectioninmanipulat-
ing separation behavior; the actual separation process is irrelevant. Since the circuit analysis
principlesarefundamentallyderivedforgenericbinaryseparations,theresultsareuniversally
applicable to any inefficient particulate separation device, regardless of the process model or
operation. Fromthemineralprocessingperspective, theprinciplesandimplicationsofcircuit
analysis are applicable to any unit operation which generates a naturally inefficient binary
separation, includingdense-mediavessels, hydrocyclones, magneticseparators, screens, flota-
tion cells, etc. As a result, the circuit analysis methodology can be validated by considering
any separation device, even a virtual device, provided that the separation is probabilistic
and that the unit can be arranged in a circuit.
The inclusion of the forced bypass element provides an additional non-selective re-
covery mechanism creating systemic bias. Many real separation devices undergo similar
inefficiencies, often denoted entrainment or entrapment. While these non-selective recov-
ery mechanisms are ever-present in real separation devices, the magnitude of the bypass
is often proportional to some other operational parameter. For example, entrainment is a
non-selective recovery mechanism inherent to flotation and hydrocyclones, whereby fine par-
ticles are carried into products by hydraulic flow rather than by the separation mechanism.
In these cases, the amount of entrained material is often proportional to the amount of re-
covered water. The virtual separator mimics this phenomenon through the forced bypass
element. The proportion of the bypass is independently controlled by the geometry of the
device.
From a pragmatic perspective, practitioners understand that bypass inefficiencies can
be mitigated by the circuit design. Circuit analysis provides a mechanism to incorporate
non-selective recovery, provided that the magnitude of bypass is known a priori. Given the
designofthevirtualseparator, thebypassparametermaybeadjustedindependentlywithout
influencingotherprocessvariables. Thisfactorishardtopredictandcontrolinrealprocesses
since the factors controlling bypass often influence other selective recovery mechanisms as
well. For example, water recovery in flotation may be increase or decreased by adjusting
froth level, air flow rates, or frother dosages; however, all three of these parameters are also
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known to influence true flotation rates as well.
Finally, the Working Model platform enabled the efficient analysis of numerous exper-
imental runs, including repeat trials to gauge experimental consistency. In this study, 255
independent trials were conducted. The resource required to pursue this degree of experi-
mental work exorbitant in the laboratory scale and absolutely prohibitive in the industrial
scale, especially when one considers the analytical costs, including sample preparation and
assays. Furthermore, the lack of feed consistency in any real experiment would likely invali-
date the work since this variability could likely exceed the measured performance difference
between some circuits.
8.4.2 Circuit Yield Rankings
The ultimate goal of circuit analysis is to provide holistic, semi-quantitative rankings
of circuit configurations in order to distinguish more efficient from less efficient designs.
While direct correlation is beneficial, a direct mathematical relationship between the circuit
analysis parameters and the true separation performance is not necessary to form general
rankings. Ultimately, these rankings may be used to select a limited number of top circuit
candidates which may then be subject to downstream experimental analysis, modeling, and
circuit simulation.
Applying this principle, the results from the Working model experimental study must
be interpreted holistically. Rather than forming a direct correlation between the various
parameters and scrutinizing every point, a better aggregate analytical method looks at all
the rankings simultaneously. While ignoring the respective values, this type of analysis
questions whether circuit analysis produces the same conclusions as the experimental data
in terms of identifying good, moderate, and bad configurations.
In this section, polar plots are used side-by-side to compare the experimental results
and the circuit analysis parameters. A generic, instructional polar plot is shown in Figure
8.14 which explains how to interpret the remaining polar plots in this section. By analogy,
the polar plot is a bar graph that has been rolled around a single point. Angular positions
on the polar graph indicate different circuit configurations, while the radial distance from
the central point indicates the magnitude of the given value. The three circular segments of
the graph distinguish the three general types of circuits considered in this study. Finally a
concentric red circuit indicates the average parameter value for all 17 configurations. Figure
8.14 includes angular labels which identifying circuit position. All other polar plots follow
this same angular convention.
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Thepolarplotisusedtoprovidequickside-by-sidecomparisonsbetweentheexperimen-
tal parameter and the corresponding circuit analysis parameter(s). Since qualitative conclu-
sions are desired, the magnitude difference between the individual polar plots is irrelevant.
Instead, a “good” comparison produces a similar shape with similar circuit configurations
exceeding the average. This condition indicates that the circuit analysis parameter produces
results which agree with the experimental data. Given the number of circuit configurations
in this study, the polar plot holds a visual advantage over side-by-side or overlapping bar
graphs. The angular positions and relative magnitude are easily identified and compared
by examining the shape of the polar plot. Alternatively, the bar graph begins to loose
convenience beyond the comparison of six to ten elements.
The circuit mass yield is analyzed first. Figures 8.15, 8.16, and 8.17 show the compar-
ative polar plots for the no, medium, and high forced bypass conditions, respectively. In
each of these plots, the experimental yield is shown in the left plot, while the circuit analysis
yield score is shown in the right plot. The circuit analysis bypass values selected for the
comparison were based off of the partition curve for the single unit circuit (C1). For the
single unit, the natural unforced bypass was determined to be approximately 10%. As a
result, this experimental data was compared to the circuit analysis parameters calculated
with a unit bypass of 10%. Following the same logic, 25% and 35% were selected as the unit
bypass levels for the medium and high bypass conditions, respectively.
In all three bypass cases, the experimental yield shows good correlation to the circuit
analysis yield score. C2, C5, C27, and C29 consistently show the highest yield values,
as well as elevated yield scores. All of these circuits represent a net scavenging condition
which logically leads to increased yield values. All of the rougher-cleaner-recleaner circuits
show reduced yield values which also correspond to expectation, since additional cleaning
stages increase product quality at the expense of reduced yield. As anticipated, C12 (the
rougher-cleaner-recleaner open circuit) shows the lowest yield value, which is additionally
corroborated by the lowest yield score in all three bypass conditions.
Figure 8.18 summarizes the yield-yield score comparison for all the available data. Each
data point indicates a unique circuit conditions with the bypass level noted in the legend.
This plot shows exceptional agreement between the yield and the yield score. The Pearson’s
correlation between the two data sets is 0.936, statistically confirming the strong positive
correlation. While the trend is not purely linear, the relationship indicates that the yield
score is a good indicator of circuit yield for the purpose of comparison.
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Experimental Data Circuit Analysis SE Circuit Analysis MOI
C12 15C5 C12 C5 C12 3C5
C4 C4 C4
C13 C13 1.5 C13
C3 C3 C3
C14 C14 C14
5 C2 C2 1 C2
0.5
C15 C15 C15
C1 C1 C1
C16 C16 C16
C31 C31 C31
C17 C17 C17
C30 C30 C30
C26 C26 C26
C29 C29 C29
C27 C28 C27 C28 C27 C28
Recovered Charge x10−7 C Circuit SE 100 / Circuit MOI
Figure 8.19: Polar selectivity plot for no forced bypass condition. Actual bypass value used
for the circuit analysis calculation is 10%.
8.4.3 Circuit Selectivity Rankings
In addition to circuit mass yield, the circuit selectivity can be analyzed via circuit
analysis and the experimental data. In this case, two circuit analysis parameters were used
as indicators of circuit selectivity: the separation sharpness (SE) and the circuit moment
of inertia (MOI). Once again, the side-by-side polar plots are used to compare the data
sets. Figures 8.19, 8.20, and 8.21 show the comparisons for the no, medium, and high
bypass data. As in the yield score calculation, the circuit MOI and circuit SE values were
calculated at unit bypass levels of 10%, 25% and 35% for the no, medium, and high bypass
levels, respectively. These values were determined from the partition curve analysis of the
single unit circuit (C1).
IntheMOI polarplot,theradialaxisisplottedas100/circuitMOI. Thismanipulation
was implemented to ease the visual comparison between the plots. In the calculation of
circuit MOI, smaller values correspond to more selective circuits. In this data set, the other
two parameters (recovered charge and circuit SE) reflect inverse behavior: larger values
correspond to more selective circuits. To ensure consistent comparisons, the inverse of the
circuit MOI is used so that larger values correspond to more selective circuits.
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Virginia Tech | CHAPTER 8. EXPERIMENTAL VALIDATION OF ANALYTICAL CIRCUIT DESIGN
METHODOLOGIES
In general, the experimental data compares well to both circuit analysis parameters. In
the no bypass condition (Figure 8.19), both circuit analysis parameters show high preference
to C16, C28, and C30, while largely dismissing C1 and C5. The experimental data agrees
on both accounts. Under the no bypass condition, C16 and C30 show the best selectivity in
terms of recovered charge, while C1 and C5 are the worst circuits, experimentally.
As the unit bypass was increased, the circuit analysis parameters (especially, the circuit
MOI) show stronger preference to the rougher-cleaner-recleaner circuits. This result is
confirmed experimentally, as many of these circuits show similar elevated performance at
the medium and high bypass conditions (Figures 8.20 and 8.21).
Figure 8.22 summarizes the recovered charge-SE and recovered charge-MOI compar-
isons for all of the available data. Each data point indicates a unique circuit configuration
at different bypass levels. In both cases, the overall trends strongly correlate, implying that
both factors are applicable for circuit ranking. Quantitatively, the Pearson’s correlation be-
tween the recovered charge and SE is 0.805, while the correlation between the recovered
charge and MOI is -0.916.
Further analysis of Figure 8.22 shows regions where the circuit analysis parameters are
better predictors of actual performance. In both case, intermediate selectivity values (recov-
ered charge between 8 and 13 ×10−7C) are predicted well by both parameters, as indicated
by the consistent linearity in this region. Alternatively, neither parameter predicts perfor-
mance of poor circuits well, as the points strongly deviate from the otherwise overwhelming
trend. Finally, highly selective circuits are better predicted by the MOI parameters. This
graph retains linearity through this region, while the SE parameter shows substantial de-
viation. This result indicates that the MOI parameter values are scalable and comparable
along most levels of meaningful selectivity. Since few instances will require selection between
two poor circuits, the deviation in this region is unsubstantial.
8.4.4 Three-Unit Utilization
One criticism of traditional circuit analysis is evident in a common three-unit utiliza-
tion problem. Without consideration of unit bypass, the circuit analysis SE parameter
dictates that the rougher-scavenger-cleaner, closed circuit is the best utilization of three
units. This configuration (C30) produces an SE value of 2.0, with the next best option
being the rougher-scavenger-cleaner double cross (C28, SE = 1.667). However, many indus-
trial flotation applications often utilize rougher-cleaner-recleaner circuits to pursue various
process objectives and meet quality specifications. Thus a discrepancy is produced between
239 |
Virginia Tech | CHAPTER 8. EXPERIMENTAL VALIDATION OF ANALYTICAL CIRCUIT DESIGN
METHODOLOGIES
15
14
13
12
11
10
9
8
7
6
0 50 100 150 200 250 300
Circuit MOI
)C
7−E(
egrahC
derevoceR
latoT
15
14
13
12
11
10
9
8
7
6
0 0.5 1 1.5 2
Circuit SE
)C
7−E(
egrahC
derevoceR
latoT
C1
C2
C3
C4 C5
C12
C13
C14
C15
C16
C17
C26
C27
C28
C29
C30
C31
Figure 8.22: Aggregate experimental results: total recovered charge plotted against MOI
and SE.
the suggestion of circuit analysis and the industrial trend.
One method that overcomes this disagreement is the inclusion of unit bypass in the
circuit analysis SE calculation. As unit bypass is increased, the selection of a recleaning
unit over a scavenging unit becomes intuitively more favorable, and this intuition is matched
by the circuit analysis projection. While the SE parameter is slightly influenced by unit
bypass, theoverallparameteronlyinherentlyreflectstheseparationofmiddlingparticles. As
a result, the SE parameter will tend to undervalue the bypass misplacement, thus retaining
a scavenging unit at elevated bypass values that are not justified economically.
To overcome this limitation of the SE parameter, the circuit MOI value was derived
(seeChapter7). Thisparameterplacedadditionalweightonpurematerialthathasbypassed
the separation stage. Consequently, the MOI parameter should be a better indicator of real
separation performance as unit bypass becomes more significant.
The experimental Working Model data verifies this hypothesis. As an isolated compari-
son, the performance of C1 (single unit), C17 (rougher-cleaner-recleaner), and C30 (rougher-
scavenger-cleaner)iscompared. Figure8.23showsthepartitioncurvesforthesethreecircuits
at the three levels of forced bypass. In all three instances, C17 shows the most substantial
bypass reduction, while C30 shows the greatest separation sharpness increase. Finally, C17
240 |
Virginia Tech | CHAPTER 8. EXPERIMENTAL VALIDATION OF ANALYTICAL CIRCUIT DESIGN
METHODOLOGIES
100
90
80
70
60
50
40
30
20
10
0
−1 −0.5 0 0.5 1
Particle Charge (E−7 C)
)%(
yrevoceR
No Bypass
100
90
80
70
60
50
40
30
20
10
0
−1 −0.5 0 0.5 1
Particle Charge (E−7 C)
)%(
yrevoceR
Medium Bypass
100
90
80
70
60
50
40
30
20
10
0
−1 −0.5 0 0.5 1
Particle Charge (E−7 C)
)%(
yrevoceR
High Bypass
C1
C30
C17
Figure8.23: ExperimentaldataandfittedpartitioncurvesforC1, C17, andC30. C1=single
unit; C17 = rougher-cleaner-recleaner, all recycles to feed; C30 = rougher-scavenger-cleaner,
all recycles to feed;.
shows the greatest cut-point shift, while C30 tends to retain the cut-point of a single unit.
The overall performance of these three circuits is compared as a function of unit bypass
(Figure 8.24). The top plot shows the experimental selectivity expressed as total recovered
charge. Obviously, as the bypass increases, the total recovered charge diminishes due to
additional recovery of negatively charged material. The lower two plots show the circuit SE
parameter and the circuit MOI parameter calculated as a function of unit bypass. In all
three cases, the single unit circuit is not a valid alternative for selection, but is shown as a
baselineforothertwodatasets. Theexperimentaldataindicatesthattherougher-scavenger-
cleaner circuit is preferred for unit bypass values less than 15%. At higher bypass values,
the scavenger should be alternatively utilized as a recleaner. The SE parameter predicts
this crossover occurring at 22%, while the MOI predicts it at 18%. While neither parameter
strictly corresponds to the experimental data, this example does confirm the circuit SE’s
tendency to undervalue unit bypass. Instead, the moment of inertia parameter is better
formulated to account for this phenomenon.
As an additional note, C17 and C30 were experimentally and analytically selected as
the best circuits (out of the original 17) over the given bypass range. The only distinction
between the three data sets is the point of the crossover.
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Virginia Tech | CHAPTER 8. EXPERIMENTAL VALIDATION OF ANALYTICAL CIRCUIT DESIGN
METHODOLOGIES
8.4.5 Comparison of Performance Measures
The aggregated circuit analysis data for all 17 circuit configurations is summarized in
Figure 8.25. In this plot, circuit MOI is plotted against circuit SE for each unique circuit
configuration and unit bypass level. The circuit configurations are grouped explicitly by the
series color and marker style, while the unit bypass levels are implied by the point position.
For each data series, the general rightmost point (the highest SE value or the lowest MOI
value) corresponds to the 0% bypass condition. Each successive data point (moving left and
up) corresponds to increasing increments of unit bypass.
The aggregated circuit MOI−SE trend mirrors the theoretical behavior exemplified in
the single-unit partition function analysis (see Chapter 7). At elevated SE values, incremen-
tal SE gains yield diminished reductions in MOI when compared to similar gains at reduced
SE values. Furthermore, this general trend does not produce a one-to-one comparison for
all data points, implying that the MOI performance measure will not always produce the
same rankings as the SE parameter.
To further illustrate the discrepancies between the two circuit analysis measures, Figure
8.25 also includes quadrant designations centered around the single unit circuit (1,100).
These four quadrants indicate differences in comparative behavior. Quadrants I and III
indicatedivergentconclusions,whileQuadrantsIIandIVshowsimilarconclusions; although,
the magnitude of the improvement may not correlate with the magnitude of the value. For
example, in Quadrant I, SE shows separation improvement while MOI shows separation
deterioration. Alternatively, in Quadrant IV, both parameters show circuit improvement.
In principle, these quadrant axes may be centered around any point on the plot to illustrate
the differences in comparisons to that point.
Given these apparent discrepancies, both performance measures cannot simultaneous
predict real circuit performance to the same precision. However, the working model exper-
imental data has repeatedly shown superior correlation to the MOI parameter, especially
when unit bypass is substantial. Both the consistent linear trend, even for highly selective
circuits (Figure 8.22), and the improved ability to select the appropriate three-unit utiliza-
tion (Figure 8.24) support utilization of the circuit MOI in addition to, or in favor of, the
circuit SE.
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Virginia Tech | CHAPTER 8. EXPERIMENTAL VALIDATION OF ANALYTICAL CIRCUIT DESIGN
METHODOLOGIES
8.5 Summary and Conclusions
This paper has described a virtual experimental analysis that was used to evaluate the
separation efficiency of various circuit designs. The Working Model 2D dynamic discreet
element modeling environment was used to construct and analyze a virtual electrostatic
separation device. This device was incorporated into 17 distinct circuit configurations and
tested at three levels of forced unit bypass. The results of the experimental analysis were
compared to the circuit analysis projections, using the yield score (YS), relative separation
sharpness (SE), and partition moment of inertia (MOI) as analytical performance indices.
The results of this study yield four conclusions:
1. The Working Model 2D program is suitable for circuit comparisons. Though the spe-
cific geometry of the device used in this study did not mimic any real separation
process, the virtual separator showed generally stochastic behavior and mimicked real
separation phenomenon. Particle interaction and feed loading influence the final sepa-
ration performance; however, the general behavior fits standard partition curves very
well.
2. The selection of an optimal separation circuit is often dependent upon the degree of
unit bypass. For the two unit case, the circuit analysis and the experimental data show
that C3 (rougher-cleaner with recycle) and C4 (rougher-cleaner open circuit) are the
best configurations in this study. Had the direction of the forced bypass been reversed
to the tailings streams, the rougher-scavenger circuits would likely have been selected
instead.
3. For the three unit-case, the selection between C30 (rougher-scavenger-cleaner, all re-
cycled to feed) and C17 (rougher-cleaner-recleaner, all recycle to the feed) is strongly
governed by the unit bypass. The experimental data shows that for bypass levels lower
than 14-16% the rougher-scavenger cleaner is the better option, while the decision is
reversed for greater bypass values. The circuit SE shows this crossover occurring at
22%, while the circuit MOI shows the crossover at 18%.
4. The bulk results indicate that both circuit analysis methods correspond very well with
real circuit performance. While the comparisons are not always direct correlations,the
overall results show that the circuit analysis methodology is suitable for making gross
comparisons which are needed in the preliminary circuit design stages. The yield score
shows excellent capacity to predict and compare circuit yield, while the MOI value
245 |
Virginia Tech | Chapter 9
Conclusions and Recommendations
Most generally, mineral processing is the art and science of particulate separation as
applied to mining products. Run-of-mine material is typically not of sufficient quality to
justify shipment to downstream users, including smelters, utilities, and other end-users.
Mineral processing separation systems seek to upgrade this material such that the value of
the final product justifies the cost of beneficiation. Often single unit operations are not
capable of producing products which meet these specifications, both in terms of product
quality and product loss. Consequently, operators utilize staged units in various circuit
configurations to produce synergistic improvements which eventually produce a sufficient
product.
Thisworkhasprovidedatreatiseoncircuitdesign. Despitethedevelopmentofsophisti-
catedprocessmodels, astuteanalyticalmethods, androbustnumericoptimizationstrategies,
the industrial design of process circuits is still largely based on empirical knowledge as well
as trial-and-error approaches. As a result, many greenfield designs and circuit modifications
pursue “better” designs often at the expense of man hours and design resources. In many
cases, the optimal circuit design remains elusive. Given the plurality of modern engineer-
ing design tools, the circuit design process should be streamlined while producing optimal
results.
Many circuit design tools have been refined, reanalyzed, and developed as a part of
this work. First, a robust, graphically-based flotation modeling and simulation software
package was developed as a means to quantitatively define circuit performance. This pro-
gram provides the framework to use laboratory, pilot-plant, or full-scale data sets to predict
performance in a user-defined circuit configuration.
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Virginia Tech | CHAPTER 9. CONCLUSIONS AND RECOMMENDATIONS
Second, rate-compositing equations were derived to assist in flotation data inquiry and
error analysis. Most contemporary flotation models are based on a distributed rate model.
Traditional methods do not provide a means to composite multiple rate constants in order
to form an average; however, these new compositing equations describe the mathematical
approach necessary to determine weighted averages for rate values. These equations are
useful in rate fitting and error prediction applications.
Third, a unique algorithm was developed to derive analytical circuit solutions when
given the circuit configuration. The analytical solution may then be used for non-iterative
circuit simulation as well as circuit analysis, via the separation sharpness parameter. This
value can be used to rank various circuit configurations based on the circuits’ inherent
ability to distinguish middlings. The evaluation of this parameter only requires the circuit
configuration; extensive feed and performance data is not necessary.
Fourth, an optimization algorithm was developed to define the ideal circuit location of
an additional unit when given an existing configuration. This optimization is based on the
aforementionedanalyticalcircuitsolutionalgorithmandtheseparationsharpnessparameter.
Fifth, a new technical-economic separation performance measure, the partition moment
of inertia, was developed to incorporate micro-pricing and incremental value concepts into
traditional partition analysis. The resultant parameter inherently reflects the technical-
economic capacity of the separation and is defined for both individual units as well as circuit
analysis applications. This parameter becomes increasingly useful as unit bypass becomes
more pronounced.
Both circuit analysis indicators were evaluated and compared in an extensive experi-
mental investigation. A virtual electrostatic separator was generated in the Working Model
2D program and tested in 17 circuit configurations. The virtual experiments were used to de-
termine the real separation capacity of the various circuits, and these results were compared
to the general rankings derived from circuit analysis. In general, the moment of inertia pa-
rameter provided slightly better predictions; however, both methodologies showed excellent
agreement with the experimental results.
Collectively,thesetoolsmaybeutilizedinterdependentlytostreamlinethecircuitdesign
process while pursuing optimal circuit designs for physical separation systems. During the
preliminary greenfield design stages, when extensive data sets are costly and largely non-
existent, the circuit analysis and incremental optimization tools may be used to define a
small number of potential alternatives. Further experimental and simulation work may
then be used to evaluate this constrained solution space, thus eliminating trial-and-error
249 |
Virginia Tech | CHAPTER 9. CONCLUSIONS AND RECOMMENDATIONS
while ensuring only near-optimal solutions are considered. Since circuit analysis can only be
used for relative comparisons, the final simulation is necessary to quantitatively define the
anticipated circuit performance. Finally, error analysis and rate-constant compositing may
be used to determine the inherent degree of uncertainty in the simulation.
For circuit redesign and modification problems, the incremental optimization tool as
well as the analytical circuit solution may be used to evaluate circuit sensitivity and define
the units in the circuit that provide the best opportunity for growth at the lowest cost.
Further experimental work can then be focused in these areas, and eventually incorporated
into detailed circuit simulations. Once again, the final simulation is necessarily to provide
quantitative justification for the circuit modification.
To the author’s knowledge, seven original contributions have been presented in this
work:
1. A four-reactor flotation model and resulting simulation package. This model uniquely
considers process kinetics as well as carrying capacity, thus placing a large significance
on the machine parameters and characteristics.
2. Generic rate compositing equations. While distributed rate models are common for
flotation modeling, no simple formula has previously been presented which predicts
apparent rate values from a truncated distribution. Such formulas have been derived
analytically for the plug-flow and perfectly-mixed reactor models. A numeric method-
ology has been proposed for the axially-dispersed reactor model.
3. The matrix reduction algorithm to determine analytical circuit solutions. Though
many researchers utilize analytical solutions in their theory and analysis, no stream-
lined methodology exists which simultaneously provides undoubted accuracy and time
efficiency. The author’s algorithm meets both specifications, while incorporating a
graphical interface for convenient circuit input.
4. The optimization algorithm for circuit modification based on the separation sharpness
parameter. Several circuit superstructure optimization algorithms have been reported;
however, this is the only one which uses the analytical solution and the separation
sharpness parameter as the objective function. Furthermore, the search algorithm
easily accommodates other performance measures derived from the analytical solution
(i.e. MOI, YS).
5. The partition moment of inertia as a technical-economic separation performance in-
dicator. No other derived partition curve parameter inherently accounts for process
250 |
Virginia Tech | CHAPTER 9. CONCLUSIONS AND RECOMMENDATIONS
economics and the biased influence pure particles have on final revenue. The moment
of inertia value provides mechanisms for this type of analysis with or without detailed
contract data.
6. The yield score as a derived circuit analysis indicator. While circuit analysis has been
usedinthepasttoassescircuitselectivity, nocurrentmethodologyaccuratelyaccounts
for yield variations between circuit configurations. The yield score has been proven as
an exceptional indicator and ranking parameter by both fundamental derivation and
empirical evidence.
7. The large-scale empirical evaluation of separation circuits using virtual experiments.
Most literature studies of circuit performance are restricted to small sample sizes (usu-
ally two or three configurations) or purely mathematical treatment. Both of these
types of investigations fail to account for the natural and probabilistic inefficiency of
real separators as well as the multitude of available configurations. This research has
included a virtual experiments which analyzed a significant breadth of configuration
types, while validating much of the proposed methodology.
Finally, the author of this dissertation recommends the following items for continued
study:
1. Refined data fitting and analysis modules for flotation model building. The current
data fitting approach limits the data set to three floatability classes. While this routine
performs adequately in most situations, not all flotation systems are optimally defined
bythreerigidfloatabilityclasses. Furthermore,thecurrentfittingroutineisnotcapable
of analyzing data which has experienced a chemical or physical change during the
experiment. Advanced fitting routines may also use an assumed or known floatability
distribution to derive multiple rates from single-residence time data sets (i.e. pilot-
plant data).
2. Evaluation of simulation confidence. All current commercial simulation packages as
well as the simulation package described in this dissertation produce deterministic so-
lutions, with no means to incorporate the uncertainty inherent to the experimental
data. As a result, users cannot reasonably estimate the confidence and precision asso-
ciated with a particular deterministic solution. The circuit analytical solution, Monte
Carlo simulation, and sensitivity analysis may be used to provide better indications of
simulation confidence.
251 |
Virginia Tech | CHAPTER 9. CONCLUSIONS AND RECOMMENDATIONS
3. Quantification of uncertainty reduction as a function of circuit configuration. As em-
pirically observed in Chapter 8, changes in the circuit configuration produce changes
in the experimental uncertainty related to individual class recovery. Intuitively, this
circuit is a function of the circuit configuration, dictating that some circuits possess
a natural ability to reduce uncertainty, just as some circuits have a natural ability to
distinguish middlings or increase yield. Mathematical manipulation of the analytical
solution will likely produce a function capable of predicting this reduction.
4. Further validation of the circuit analysis methodology in real mineral processing sys-
tems. While the virtual experiments provide numerous qualitative and quantitative
benefits, the ultimate measure of the methodology must be proven in real systems.
Such experiments require due consideration in order to isolate the circuit configura-
tion as the single cause of performance changes. In the virtual experiment, extraneous
influences are mitigated since the feed condition was fixed. However, in physical ex-
periments, feed degradation, environmental factors, as well as unknown factors are all
difficult to control but contribute to the final outcome.
5. Quantified correlation of circuit analysis parameters to real separation performance
measures. Currently, the circuit analysis parameters are ill-defined in terms of real
performance gain and only applicable for comparative analysis. However, further em-
pirical testing, especially in real separation systems, may produce general rules relating
the incremental circuit analysis parameter change to real changes in measured perfor-
mance.
252 |
Virginia Tech | Working Model Electrostatic Simulation
Simulation Information Summary
100
Date: 3.13.2013 Color Charge Feed No BP Med BP High BP
Time: 7:00 PM -- (*10^-7 C) Number Recovery Recovery Recovery 90
File Name1 Electrostatic_C1a_noBP.wm2d Pink -1 5 8 36 24 80 16 14.96663 14.96663
File Name2 Electrostatic_C1a_withBP1.wm2d Blue -0.8 5 24 16 28 14.96663 8 9.797959
File Name 3 Electrostatic_C1a_withBP2.wm2d Purple -0.6 5 12 8 52 70 9.797959 9.797959 27.12932
Circuit Name: Single Unit Maroon -0.4 5 32 36 64 60 27.12932 23.32381 8
Circuit No: 1 Green -0.2 5 24 36 28 8 14.96663 20.39608 By: Noble Yellow 0 5 56 52 60 50 14.96663 20.39608 21.9089
Red 0.2 5 68 68 56 40 16 24 23.32381
Orange 0.4 5 72 72 60 20.39608 9.797959 25.29822 Dark Green 0.6 5 100 88 92 30 0 9.797959 9.797959
Light Blue 0.8 5 100 84 88 20 0 23.32381 16
Cyan 1 5 100 100 92 0 0 9.797959
10
Recovered Charge: 0 11.52 9.36 7.2
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Condition 1
Separation Plate 0.00006C Total Recovered Charge: 11.52
BP board NAm
Bypass Motor NArad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 2 0 0 0 0.4 25 Pink 5 3 5 5 5 4.6
Blue -0.8 1 2 1 0 2 1.2 25 Blue 4 3 4 5 3 3.8
Purple -0.6 1 1 0 0 1 0.6 25 Purple 4 4 5 5 4 4.4
Maroon -0.4 4 1 1 0 2 1.6 25 Maroon 1 4 4 5 3 3.4
Green -0.2 1 1 1 1 2 1.2 25 Green 4 4 4 4 3 3.8
Yellow 0 2 3 3 2 4 2.8 25 Yellow 3 2 2 3 1 2.2
Red 0.2 3 4 4 2 4 3.4 25 Red 2 1 1 3 1 1.6
Orange 0.4 3 4 5 2 4 3.6 25 Orange 2 1 0 3 1 1.4
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 2
Separation Plate 0.00006C Total Recovered Charge: 9.36
BP board 0.3m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 2 2 1 1 3 1.8 25 Pink 3 3 4 4 2 3.2
Blue -0.8 1 1 1 1 0 0.8 25 Blue 4 4 4 4 5 4.2
Purple -0.6 0 1 0 0 1 0.4 25 Purple 5 4 5 5 4 4.6
Maroon -0.4 3 0 1 2 3 1.8 25 Maroon 2 5 4 3 2 3.2
Green -0.2 1 1 2 3 2 1.8 25 Green 4 4 3 2 3 3.2
Yellow 0 1 4 3 2 3 2.6 25 Yellow 4 1 2 3 2 2.4
Red 0.2 5 2 4 2 4 3.4 25 Red 0 3 1 3 1 1.6
Orange 0.4 4 3 4 4 3 3.6 25 Orange 1 2 1 1 2 1.4
Dark Green 0.6 4 4 5 5 4 4.4 25 Dark Green 1 1 0 0 1 0.6
Light Blue 0.8 4 2 5 5 5 4.2 25 Light Blue 1 3 0 0 0 0.8
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 3
Separation Block 0.00006C Total Recovered Charge: 7.2
BP board 0.5m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 2 1 2 1 0 1.2 25 Pink 3 4 3 4 5 3.8
Blue -0.8 1 1 2 1 2 1.4 25 Blue 4 4 3 4 3 3.6
Purple -0.6 2 2 3 1 5 2.6 25 Purple 3 3 2 4 0 2.4
Maroon -0.4 3 3 3 3 4 3.2 25 Maroon 2 2 2 2 1 1.8
Green -0.2 2 0 1 1 3 1.4 25 Green 3 5 4 4 2 3.6
Yellow 0 2 5 2 3 3 3 25 Yellow 3 0 3 2 2 2
Red 0.2 2 5 2 2 3 2.8 25 Red 3 0 3 3 2 2.2
Orange 0.4 4 1 4 2 4 3 25 Orange 1 4 1 3 1 2
Dark Green 0.6 5 4 5 4 5 4.6 25 Dark Green 0 1 0 1 0 0.4
Light Blue 0.8 5 4 5 5 3 4.4 25 Light Blue 0 1 0 0 2 0.6
Cyan 1 5 4 5 5 4 4.6 25 Cyan 0 1 0 0 1 0.4
)%(
yrevoceR
No BP
Med BP
High BP
Particle Charge (E-7 C) |
Virginia Tech | Working Model Electrostatic Simulation
Simulation Information Summary 100
Date: 3.20.2013 Color Charge Feed No BP Med BP High BP
Time: 10:00 AM -- (*10^-7 C) Number Recovery Recovery Recovery 90
File Name1 Electrostatic_C2a_noBP.wm2d Pink -1 5 12 56 36 80
File Name2 Electrostatic_C2a_withBP1.wm2d Blue -0.8 5 16 56 60
File Name 3 Electrostatic_C2a_withBP2.wm2d Purple -0.6 5 24 32 68 70
Circuit Name: Scavenger with Recycle Maroon -0.4 5 28 48 92 60
Circuit No: 2 Green -0.2 5 28 64 64 By: Noble Yellow 0 5 60 64 84 50
Red 0.2 5 92 92 96 40
Orange 0.4 5 96 96 96 30
Dark Green 0.6 5 100 100 100
Light Blue 0.8 5 100 100 100 20
Cyan 1 5 100 100 100
10
Total Charge: 0 12.04 7.24 6.16
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Condition 1
Separation Plate 0.00006C Total Recovered Charge: 12.04
BP board NAm
Bypass Motor NArad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 1 0 0 0 2 0.6 25 Pink 4 5 5 5 3 4.4
Blue -0.8 1 2 1 0 0 0.8 25 Blue 4 3 4 5 5 4.2
Purple -0.6 1 1 3 0 1 1.2 25 Purple 4 4 2 5 4 3.8
Maroon -0.4 1 2 1 3 0 1.4 25 Maroon 4 3 4 2 5 3.6
Green -0.2 0 3 3 1 0 1.4 25 Green 5 2 2 4 5 3.6
Yellow 0 5 4 3 2 1 3 25 Yellow 0 1 2 3 4 2
Red 0.2 5 5 5 5 3 4.6 25 Red 0 0 0 0 2 0.4
Orange 0.4 4 5 5 5 5 4.8 25 Orange 1 0 0 0 0 0.2
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 2
Separation Plate 0.00006C Total Recovered Charge: 7.24
BP board 0.3m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 4 2 4 2 2 2.8 25 Pink 1 3 1 3 3 2.2
Blue -0.8 4 2 4 4 0 2.8 25 Blue 1 3 1 1 5 2.2
Purple -0.6 1 2 1 4 0 1.6 25 Purple 4 3 4 1 5 3.4
Maroon -0.4 3 2 1 5 1 2.4 25 Maroon 2 3 4 0 4 2.6
Green -0.2 3 3 3 4 3 3.2 25 Green 2 2 2 1 2 1.8
Yellow 0 3 4 3 4 2 3.2 25 Yellow 2 1 2 1 3 1.8
Red 0.2 5 4 5 5 4 4.6 25 Red 0 1 0 0 1 0.4
Orange 0.4 5 4 5 5 5 4.8 25 Orange 0 1 0 0 0 0.2
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 3
Separation Block 0.00006C Total Recovered Charge: 6.16
BP board 0.5m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 1 1 1 3 3 1.8 25 Pink 4 4 4 2 2 3.2
Blue -0.8 2 4 2 4 3 3 25 Blue 3 1 3 1 2 2
Purple -0.6 3 1 5 4 4 3.4 25 Purple 2 4 0 1 1 1.6
Maroon -0.4 5 4 5 4 5 4.6 25 Maroon 0 1 0 1 0 0.4
Green -0.2 1 3 3 5 4 3.2 25 Green 4 2 2 0 1 1.8
Yellow 0 4 4 5 4 4 4.2 25 Yellow 1 1 0 1 1 0.8
Red 0.2 5 5 4 5 5 4.8 25 Red 0 0 1 0 0 0.2
Orange 0.4 5 5 5 4 5 4.8 25 Orange 0 0 0 1 0 0.2
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
)%(
yrevoceR
No BP
Med BP
High BP
Particle Charge (E-7 C) |
Virginia Tech | Working Model Electrostatic Simulation
Simulation Information Summary
100
Date: 3.19.2013 Color Charge Feed No BP Med BP High BP
Time: 7:30 PM -- (*10^-7 C) Number Recovery Recovery Recovery 90
File Name1 Electrostatic_C3a_noBP.wm2d Pink -1 5 0 4 8 80
File Name2 Electrostatic_C3a_withBP1.wm2d Blue -0.8 5 8 4 12
File Name 3 Electrostatic_C3a_withBP2.wm2d Purple -0.6 5 8 12 24 70
Circuit Name: Cleaner with Recycle Maroon -0.4 5 28 8 8 60
Circuit No: 3 Green -0.2 5 20 12 20 By: Noble Yellow 0 5 28 12 40 50
Red 0.2 5 52 40 64 40
Orange 0.4 5 80 72 68 Dark Green 0.6 5 100 96 100 30
Light Blue 0.8 5 96 96 100 20
Cyan 1 5 100 88 96
10
Total Charge: 0 12.64 11.96 11.84
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Condition 1
Separation Plate 0.00006C Total Recovered Charge: 12.64
BP board NAm
Bypass Motor NArad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 0 0 0 25 Pink 5 5 5 5 5 5
Blue -0.8 0 0 0 2 0 0.4 25 Blue 5 5 5 3 5 4.6
Purple -0.6 1 0 0 1 0 0.4 25 Purple 4 5 5 4 5 4.6
Maroon -0.4 1 0 3 1 2 1.4 25 Maroon 4 5 2 4 3 3.6
Green -0.2 0 2 2 1 0 1 25 Green 5 3 3 4 5 4
Yellow 0 3 1 1 1 1 1.4 25 Yellow 2 4 4 4 4 3.6
Red 0.2 4 3 1 3 2 2.6 25 Red 1 2 4 2 3 2.4
Orange 0.4 5 2 3 5 5 4 25 Orange 0 3 2 0 0 1
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 4 5 5 5 5 4.8 25 Light Blue 1 0 0 0 0 0.2
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 2
Separation Plate 0.00006C Total Recovered Charge: 11.96
BP board 0.3m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 1 0 0 0 0.2 25 Pink 5 4 5 5 5 4.8
Blue -0.8 0 1 0 0 0 0.2 25 Blue 5 4 5 5 5 4.8
Purple -0.6 0 0 0 1 2 0.6 25 Purple 5 5 5 4 3 4.4
Maroon -0.4 0 0 0 1 1 0.4 25 Maroon 5 5 5 4 4 4.6
Green -0.2 1 0 0 2 0 0.6 25 Green 4 5 5 3 5 4.4
Yellow 0 1 0 1 0 1 0.6 25 Yellow 4 5 4 5 4 4.4
Red 0.2 1 2 2 3 2 2 25 Red 4 3 3 2 3 3
Orange 0.4 4 3 2 5 4 3.6 25 Orange 1 2 3 0 1 1.4
Dark Green 0.6 5 5 5 4 5 4.8 25 Dark Green 0 0 0 1 0 0.2
Light Blue 0.8 4 5 5 5 5 4.8 25 Light Blue 1 0 0 0 0 0.2
Cyan 1 5 4 5 5 3 4.4 25 Cyan 0 1 0 0 2 0.6
Condition 3
Separation Block 0.00006C Total Recovered Charge: 11.84
BP board 0.5m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 1 1 0 0 0 0.4 25 Pink 4 4 5 5 5 4.6
Blue -0.8 0 1 1 0 1 0.6 25 Blue 5 4 4 5 4 4.4
Purple -0.6 0 3 1 1 1 1.2 25 Purple 5 2 4 4 4 3.8
Maroon -0.4 1 0 0 0 1 0.4 25 Maroon 4 5 5 5 4 4.6
Green -0.2 2 1 1 1 0 1 25 Green 3 4 4 4 5 4
Yellow 0 4 1 2 2 1 2 25 Yellow 1 4 3 3 4 3
Red 0.2 4 2 3 5 2 3.2 25 Red 1 3 2 0 3 1.8
Orange 0.4 5 5 4 2 1 3.4 25 Orange 0 0 1 3 4 1.6
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 4 5 5 4.8 25 Cyan 0 0 1 0 0 0.2
)%(
yrevoceR
No BP
Med BP
High BP
Particle Charge (E-7 C) |
Virginia Tech | Working Model Electrostatic Simulation
Simulation Information Summary
100
Date: 3.19.2013 Color Charge Feed No BP Med BP High BP
Time: 5:20 PM -- (*10^-7 C) Number Recovery Recovery Recovery 90
File Name1 Electrostatic_C4a_noBP.wm2d Pink -1 5 0 4 8 80
File Name2 Electrostatic_C4a_withBP1.wm2d Blue -0.8 5 0 0 12
File Name 3 Electrostatic_C4a_withBP2.wm2d Purple -0.6 5 8 0 0 70
Circuit Name: Cleaner no Recycle Maroon -0.4 5 4 0 4 60
Circuit No: 4 Green -0.2 5 4 20 16 By: Noble Yellow 0 5 16 12 12 50
Red 0.2 5 16 24 32 40
Orange 0.4 5 84 52 52 Dark Green 0.6 5 80 92 84 30
Light Blue 0.8 5 92 96 80 20
Cyan 1 5 100 96 96
10
Total Charge: 0 12.56 12.28 10.76
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Condition 1
Separation Plate 0.00006C Total Recovered Charge: 12.56
BP board NAm
Bypass Motor NArad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 0 0 0 25 Pink 5 5 5 5 5 5
Blue -0.8 0 0 0 0 0 0 25 Blue 5 5 5 5 5 5
Purple -0.6 0 0 1 0 1 0.4 25 Purple 5 5 4 5 4 4.6
Maroon -0.4 0 0 1 0 0 0.2 25 Maroon 5 5 4 5 5 4.8
Green -0.2 1 0 0 0 0 0.2 25 Green 4 5 5 5 5 4.8
Yellow 0 0 1 2 0 1 0.8 25 Yellow 5 4 3 5 4 4.2
Red 0.2 1 0 0 2 1 0.8 25 Red 4 5 5 3 4 4.2
Orange 0.4 4 4 4 5 4 4.2 25 Orange 1 1 1 0 1 0.8
Dark Green 0.6 4 4 4 3 5 4 25 Dark Green 1 1 1 2 0 1
Light Blue 0.8 5 4 5 5 4 4.6 25 Light Blue 0 1 0 0 1 0.4
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 2
Separation Plate 0.00006C Total Recovered Charge: 12.28
BP board 0.3m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 1 0 0 0.2 25 Pink 5 5 4 5 5 4.8
Blue -0.8 0 0 0 0 0 0 25 Blue 5 5 5 5 5 5
Purple -0.6 0 0 0 0 0 0 25 Purple 5 5 5 5 5 5
Maroon -0.4 0 0 0 0 0 0 25 Maroon 5 5 5 5 5 5
Green -0.2 2 0 2 1 0 1 25 Green 3 5 3 4 5 4
Yellow 0 2 0 0 0 1 0.6 25 Yellow 3 5 5 5 4 4.4
Red 0.2 2 2 1 0 1 1.2 25 Red 3 3 4 5 4 3.8
Orange 0.4 3 1 3 3 3 2.6 25 Orange 2 4 2 2 2 2.4
Dark Green 0.6 5 3 5 5 5 4.6 25 Dark Green 0 2 0 0 0 0.4
Light Blue 0.8 5 5 5 5 4 4.8 25 Light Blue 0 0 0 0 1 0.2
Cyan 1 5 5 4 5 5 4.8 25 Cyan 0 0 1 0 0 0.2
Condition 3
Separation Block 0.00006C Total Recovered Charge: 10.76
BP board 0.5m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 1 0 1 0.4 25 Pink 5 5 4 5 4 4.6
Blue -0.8 0 0 1 1 1 0.6 25 Blue 5 5 4 4 4 4.4
Purple -0.6 0 0 0 0 0 0 25 Purple 5 5 5 5 5 5
Maroon -0.4 0 1 0 0 0 0.2 25 Maroon 5 4 5 5 5 4.8
Green -0.2 0 2 2 0 0 0.8 25 Green 5 3 3 5 5 4.2
Yellow 0 1 1 1 0 0 0.6 25 Yellow 4 4 4 5 5 4.4
Red 0.2 2 2 2 0 2 1.6 25 Red 3 3 3 5 3 3.4
Orange 0.4 3 1 4 2 3 2.6 25 Orange 2 4 1 3 2 2.4
Dark Green 0.6 5 4 3 5 4 4.2 25 Dark Green 0 1 2 0 1 0.8
Light Blue 0.8 4 3 3 5 5 4 25 Light Blue 1 2 2 0 0 1
Cyan 1 5 5 5 5 4 4.8 25 Cyan 0 0 0 0 1 0.2
)%(
yrevoceR
No BP
Med BP
High BP
Particle Charge (E-7 C) |
Virginia Tech | Working Model Electrostatic Simulation
Simulation Information Summary
100
Date: 3.19.2013 Color Charge Feed No BP Med BP High BP
Time: 11:20 AM -- (*10^-7 C) Number Recovery Recovery Recovery 90
File Name1 Electrostatic_C5a_noBP.wm2d Pink -1 5 8 36 28 80
File Name2 Electrostatic_C5a_withBP1.wm2d Blue -0.8 5 32 32 56
File Name 3 Electrostatic_C5a_withBP2.wm2d Purple -0.6 5 32 24 56 70
Circuit Name: Scavenger no Recycle Maroon -0.4 5 52 56 76 60
Circuit No: 5 Green -0.2 5 44 40 60 By: Noble Yellow 0 5 88 68 72 50
Red 0.2 5 96 88 80 40
Orange 0.4 5 96 88 96 Dark Green 0.6 5 100 96 100 30
Light Blue 0.8 5 100 100 96 20
Cyan 1 5 100 100 100
10
Total Charge: 15 10.76 9.2 7.12
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Condition 1
Separation Plate 0.00006C Total Recovered Charge: 10.76
BP board NAm
Bypass Motor NArad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 1 0 0 1 0.4 25 Pink 5 4 5 5 4 4.6
Blue -0.8 1 1 2 3 1 1.6 25 Blue 4 4 3 2 4 3.4
Purple -0.6 1 3 1 2 1 1.6 25 Purple 4 2 4 3 4 3.4
Maroon -0.4 4 3 0 3 3 2.6 25 Maroon 1 2 5 2 2 2.4
Green -0.2 3 2 3 2 1 2.2 25 Green 2 3 2 3 4 2.8
Yellow 0 5 3 4 5 5 4.4 25 Yellow 0 2 1 0 0 0.6
Red 0.2 5 4 5 5 5 4.8 25 Red 0 1 0 0 0 0.2
Orange 0.4 5 5 5 5 4 4.8 25 Orange 0 0 0 0 1 0.2
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 2
Separation Plate 0.00006C Total Recovered Charge: 9.2
BP board 0.3m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 1 3 0 2 3 1.8 25 Pink 4 2 5 3 2 3.2
Blue -0.8 0 2 1 0 5 1.6 25 Blue 5 3 4 5 0 3.4
Purple -0.6 2 2 0 2 0 1.2 25 Purple 3 3 5 3 5 3.8
Maroon -0.4 4 3 3 2 2 2.8 25 Maroon 1 2 2 3 3 2.2
Green -0.2 2 3 0 3 2 2 25 Green 3 2 5 2 3 3
Yellow 0 3 5 3 1 5 3.4 25 Yellow 2 0 2 4 0 1.6
Red 0.2 5 5 4 4 4 4.4 25 Red 0 0 1 1 1 0.6
Orange 0.4 5 4 5 4 4 4.4 25 Orange 0 1 0 1 1 0.6
Dark Green 0.6 4 5 5 5 5 4.8 25 Dark Green 1 0 0 0 0 0.2
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 3
Separation Block 0.00006C Total Recovered Charge: 7.12
BP board 0.5m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 1 1 1 2 2 1.4 25 Pink 4 4 4 3 3 3.6
Blue -0.8 4 2 2 2 4 2.8 25 Blue 1 3 3 3 1 2.2
Purple -0.6 2 2 3 4 3 2.8 25 Purple 3 3 2 1 2 2.2
Maroon -0.4 3 5 4 5 2 3.8 25 Maroon 2 0 1 0 3 1.2
Green -0.2 2 2 2 4 5 3 25 Green 3 3 3 1 0 2
Yellow 0 4 2 5 3 4 3.6 25 Yellow 1 3 0 2 1 1.4
Red 0.2 3 4 4 4 5 4 25 Red 2 1 1 1 0 1
Orange 0.4 5 5 5 5 4 4.8 25 Orange 0 0 0 0 1 0.2
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 5 4 5 5 5 4.8 25 Light Blue 0 1 0 0 0 0.2
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
)%(
yrevoceR
No BP
Med BP
High BP
Particle Charge (E-7 C) |
Virginia Tech | Working Model Electrostatic Simulation
Simulation Information Summary
100
Date: 4.5.2013 Color Charge Feed No BP Med BP High BP
Time: 4:11 PM -- (*10^-7 C) Number Recovery Recovery Recovery 90
File Name1 Electrostatic_C12a_noBP.wm2d Pink -1 5 0 0 4 80
File Name2 Electrostatic_C12a_withBP1.wm2d Blue -0.8 5 0 0 8
File Name 3 Electrostatic_C12a_withBP2.wm2d Purple -0.6 5 0 0 4 70
Circuit Name: Cleaner-Recleaner Open Maroon -0.4 5 0 4 8 60
Circuit No: 12 Green -0.2 5 0 0 4 By: Noble Yellow 0 5 4 8 20 50
Red 0.2 5 0 8 20 40
Orange 0.4 5 68 52 52 Dark Green 0.6 5 96 88 84 30
Light Blue 0.8 5 100 100 96 20
Cyan 1 5 100 96 100
10
Total Charge: 15 13.24 12.48 11.76
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Condition 1
Separation Plate 0.00006C Total Recovered Charge: 13.24
BP board NAm
Bypass Motor NArad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 0 0 0 25 Pink 5 5 5 5 5 5
Blue -0.8 0 0 0 0 0 0 25 Blue 5 5 5 5 5 5
Purple -0.6 0 0 0 0 0 0 25 Purple 5 5 5 5 5 5
Maroon -0.4 0 0 0 0 0 0 25 Maroon 5 5 5 5 5 5
Green -0.2 0 0 0 0 0 0 25 Green 5 5 5 5 5 5
Yellow 0 0 1 0 0 0 0.2 25 Yellow 5 4 5 5 5 4.8
Red 0.2 0 0 0 0 0 0 25 Red 5 5 5 5 5 5
Orange 0.4 3 3 4 3 4 3.4 25 Orange 2 2 1 2 1 1.6
Dark Green 0.6 5 5 4 5 5 4.8 25 Dark Green 0 0 1 0 0 0.2
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 2
Separation Plate 0.00006C Total Recovered Charge: 12.48
BP board 0.3m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 0 0 0 25 Pink 5 5 5 5 5 5
Blue -0.8 0 0 0 0 0 0 25 Blue 5 5 5 5 5 5
Purple -0.6 0 0 0 0 0 0 25 Purple 5 5 5 5 5 5
Maroon -0.4 0 1 0 0 0 0.2 25 Maroon 5 4 5 5 5 4.8
Green -0.2 0 0 0 0 0 0 25 Green 5 5 5 5 5 5
Yellow 0 0 0 1 0 1 0.4 25 Yellow 5 5 4 5 4 4.6
Red 0.2 0 0 1 0 1 0.4 25 Red 5 5 4 5 4 4.6
Orange 0.4 2 2 4 3 2 2.6 25 Orange 3 3 1 2 3 2.4
Dark Green 0.6 5 5 3 4 5 4.4 25 Dark Green 0 0 2 1 0 0.6
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 4 4.8 25 Cyan 0 0 0 0 1 0.2
Condition 3
Separation Block 0.00006C Total Recovered Charge: 11.76
BP board 0.5m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 0 1 0.2 21 Pink 1 5 5 5 4 4
Blue -0.8 1 0 0 1 0 0.4 25 Blue 4 5 5 4 5 4.6
Purple -0.6 0 0 0 1 0 0.2 25 Purple 5 5 5 4 5 4.8
Maroon -0.4 0 0 0 1 1 0.4 25 Maroon 5 5 5 4 4 4.6
Green -0.2 0 0 1 0 0 0.2 25 Green 5 5 4 5 5 4.8
Yellow 0 0 0 0 3 2 1 25 Yellow 5 5 5 2 3 4
Red 0.2 0 1 1 2 1 1 25 Red 5 4 4 3 4 4
Orange 0.4 3 2 4 3 1 2.6 25 Orange 2 3 1 2 4 2.4
Dark Green 0.6 5 4 4 4 4 4.2 25 Dark Green 0 1 1 1 1 0.8
Light Blue 0.8 5 5 4 5 5 4.8 25 Light Blue 0 0 1 0 0 0.2
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
)%(
yrevoceR
No BP
Med BP
High BP
Particle Charge (E-7 C) |
Virginia Tech | Working Model Electrostatic Simulation
Simulation Information Summary
100
Date: 4.5.2013 Color Charge Feed No BP Med BP High BP
Time: 4:11 PM -- (*10^-7 C) Number Recovery Recovery Recovery 90
File Name1 Electrostatic_C12a_noBP.wm2d Pink -1 5 0 0 4 80
File Name2 Electrostatic_C12a_withBP1.wm2d Blue -0.8 5 0 4 8
File Name 3 Electrostatic_C12a_withBP2.wm2d Purple -0.6 5 0 0 8 70
Circuit Name: Cleaner-Recleaner/Recleaner Back 1 Maroon -0.4 5 0 4 8 60
Circuit No: 13 Green -0.2 5 0 0 8 By: Noble Yellow 0 5 4 16 20 50
Red 0.2 5 8 12 28 40
Orange 0.4 5 80 60 56 Dark Green 0.6 5 96 88 84 30
Light Blue 0.8 5 100 100 96 20
Cyan 1 5 100 100 100
10
Total Charge: 15 13.56 12.72 11.76
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Condition 1
Separation Plate 0.00006C Total Recovered Charge: 13.56
BP board NAm
Bypass Motor NArad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 0 0 0 25 Pink 5 5 5 5 5 5
Blue -0.8 0 0 0 0 0 0 25 Blue 5 5 5 5 5 5
Purple -0.6 0 0 0 0 0 0 25 Purple 5 5 5 5 5 5
Maroon -0.4 0 0 0 0 0 0 25 Maroon 5 5 5 5 5 5
Green -0.2 0 0 0 0 0 0 25 Green 5 5 5 5 5 5
Yellow 0 0 1 0 0 0 0.2 25 Yellow 5 4 5 5 5 4.8
Red 0.2 0 1 0 1 0 0.4 24 Red 5 4 5 4 4 4.4
Orange 0.4 5 3 4 3 5 4 26 Orange 0 2 1 2 1 1.2
Dark Green 0.6 5 5 4 5 5 4.8 25 Dark Green 0 0 1 0 0 0.2
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 2
Separation Plate 0.00006C Total Recovered Charge: 12.72
BP board 0.3m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 0 0 0 25 Pink 5 5 5 5 5 5
Blue -0.8 0 0 0 1 0 0.2 25 Blue 5 5 5 4 5 4.8
Purple -0.6 0 0 0 0 0 0 25 Purple 5 5 5 5 5 5
Maroon -0.4 0 1 0 0 0 0.2 25 Maroon 5 4 5 5 5 4.8
Green -0.2 0 0 0 0 0 0 25 Green 5 5 5 5 5 5
Yellow 0 1 1 1 0 1 0.8 25 Yellow 4 4 4 5 4 4.2
Red 0.2 1 0 1 0 1 0.6 25 Red 4 5 4 5 4 4.4
Orange 0.4 3 3 4 3 2 3 25 Orange 2 2 1 2 3 2
Dark Green 0.6 5 5 3 4 5 4.4 25 Dark Green 0 0 2 1 0 0.6
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 3
Separation Block 0.00006C Total Recovered Charge: 11.76
BP board 0.5m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 0 1 0.2 21 Pink 1 5 5 5 4 4
Blue -0.8 1 0 0 1 0 0.4 25 Blue 4 5 5 4 5 4.6
Purple -0.6 0 0 1 1 0 0.4 25 Purple 5 5 4 4 5 4.6
Maroon -0.4 0 0 0 1 1 0.4 25 Maroon 5 5 5 4 4 4.6
Green -0.2 0 0 2 0 0 0.4 25 Green 5 5 3 5 5 4.6
Yellow 0 0 0 0 3 2 1 25 Yellow 5 5 5 2 3 4
Red 0.2 0 2 2 2 1 1.4 25 Red 5 3 3 3 4 3.6
Orange 0.4 4 2 4 3 1 2.8 25 Orange 1 3 1 2 4 2.2
Dark Green 0.6 5 4 4 4 4 4.2 25 Dark Green 0 1 1 1 1 0.8
Light Blue 0.8 5 5 4 5 5 4.8 25 Light Blue 0 0 1 0 0 0.2
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
)%(
yrevoceR
No BP
Med BP
High BP
Particle Charge (E-7 C) |
Virginia Tech | Working Model Electrostatic Simulation
Simulation Information Summary
100
Date: 4.5.2013 Color Charge Feed No BP Med BP High BP
Time: 4:11 PM -- (*10^-7 C) Number Recovery Recovery Recovery 90
File Name1 Electrostatic_C12a_noBP.wm2d Pink -1 5 0 0 4 80
File Name2 Electrostatic_C12a_withBP1.wm2d Blue -0.8 5 0 0 8
File Name 3 Electrostatic_C12a_withBP2.wm2d Purple -0.6 5 0 0 8 70
Circuit Name: Cleaner-Recleaner/Recleaner Back 2 Maroon -0.4 5 0 8 8 60
Circuit No: 14 Green -0.2 5 0 0 8 By: Noble Yellow 0 5 4 12 20 50
Red 0.2 5 4 12 28 40
Orange 0.4 5 76 60 56 Dark Green 0.6 5 96 88 84 30
Light Blue 0.8 5 100 100 96 20
Cyan 1 5 100 100 100
10
Total Charge: 15 13.44 12.8 11.76
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Condition 1
Separation Plate 0.00006C Total Recovered Charge: 13.44
BP board NAm
Bypass Motor NArad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 0 0 0 25 Pink 5 5 5 5 5 5
Blue -0.8 0 0 0 0 0 0 25 Blue 5 5 5 5 5 5
Purple -0.6 0 0 0 0 0 0 25 Purple 5 5 5 5 5 5
Maroon -0.4 0 0 0 0 0 0 25 Maroon 5 5 5 5 5 5
Green -0.2 0 0 0 0 0 0 25 Green 5 5 5 5 5 5
Yellow 0 0 1 0 0 0 0.2 25 Yellow 5 4 5 5 5 4.8
Red 0.2 0 1 0 0 0 0.2 25 Red 5 4 5 5 5 4.8
Orange 0.4 4 3 4 3 5 3.8 25 Orange 1 2 1 2 0 1.2
Dark Green 0.6 5 5 4 5 5 4.8 25 Dark Green 0 0 1 0 0 0.2
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 2
Separation Plate 0.00006C Total Recovered Charge: 12.8
BP board 0.3m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 0 0 0 25 Pink 5 5 5 5 5 5
Blue -0.8 0 0 0 0 0 0 25 Blue 5 5 5 5 5 5
Purple -0.6 0 0 0 0 0 0 25 Purple 5 5 5 5 5 5
Maroon -0.4 0 1 0 1 0 0.4 25 Maroon 5 4 5 4 5 4.6
Green -0.2 0 0 0 0 0 0 25 Green 5 5 5 5 5 5
Yellow 0 0 0 1 0 2 0.6 25 Yellow 5 5 4 5 3 4.4
Red 0.2 0 0 2 0 1 0.6 25 Red 5 5 3 5 4 4.4
Orange 0.4 3 3 4 3 2 3 25 Orange 2 2 1 2 3 2
Dark Green 0.6 5 5 3 4 5 4.4 25 Dark Green 0 0 2 1 0 0.6
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 3
Separation Block 0.00006C Total Recovered Charge: 11.76
BP board 0.5m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 0 1 0.2 21 Pink 1 5 5 5 4 4
Blue -0.8 1 0 0 1 0 0.4 25 Blue 4 5 5 4 5 4.6
Purple -0.6 0 0 1 1 0 0.4 25 Purple 5 5 4 4 5 4.6
Maroon -0.4 0 0 0 1 1 0.4 25 Maroon 5 5 5 4 4 4.6
Green -0.2 0 0 2 0 0 0.4 25 Green 5 5 3 5 5 4.6
Yellow 0 0 0 0 3 2 1 25 Yellow 5 5 5 2 3 4
Red 0.2 0 2 2 2 1 1.4 25 Red 5 3 3 3 4 3.6
Orange 0.4 4 2 4 3 1 2.8 25 Orange 1 3 1 2 4 2.2
Dark Green 0.6 5 4 4 4 4 4.2 25 Dark Green 0 1 1 1 1 0.8
Light Blue 0.8 5 5 4 5 5 4.8 25 Light Blue 0 0 1 0 0 0.2
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
)%(
yrevoceR
No BP
Med BP
High BP
Particle Charge (E-7 C) |
Virginia Tech | Working Model Electrostatic Simulation
Simulation Information Summary
100
Date: 4.5.2013 Color Charge Feed No BP Med BP High BP
Time: 4:11 PM -- (*10^-7 C) Number Recovery Recovery Recovery 90
File Name1 Electrostatic_C12a_noBP.wm2d Pink -1 5 0 0 4 80
File Name2 Electrostatic_C12a_withBP1.wm2d Blue -0.8 5 0 0 8
File Name 3 Electrostatic_C12a_withBP2.wm2d Purple -0.6 5 0 0 8 70
Circuit Name: Cleaner-Recleaner/Cleaner Back Maroon -0.4 5 0 4 8 60
Circuit No: 15 Green -0.2 5 4 0 12 By: Noble Yellow 0 5 4 8 28 50
Red 0.2 5 8 16 24 40
Orange 0.4 5 68 60 64 Dark Green 0.6 5 96 88 84 30
Light Blue 0.8 5 100 100 96 20
Cyan 1 5 100 96 100
10
Total Charge: 15 13.28 12.72 11.84
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Condition 1
Separation Plate 0.00006C Total Recovered Charge: 13.28
BP board NAm
Bypass Motor NArad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 0 0 0 25 Pink 5 5 5 5 5 5
Blue -0.8 0 0 0 0 0 0 25 Blue 5 5 5 5 5 5
Purple -0.6 0 0 0 0 0 0 25 Purple 5 5 5 5 5 5
Maroon -0.4 0 0 0 0 0 0 25 Maroon 5 5 5 5 5 5
Green -0.2 0 1 0 0 0 0.2 25 Green 5 4 5 5 5 4.8
Yellow 0 0 1 0 0 0 0.2 25 Yellow 5 4 5 5 5 4.8
Red 0.2 1 1 0 0 0 0.4 25 Red 4 4 5 5 5 4.6
Orange 0.4 3 3 4 3 4 3.4 25 Orange 2 2 1 2 1 1.6
Dark Green 0.6 5 5 4 5 5 4.8 25 Dark Green 0 0 1 0 0 0.2
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 2
Separation Plate 0.00006C Total Recovered Charge: 12.72
BP board 0.3m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 0 0 0 25 Pink 5 5 5 5 5 5
Blue -0.8 0 0 0 0 0 0 25 Blue 5 5 5 5 5 5
Purple -0.6 0 0 0 0 0 0 25 Purple 5 5 5 5 5 5
Maroon -0.4 0 1 0 0 0 0.2 25 Maroon 5 4 5 5 5 4.8
Green -0.2 0 0 0 0 0 0 25 Green 5 5 5 5 5 5
Yellow 0 0 0 1 0 1 0.4 25 Yellow 5 5 4 5 4 4.6
Red 0.2 1 0 1 0 2 0.8 25 Red 4 5 4 5 3 4.2
Orange 0.4 3 3 4 3 2 3 25 Orange 2 2 1 2 3 2
Dark Green 0.6 5 5 3 4 5 4.4 25 Dark Green 0 0 2 1 0 0.6
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 4 4.8 25 Cyan 0 0 0 0 1 0.2
Condition 3
Separation Block 0.00006C Total Recovered Charge: 11.84
BP board 0.5m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 0 1 0.2 21 Pink 1 5 5 5 4 4
Blue -0.8 1 0 0 1 0 0.4 25 Blue 4 5 5 4 5 4.6
Purple -0.6 0 0 1 1 0 0.4 25 Purple 5 5 4 4 5 4.6
Maroon -0.4 0 0 0 1 1 0.4 25 Maroon 5 5 5 4 4 4.6
Green -0.2 1 0 1 1 0 0.6 25 Green 4 5 4 4 5 4.4
Yellow 0 0 0 1 3 3 1.4 25 Yellow 5 5 4 2 2 3.6
Red 0.2 0 2 1 2 1 1.2 25 Red 5 3 4 3 4 3.8
Orange 0.4 3 2 5 3 3 3.2 25 Orange 2 3 0 2 2 1.8
Dark Green 0.6 5 4 4 4 4 4.2 25 Dark Green 0 1 1 1 1 0.8
Light Blue 0.8 5 5 4 5 5 4.8 25 Light Blue 0 0 1 0 0 0.2
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
)%(
yrevoceR
No BP
Med BP
High BP
Particle Charge (E-7 C) |
Virginia Tech | Working Model Electrostatic Simulation
Simulation Information Summary
100
Date: 4.5.2013 Color Charge Feed No BP Med BP High BP
Time: 4:11 PM -- (*10^-7 C) Number Recovery Recovery Recovery 90
File Name1 Electrostatic_C12a_noBP.wm2d Pink -1 5 0 0 4 80
File Name2 Electrostatic_C12a_withBP1.wm2d Blue -0.8 5 0 4 8
File Name 3 Electrostatic_C12a_withBP2.wm2d Purple -0.6 5 0 0 12 70
Circuit Name: Cleaner-Recleaner/Countercurrent Maroon -0.4 5 0 8 8 60
Circuit No: 16 Green -0.2 5 4 4 16 By: Noble Yellow 0 5 8 20 36 50
Red 0.2 5 28 44 52 40
Orange 0.4 5 92 68 68 Dark Green 0.6 5 96 88 84 30
Light Blue 0.8 5 100 100 96 20
Cyan 1 5 100 100 100
10
Total Charge: 15 13.96 13.08 12.04
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Condition 1
Separation Plate 0.00006C Total Recovered Charge: 13.96
BP board NAm
Bypass Motor NArad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 0 0 0 25 Pink 5 5 5 5 5 5
Blue -0.8 0 0 0 0 0 0 25 Blue 5 5 5 5 5 5
Purple -0.6 0 0 0 0 0 0 25 Purple 5 5 5 5 5 5
Maroon -0.4 0 0 0 0 0 0 25 Maroon 5 5 5 5 5 5
Green -0.2 0 1 0 0 0 0.2 25 Green 5 4 5 5 5 4.8
Yellow 0 0 1 0 0 1 0.4 25 Yellow 5 4 5 5 4 4.6
Red 0.2 1 2 1 2 1 1.4 25 Red 4 3 4 3 4 3.6
Orange 0.4 5 5 4 4 5 4.6 25 Orange 0 0 1 1 0 0.4
Dark Green 0.6 5 5 4 5 5 4.8 25 Dark Green 0 0 1 0 0 0.2
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 2
Separation Plate 0.00006C Total Recovered Charge: 13.08
BP board 0.3m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 0 0 0 25 Pink 5 5 5 5 5 5
Blue -0.8 0 0 0 1 0 0.2 25 Blue 5 5 5 4 5 4.8
Purple -0.6 0 0 0 0 0 0 25 Purple 5 5 5 5 5 5
Maroon -0.4 0 1 0 0 1 0.4 25 Maroon 5 4 5 5 4 4.6
Green -0.2 0 0 1 0 0 0.2 25 Green 5 5 4 5 5 4.8
Yellow 0 2 1 1 0 1 1 25 Yellow 3 4 4 5 4 4
Red 0.2 4 2 2 1 2 2.2 25 Red 1 3 3 4 3 2.8
Orange 0.4 4 4 4 3 2 3.4 25 Orange 1 1 1 2 3 1.6
Dark Green 0.6 5 5 3 4 5 4.4 25 Dark Green 0 0 2 1 0 0.6
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 3
Separation Block 0.00006C Total Recovered Charge: 12.04
BP board 0.5m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 0 1 0.2 21 Pink 1 5 5 5 4 4
Blue -0.8 1 0 0 1 0 0.4 25 Blue 4 5 5 4 5 4.6
Purple -0.6 0 0 2 1 0 0.6 25 Purple 5 5 3 4 5 4.4
Maroon -0.4 0 0 0 1 1 0.4 24 Maroon 4 5 5 4 4 4.4
Green -0.2 1 0 2 1 0 0.8 26 Green 5 5 3 4 5 4.4
Yellow 0 0 1 2 3 3 1.8 25 Yellow 5 4 3 2 2 3.2
Red 0.2 2 4 3 2 2 2.6 25 Red 3 1 2 3 3 2.4
Orange 0.4 4 2 5 3 3 3.4 25 Orange 1 3 0 2 2 1.6
Dark Green 0.6 5 4 4 4 4 4.2 25 Dark Green 0 1 1 1 1 0.8
Light Blue 0.8 5 5 4 5 5 4.8 25 Light Blue 0 0 1 0 0 0.2
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
)%(
yrevoceR
No BP
Med BP
High BP
Particle Charge (E-7 C) |
Virginia Tech | Working Model Electrostatic Simulation
Simulation Information Summary
100
Date: 4.5.2013 Color Charge Feed No BP Med BP High BP
Time: 4:11 PM -- (*10^-7 C) Number Recovery Recovery Recovery 90
File Name1 Electrostatic_C12a_noBP.wm2d Pink -1 5 0 0 4 80 0 0 8
File Name2 Electrostatic_C12a_withBP1.wm2d Blue -0.8 5 0 0 8 0 0 9.797959
File Name 3 Electrostatic_C12a_withBP2.wm2d Purple -0.6 5 0 0 8 70 0 0 9.797959
Circuit Name: Cleaner-Recleaner/All to Head Maroon -0.4 5 0 12 8 60 0 16 9.797959
Circuit No: 17 Green -0.2 5 4 0 16 8 0 14.96663 By: Noble Yellow 0 5 4 24 32 50 8 19.59592 24
Red 0.2 5 32 24 48 40 20.39608 14.96663 9.797959
Orange 0.4 5 80 68 68 12.64911 16 20.39608 Dark Green 0.6 5 96 88 84 30 8 16 8
Light Blue 0.8 5 100 100 96 20 0 0 8
Cyan 1 5 100 100 100 0 0 0
10
Total Charge: 15 13.76 13 12.12
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Condition 1
Separation Plate 0.00006C Total Recovered Charge: 13.76
BP board NAm
Bypass Motor NArad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 0 0 0 25 Pink 5 5 5 5 5 5
Blue -0.8 0 0 0 0 0 0 25 Blue 5 5 5 5 5 5
Purple -0.6 0 0 0 0 0 0 25 Purple 5 5 5 5 5 5
Maroon -0.4 0 0 0 0 0 0 25 Maroon 5 5 5 5 5 5
Green -0.2 0 1 0 0 0 0.2 25 Green 5 4 5 5 5 4.8
Yellow 0 0 1 0 0 0 0.2 25 Yellow 5 4 5 5 5 4.8
Red 0.2 1 3 0 2 2 1.6 25 Red 4 2 5 3 3 3.4
Orange 0.4 4 3 4 4 5 4 25 Orange 1 2 1 1 0 1
Dark Green 0.6 5 5 4 5 5 4.8 25 Dark Green 0 0 1 0 0 0.2
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 2
Separation Plate 0.00006C Total Recovered Charge: 13
BP board 0.3m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 0 0 0 25 Pink 5 5 5 5 5 5
Blue -0.8 0 0 0 0 0 0 25 Blue 5 5 5 5 5 5
Purple -0.6 0 0 0 0 0 0 25 Purple 5 5 5 5 5 5
Maroon -0.4 0 1 0 2 0 0.6 25 Maroon 5 4 5 3 5 4.4
Green -0.2 0 0 0 0 0 0 25 Green 5 5 5 5 5 5
Yellow 0 1 1 1 0 3 1.2 25 Yellow 4 4 4 5 2 3.8
Red 0.2 1 1 2 0 2 1.2 25 Red 4 4 3 5 3 3.8
Orange 0.4 4 4 4 3 2 3.4 25 Orange 1 1 1 2 3 1.6
Dark Green 0.6 5 5 3 4 5 4.4 25 Dark Green 0 0 2 1 0 0.6
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 3
Separation Block 0.00006C Total Recovered Charge: 12.12
BP board 0.5m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 0 1 0.2 25 Pink 5 5 5 5 4 4.8
Blue -0.8 1 0 0 1 0 0.4 25 Blue 4 5 5 4 5 4.6
Purple -0.6 0 0 1 1 0 0.4 25 Purple 5 5 4 4 5 4.6
Maroon -0.4 0 0 0 1 1 0.4 25 Maroon 5 5 5 4 4 4.6
Green -0.2 1 0 2 1 0 0.8 25 Green 4 5 3 4 5 4.2
Yellow 0 1 0 1 3 3 1.6 25 Yellow 4 5 4 2 2 3.4
Red 0.2 2 3 3 2 2 2.4 25 Red 3 2 2 3 3 2.6
Orange 0.4 4 2 5 3 3 3.4 25 Orange 1 3 0 2 2 1.6
Dark Green 0.6 5 4 4 4 4 4.2 25 Dark Green 0 1 1 1 1 0.8
Light Blue 0.8 5 5 4 5 5 4.8 25 Light Blue 0 0 1 0 0 0.2
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
)%(
yrevoceR
No BP
Med BP
High BP
Particle Charge (E-7 C) |
Virginia Tech | Working Model Electrostatic Simulation
Simulation Information Summary
100
Date: 3.21.2013 Color Charge Feed No BP Med BP High BP
Time: 10:30 AM -- (*10^-7 C) Number Recovery Recovery Recovery 90
File Name1 Electrostatic_C26a_noBP.wm2d Pink -1 5 4 4 8 80
File Name2 Electrostatic_C26a_withBP1.wm2d Blue -0.8 5 0 12 8
File Name 3 Electrostatic_C26a_withBP2.wm2d Purple -0.6 5 4 4 16 70
Circuit Name: RSC - Scav. Con to cleaner Maroon -0.4 5 8 24 20 60
Circuit No: 26 Green -0.2 5 16 8 20 By: Noble Yellow 0 5 8 24 28 50
Red 0.2 5 24 28 40 40
Orange 0.4 5 80 72 64 Dark Green 0.6 5 96 100 100 30
Light Blue 0.8 5 100 100 100 20
Cyan 1 5 100 96 96
10
Total Charge: 15 13.08 12.16 11.68
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Condition 1
Separation Plate 0.00006C Total Recovered Charge: 13.08
BP board NAm
Bypass Motor NArad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 1 0 0.2 25 Pink 5 5 5 4 5 4.8
Blue -0.8 0 0 0 0 0 0 25 Blue 5 5 5 5 5 5
Purple -0.6 1 0 0 0 0 0.2 25 Purple 4 5 5 5 5 4.8
Maroon -0.4 1 1 0 0 0 0.4 25 Maroon 4 4 5 5 5 4.6
Green -0.2 1 1 1 1 0 0.8 25 Green 4 4 4 4 5 4.2
Yellow 0 1 1 0 0 0 0.4 25 Yellow 4 4 5 5 5 4.6
Red 0.2 1 0 3 2 0 1.2 25 Red 4 5 2 3 5 3.8
Orange 0.4 3 3 4 5 5 4 25 Orange 2 2 1 0 0 1
Dark Green 0.6 5 5 4 5 5 4.8 25 Dark Green 0 0 1 0 0 0.2
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 2
Separation Plate 0.00006C Total Recovered Charge: 12.16
BP board 0.3m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 1 0 0 0.2 25 Pink 5 5 4 5 5 4.8
Blue -0.8 1 1 0 1 0 0.6 25 Blue 4 4 5 4 5 4.4
Purple -0.6 0 0 1 0 0 0.2 25 Purple 5 5 4 5 5 4.8
Maroon -0.4 3 1 0 2 0 1.2 25 Maroon 2 4 5 3 5 3.8
Green -0.2 0 0 0 1 1 0.4 25 Green 5 5 5 4 4 4.6
Yellow 0 1 1 1 0 3 1.2 25 Yellow 4 4 4 5 2 3.8
Red 0.2 0 2 1 3 1 1.4 25 Red 5 3 4 2 4 3.6
Orange 0.4 2 3 4 5 4 3.6 25 Orange 3 2 1 0 1 1.4
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 4 4.8 25 Cyan 0 0 0 0 1 0.2
Condition 3
Separation Block 0.00006C Total Recovered Charge: 11.68
BP board 0.5m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 1 1 0 0.4 25 Pink 5 5 4 4 5 4.6
Blue -0.8 0 0 1 1 0 0.4 25 Blue 5 5 4 4 5 4.6
Purple -0.6 0 0 2 0 2 0.8 25 Purple 5 5 3 5 3 4.2
Maroon -0.4 1 2 0 2 0 1 25 Maroon 4 3 5 3 5 4
Green -0.2 1 1 0 2 1 1 25 Green 4 4 5 3 4 4
Yellow 0 1 2 1 1 2 1.4 25 Yellow 4 3 4 4 3 3.6
Red 0.2 0 1 4 2 3 2 25 Red 5 4 1 3 2 3
Orange 0.4 3 4 4 4 1 3.2 25 Orange 2 1 1 1 4 1.8
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 4 5 5 5 4.8 25 Cyan 0 1 0 0 0 0.2
)%(
yrevoceR
No BP
Med BP
High BP
Particle Charge (E-7 C) |
Virginia Tech | Working Model Electrostatic Simulation
Simulation Information Summary
100
Date: 3.21.2013 Color Charge Feed No BP Med BP High BP
Time: 11:00 AM -- (*10^-7 C) Number Recovery Recovery Recovery 90
File Name1 Electrostatic_C27a_noBP.wm2d Pink -1 5 16 24 24 80
File Name2 Electrostatic_C27a_withBP1.wm2d Blue -0.8 5 16 28 28
File Name 3 Electrostatic_C27a_withBP2.wm2d Purple -0.6 5 24 24 52 70
Circuit Name: RSC - Cleaner tail to Scavenger Maroon -0.4 5 28 40 44 60
Circuit No: 27 Green -0.2 5 32 16 44 By: Noble Yellow 0 5 36 48 56 50
Red 0.2 5 56 52 76 40
Orange 0.4 5 96 88 92 Dark Green 0.6 5 100 100 100 30
Light Blue 0.8 5 100 100 100 20
Cyan 1 5 100 100 100
10
Total Charge: 15 11.44 10.28 9.4
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Condition 1
Separation Plate 0.00006C Total Recovered Charge: 11.44
BP board NAm
Bypass Motor NArad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 1 1 2 0.8 25 Pink 5 5 4 4 3 4.2
Blue -0.8 1 2 1 0 0 0.8 25 Blue 4 3 4 5 5 4.2
Purple -0.6 3 1 1 0 1 1.2 25 Purple 2 4 4 5 4 3.8
Maroon -0.4 1 1 2 1 2 1.4 25 Maroon 4 4 3 4 3 3.6
Green -0.2 2 1 2 2 1 1.6 25 Green 3 4 3 3 4 3.4
Yellow 0 1 2 2 2 2 1.8 25 Yellow 4 3 3 3 3 3.2
Red 0.2 3 3 4 4 0 2.8 25 Red 2 2 1 1 5 2.2
Orange 0.4 5 5 4 5 5 4.8 25 Orange 0 0 1 0 0 0.2
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 2
Separation Plate 0.00006C Total Recovered Charge: 10.28
BP board 0.3m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 2 1 0 3 1.2 25 Pink 5 3 4 5 2 3.8
Blue -0.8 2 2 0 2 1 1.4 25 Blue 3 3 5 3 4 3.6
Purple -0.6 1 0 2 1 2 1.2 25 Purple 4 5 3 4 3 3.8
Maroon -0.4 5 2 0 2 1 2 25 Maroon 0 3 5 3 4 3
Green -0.2 1 1 0 1 1 0.8 25 Green 4 4 5 4 4 4.2
Yellow 0 2 2 3 2 3 2.4 25 Yellow 3 3 2 3 2 2.6
Red 0.2 3 3 2 4 1 2.6 25 Red 2 2 3 1 4 2.4
Orange 0.4 4 3 5 5 5 4.4 25 Orange 1 2 0 0 0 0.6
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 3
Separation Block 0.00006C Total Recovered Charge: 9.4
BP board 0.5m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 1 1 2 2 0 1.2 25 Pink 4 4 3 3 5 3.8
Blue -0.8 1 1 3 1 1 1.4 25 Blue 4 4 2 4 4 3.6
Purple -0.6 3 2 3 2 3 2.6 25 Purple 2 3 2 3 2 2.4
Maroon -0.4 4 2 0 2 3 2.2 25 Maroon 1 3 5 3 2 2.8
Green -0.2 4 2 1 2 2 2.2 25 Green 1 3 4 3 3 2.8
Yellow 0 3 4 1 3 3 2.8 25 Yellow 2 1 4 2 2 2.2
Red 0.2 1 5 4 4 5 3.8 25 Red 4 0 1 1 0 1.2
Orange 0.4 5 5 4 5 4 4.6 25 Orange 0 0 1 0 1 0.4
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
)%(
yrevoceR
No BP
Med BP
High BP
Particle Charge (E-7 C) |
Virginia Tech | Working Model Electrostatic Simulation
Simulation Information Summary
100
Date: 3.21.2013 Color Charge Feed No BP Med BP High BP
Time: 11:00 AM -- (*10^-7 C) Number Recovery Recovery Recovery 90
File Name1 Electrostatic_C28a_noBP.wm2d Pink -1 5 8 4 12 80
File Name2 Electrostatic_C28a_withBP1.wm2d Blue -0.8 5 4 12 12
File Name 3 Electrostatic_C28a_withBP2.wm2d Purple -0.6 5 4 12 28 70
Circuit Name: RSC - Double Cross Maroon -0.4 5 4 16 32 60
Circuit No: 28 Green -0.2 5 12 12 40 By: Noble Yellow 0 5 8 36 48 50
Red 0.2 5 40 44 60 40
Orange 0.4 5 92 96 96 Dark Green 0.6 5 100 100 100 30
Light Blue 0.8 5 100 100 100 20
Cyan 1 5 100 100 100
10
Total Charge: 15 13.36 12.88 11.56
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Condition 1
Separation Plate 0.00006C Total Recovered Charge: 13.36
BP board NAm
Bypass Motor NArad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 1 1 0.4 25 Pink 5 5 5 4 4 4.6
Blue -0.8 0 1 0 0 0 0.2 25 Blue 5 4 5 5 5 4.8
Purple -0.6 1 0 0 0 0 0.2 25 Purple 4 5 5 5 5 4.8
Maroon -0.4 0 1 0 0 0 0.2 25 Maroon 5 4 5 5 5 4.8
Green -0.2 1 0 0 1 1 0.6 25 Green 4 5 5 4 4 4.4
Yellow 0 1 1 0 0 0 0.4 25 Yellow 4 4 5 5 5 4.6
Red 0.2 3 1 2 3 1 2 25 Red 2 4 3 2 4 3
Orange 0.4 5 4 5 4 5 4.6 25 Orange 0 1 0 1 0 0.4
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 2
Separation Plate 0.00006C Total Recovered Charge: 12.88
BP board 0.3m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 1 0 0 0.2 25 Pink 5 5 4 5 5 4.8
Blue -0.8 1 1 0 1 0 0.6 25 Blue 4 4 5 4 5 4.4
Purple -0.6 0 0 1 1 1 0.6 25 Purple 5 5 4 4 4 4.4
Maroon -0.4 1 0 0 2 1 0.8 25 Maroon 4 5 5 3 4 4.2
Green -0.2 0 0 1 1 1 0.6 25 Green 5 5 4 4 4 4.4
Yellow 0 1 1 3 0 4 1.8 25 Yellow 4 4 2 5 1 3.2
Red 0.2 1 2 2 3 3 2.2 25 Red 4 3 3 2 2 2.8
Orange 0.4 4 5 5 5 5 4.8 25 Orange 1 0 0 0 0 0.2
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 3
Separation Block 0.00006C Total Recovered Charge: 11.56
BP board 0.5m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 1 2 0 0.6 25 Pink 5 5 4 3 5 4.4
Blue -0.8 0 0 2 0 1 0.6 25 Blue 5 5 3 5 4 4.4
Purple -0.6 1 0 2 2 2 1.4 25 Purple 4 5 3 3 3 3.6
Maroon -0.4 2 2 0 3 1 1.6 25 Maroon 3 3 5 2 4 3.4
Green -0.2 3 2 1 3 1 2 25 Green 2 3 4 2 4 3
Yellow 0 2 3 1 4 2 2.4 25 Yellow 3 2 4 1 3 2.6
Red 0.2 0 5 4 2 4 3 25 Red 5 0 1 3 1 2
Orange 0.4 5 5 5 5 4 4.8 25 Orange 0 0 0 0 1 0.2
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
)%(
yrevoceR
No BP
Med BP
High BP
Particle Charge (E-7 C) |
Virginia Tech | Working Model Electrostatic Simulation
Simulation Information Summary
100
Date: 3.21.2013 Color Charge Feed No BP Med BP High BP
Time: 4:00 PM -- (*10^-7 C) Number Recovery Recovery Recovery 90
File Name1 Electrostatic_C29a_noBP.wm2d Pink -1 5 16 16 32 80
File Name2 Electrostatic_C29a_withBP1.wm2d Blue -0.8 5 12 28 44
File Name 3 Electrostatic_C29a_withBP2.wm2d Purple -0.6 5 24 24 60 70
Circuit Name: RSC - Cleaner tails to Feed Maroon -0.4 5 28 48 68 60
Circuit No: 29 Green -0.2 5 48 32 56 By: Noble Yellow 0 5 28 68 60 50
Red 0.2 5 76 60 80 40
Orange 0.4 5 96 96 96 Dark Green 0.6 5 100 100 100 30
Light Blue 0.8 5 100 100 100 20
Cyan 1 5 100 100 100
10
Total Charge: 15 11.64 10.6 7.64
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Condition 1
Separation Plate 0.00006C Total Recovered Charge: 11.64
BP board NAm
Bypass Motor NArad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 1 1 2 0.8 25 Pink 5 5 4 4 3 4.2
Blue -0.8 1 1 1 0 0 0.6 25 Blue 4 4 4 5 5 4.4
Purple -0.6 3 1 1 0 1 1.2 25 Purple 2 4 4 5 4 3.8
Maroon -0.4 2 1 1 1 2 1.4 25 Maroon 3 4 4 4 3 3.6
Green -0.2 3 2 2 3 2 2.4 25 Green 2 3 3 2 3 2.6
Yellow 0 1 2 0 2 2 1.4 25 Yellow 4 3 5 3 3 3.6
Red 0.2 4 3 4 5 3 3.8 25 Red 1 2 1 0 2 1.2
Orange 0.4 4 5 5 5 5 4.8 25 Orange 1 0 0 0 0 0.2
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 2
Separation Plate 0.00006C Total Recovered Charge: 10.6
BP board 0.3m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 2 0 2 0.8 25 Pink 5 5 3 5 3 4.2
Blue -0.8 2 1 1 2 1 1.4 25 Blue 3 4 4 3 4 3.6
Purple -0.6 1 0 2 1 2 1.2 25 Purple 4 5 3 4 3 3.8
Maroon -0.4 5 2 0 4 1 2.4 25 Maroon 0 3 5 1 4 2.6
Green -0.2 2 3 1 1 1 1.6 25 Green 3 2 4 4 4 3.4
Yellow 0 4 3 3 3 4 3.4 25 Yellow 1 2 2 2 1 1.6
Red 0.2 3 3 2 5 2 3 25 Red 2 2 3 0 3 2
Orange 0.4 4 5 5 5 5 4.8 25 Orange 1 0 0 0 0 0.2
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 3
Separation Block 0.00006C Total Recovered Charge: 7.64
BP board 0.5m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 1 1 3 2 1 1.6 25 Pink 4 4 2 3 4 3.4
Blue -0.8 2 2 4 2 1 2.2 25 Blue 3 3 1 3 4 2.8
Purple -0.6 4 1 4 3 3 3 25 Purple 1 4 1 2 2 2
Maroon -0.4 5 3 1 5 3 3.4 25 Maroon 0 2 4 0 2 1.6
Green -0.2 4 3 2 3 2 2.8 25 Green 1 2 3 2 3 2.2
Yellow 0 2 2 4 3 4 3 25 Yellow 3 3 1 2 1 2
Red 0.2 2 4 5 5 4 4 25 Red 3 1 0 0 1 1
Orange 0.4 5 5 5 5 4 4.8 25 Orange 0 0 0 0 1 0.2
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
)%(
yrevoceR
No BP
Med BP
High BP
Particle Charge (E-7 C) |
Virginia Tech | Working Model Electrostatic Simulation
Simulation Information Summary
100
Date: 3.21.2013 Color Charge Feed No BP Med BP High BP
Time: 10:15 PM -- (*10^-7 C) Number Recovery Recovery Recovery 90
File Name1 Electrostatic_C30a_noBP.wm2d Pink -1 5 4 8 12 80 8 9.797959 16
File Name2 Electrostatic_C30a_withBP1.wm2d Blue -0.8 5 0 16 16 0 23.32381 14.96663
File Name 3 Electrostatic_C30a_withBP2.wm2d Purple -0.6 5 4 12 20 70 8 16 17.88854
Circuit Name: RSC - Full Recylce Maroon -0.4 5 4 20 40 60 8 21.9089 21.9089
Circuit No: 30 Green -0.2 5 16 24 40 8 19.59592 17.88854 By: Noble Yellow 0 5 40 32 52 50 25.29822 24 9.797959
Red 0.2 5 68 60 76 40 20.39608 21.9089 14.96663
Orange 0.4 5 96 92 88 8 16 9.797959 Dark Green 0.6 5 100 100 100 30 0 0 0
Light Blue 0.8 5 100 100 100 20 0 0 0
Cyan 1 5 100 100 100 0 0 0
10
Total Charge: 15 14.04 12.4 11.48
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Condition 1
Separation Plate 0.00006C Total Recovered Charge: 14.04
BP board NAm
Bypass Motor NArad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 1 0 0.2 25 Pink 5 5 5 4 5 4.8
Blue -0.8 0 0 0 0 0 0 25 Blue 5 5 5 5 5 5
Purple -0.6 1 0 0 0 0 0.2 25 Purple 4 5 5 5 5 4.8
Maroon -0.4 0 1 0 0 0 0.2 25 Maroon 5 4 5 5 5 4.8
Green -0.2 1 0 1 1 1 0.8 25 Green 4 5 4 4 4 4.2
Yellow 0 3 1 0 3 3 2 25 Yellow 2 4 5 2 2 3
Red 0.2 5 3 3 2 4 3.4 25 Red 0 2 2 3 1 1.6
Orange 0.4 5 5 5 4 5 4.8 25 Orange 0 0 0 1 0 0.2
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 2
Separation Plate 0.00006C Total Recovered Charge: 12.4
BP board 0.3m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 1 0 1 0.4 25 Pink 5 5 4 5 4 4.6
Blue -0.8 0 1 0 3 0 0.8 25 Blue 5 4 5 2 5 4.2
Purple -0.6 0 0 2 0 1 0.6 25 Purple 5 5 3 5 4 4.4
Maroon -0.4 1 1 0 3 0 1 25 Maroon 4 4 5 2 5 4
Green -0.2 3 0 1 1 1 1.2 25 Green 2 5 4 4 4 3.8
Yellow 0 3 0 1 1 3 1.6 25 Yellow 2 5 4 4 2 3.4
Red 0.2 3 4 1 4 3 3 25 Red 2 1 4 1 2 2
Orange 0.4 3 5 5 5 5 4.6 25 Orange 2 0 0 0 0 0.4
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 3
Separation Block 0.00006C Total Recovered Charge: 11.48
BP board 0.5m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 2 1 0 0.6 25 Pink 5 5 3 4 5 4.4
Blue -0.8 0 0 1 1 2 0.8 25 Blue 5 5 4 4 3 4.2
Purple -0.6 1 0 2 0 2 1 25 Purple 4 5 3 5 3 4
Maroon -0.4 3 2 0 3 2 2 25 Maroon 2 3 5 2 3 3
Green -0.2 2 1 1 3 3 2 25 Green 3 4 4 2 2 3
Yellow 0 3 3 2 2 3 2.6 25 Yellow 2 2 3 3 2 2.4
Red 0.2 3 3 5 4 4 3.8 25 Red 2 2 0 1 1 1.2
Orange 0.4 4 5 5 4 4 4.4 25 Orange 1 0 0 1 1 0.6
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
)%(
yrevoceR
No BP
Med BP
High BP
Particle Charge (E-7 C) |
Virginia Tech | Working Model Electrostatic Simulation
Simulation Information Summary
100
Date: 3.21.2013 Color Charge Feed No BP Med BP High BP
Time: 9:00 PM -- (*10^-7 C) Number Recovery Recovery Recovery 90
File Name1 Electrostatic_C31a_noBP.wm2d Pink -1 5 4 4 8 80
File Name2 Electrostatic_C31a_withBP1.wm2d Blue -0.8 5 0 8 4
File Name 3 Electrostatic_C31a_withBP2.wm2d Purple -0.6 5 8 4 16 70
Circuit Name: RSC - Scavenger con to feed Maroon -0.4 5 4 4 24 60
Circuit No: 31 Green -0.2 5 8 8 24 By: Noble Yellow 0 5 4 20 16 50
Red 0.2 5 16 24 32 40
Orange 0.4 5 76 72 64 Dark Green 0.6 5 96 100 100 30
Light Blue 0.8 5 100 100 100 20
Cyan 1 5 100 96 96
10
Total Charge: 15 12.96 12.68 11.64
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Condition 1
Separation Plate 0.00006C Total Recovered Charge: 12.96
BP board NAm
Bypass Motor NArad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 0 1 0 0.2 25 Pink 5 5 5 4 5 4.8
Blue -0.8 0 0 0 0 0 0 25 Blue 5 5 5 5 5 5
Purple -0.6 1 0 1 0 0 0.4 25 Purple 4 5 4 5 5 4.6
Maroon -0.4 0 1 0 0 0 0.2 25 Maroon 5 4 5 5 5 4.8
Green -0.2 1 0 0 1 0 0.4 25 Green 4 5 5 4 5 4.6
Yellow 0 1 0 0 0 0 0.2 25 Yellow 4 5 5 5 5 4.8
Red 0.2 1 0 2 1 0 0.8 25 Red 4 5 3 4 5 4.2
Orange 0.4 3 3 4 4 5 3.8 25 Orange 2 2 1 1 0 1.2
Dark Green 0.6 5 5 4 5 5 4.8 25 Dark Green 0 0 1 0 0 0.2
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 5 5 25 Cyan 0 0 0 0 0 0
Condition 2
Separation Plate 0.00006C Total Recovered Charge: 12.68
BP board 0.3m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 1 0 0 0.2 25 Pink 5 5 4 5 5 4.8
Blue -0.8 0 1 0 1 0 0.4 25 Blue 5 4 5 4 5 4.6
Purple -0.6 0 0 1 0 0 0.2 25 Purple 5 5 4 5 5 4.8
Maroon -0.4 0 0 0 1 0 0.2 25 Maroon 5 5 5 4 5 4.8
Green -0.2 0 0 0 1 1 0.4 25 Green 5 5 5 4 4 4.6
Yellow 0 1 1 0 0 3 1 25 Yellow 4 4 5 5 2 4
Red 0.2 0 2 0 3 1 1.2 25 Red 5 3 5 2 4 3.8
Orange 0.4 2 3 4 5 4 3.6 25 Orange 3 2 1 0 1 1.4
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 5 5 5 4 4.8 25 Cyan 0 0 0 0 1 0.2
Condition 3
Separation Block 0.00006C Total Recovered Charge: 11.64
BP board 0.5m
Bypass Motor 5rad/sec
Number Recovered to Right Side Number Recovered to Left Side
Charge Test Run Test Run
(*10^-7 C) 1 2 3 4 5 Average Sum 1 2 3 4 5 Average
Pink -1 0 0 1 1 0 0.4 25 Pink 5 5 4 4 5 4.6
Blue -0.8 0 0 1 0 0 0.2 26 Blue 5 5 5 5 5 5
Purple -0.6 0 0 2 0 2 0.8 24 Purple 5 5 2 5 3 4
Maroon -0.4 2 2 0 2 0 1.2 25 Maroon 3 3 5 3 5 3.8
Green -0.2 1 2 0 2 1 1.2 25 Green 4 3 5 3 4 3.8
Yellow 0 0 1 1 0 2 0.8 25 Yellow 5 4 4 5 3 4.2
Red 0.2 0 2 3 1 2 1.6 25 Red 5 3 2 4 3 3.4
Orange 0.4 3 4 4 4 1 3.2 25 Orange 2 1 1 1 4 1.8
Dark Green 0.6 5 5 5 5 5 5 25 Dark Green 0 0 0 0 0 0
Light Blue 0.8 5 5 5 5 5 5 25 Light Blue 0 0 0 0 0 0
Cyan 1 5 4 5 5 5 4.8 25 Cyan 0 1 0 0 0 0.2
)%(
yrevoceR
No BP
Med BP
High BP
Particle Charge (E-7 C) |
Virginia Tech | Hydrophobic Forces in Wetting Films
Lei Pan
Abstract
Flotation is an important separation process used in the mining industry. The process is based
on hydrophobizing a selected mineral using an appropriate surfactant, so that an air bubble can
spontaneously adhere on the mineral surface. The bubble-particle adhesion is possible only when
the thin film of water between the bubble and particle ruptures, just like when two colloidal
particles or air bubbles adhere with each other. Under most flotation conditions, however, both
the double-layer and dispersion forces are repulsive, which makes it difficult to model the rupture
of the wetting films using the DLVO theory.
In the present work, we have measured the kinetics of film thinning between air bubble and
flat surfaces of gold and silica. The former was hydrophobized by ex-site potassium amyl
xanthate, while the latter by in-site Octadecyltrimetylammonium chlroride. The kinetics curves
obtained with and without theses hydrophobizing agents were fitted to the Reynolds lubrication
theory by assuming that the driving force for film thinning was the sum of capillary pressure and
the disjoining pressure in a thin film. It was found that the kinetics curves obtained with
hydrophilic surfaces can be fitted to the theory with the disjoining pressure calculated from the
DLVO theory. With hydrophobized surfaces, however, the kinetics curves can be fitted only by
assuming the presence of a non-DLVO attractive force (or hydrophobic force) in the wetting
films. The results obtained in the present work shows that long-range hydrophobic forces is
responsible for the faster drainage of wetting film.
It is shown that the changes in hydrophobic forces upon the thin water film between air
bubble and hydrophobic surface is dependent on hydrophobizing agent concentration, immersion
time and the electrolyte concentration in solution. The obtained hydrophobic forces constant in
wetting film K is compared with the hydrophobic forces constant between two solid surfaces
132
K to verify the combining rule for flotation.
131 |
Virginia Tech | Acknowledgement
First and foremost, I would like to acknowledge my advisor, Dr. Roe-Hoan Yoon, for his
support and guidance throughout my first one and half year study in Virginia Tech. I would like
to express my appreciation to him for his sincere suggestion on my research life and his care on
my personal life in Blacksburg. During this period, I have learned a lot from Dr. Yoon about how
to conduct research projects, how to think, how to write. I also thank Dr. Gregory Adel and Dr.
Gerald Luttrell who served as my thesis committee member for their suggestions on my research
works.
I sincerely appreciate to Ruijia Wang for his instruction and training on thin film balance
technique, and Dr. Jialin Wang for his guidance and instruction in preparing the hydrophobic
surface in our lab. Especially, I would like to thank Mr. Zuoli Li who measured the hydrophobic
forces between gold surfaces in pure water for me to verify the combining rule.
I would like to express my appreciation to Hyunsun Do for his discussion on theoretical
model of wetting film drainage. Also I want to express my sincere gratitude to Dr. Liguang Wang
for his discussion on thinning kinetics and TFB technique.
I would like to thank past and present members in Center of Advanced Separation
Techniques (CAST), Charles Schlosser, Chad Freeland for their discussion and friendship, Chris
Hull, Kathy Flint, Dongcheol Shin and Dr. Jinming Zhang for their support and help.
Last but not least, I would like to express my deepest gratitude to my parents for their selfless
love and support. Without their love and support, I would not have my accomplishment.
iii |
Virginia Tech | Chapter 1
Introduction
1.1 General
Froth flotation has been widely used in industry for more than 100 years to separate the
different minerals from each other, ever since the gas bubble was first recognized as a means of
mineral separation by Bassel brothers in Germany in 1886.1, 2 Even today after 100 years since its
commercial inception in 1905, froth flotation is still the most successful and cheapest technique
in many separating industry, such as deinking3-5, oil recovery from oil sand6-8, and coal cleaning9-
11. The basic principle of froth flotation is rendering the target mineral hydrophobic using
appropriate surfactants and untarget mineral hydrophilic. The gas bubbles prefer to attach the
hydrophobic particles, and rise to the top of cell, forming the three-phase froth. The attachment
between the gas bubble and the particle is the fundamental process for flotation. As described in
previous literatures12, the initial approach of gas bubbles to the particles is controlled by turbulent
flow generated by the agitator. As the intervening water layers between the gas bubble and the
particle become thinner, the surface forces become an active component to control the thinning
and rupture of the water thin film between the gas bubble and particle which is usually titled
“wetting film” in colloidal science. On the hydrophobic surface, the wetting film will
spontaneously rupture forming the three-phase froth. Recognizing the significance of the
interaction between the gas bubble and the particle related to the flotation condition, many
investigators tried to better understand the interaction energy between air bubbles and particles in
order to predict the flotation behavior and improve the flotation cell design.
The DLVO theory was named after Derjaguin and Landau13, Verwey and Overbeek14, two
independent research groups who first formulated the classical standard theory of colloidal
dispersions that comprised both the repulsion and attraction forces. Classical DLVO theory
contains two additive components: van der Waals dispersion energy (V ) and electrostatic double
d
layer energy (V ). Thus the total interaction energy could be represented as follows:
e
(cid:1848) (cid:3404) (cid:1848) (cid:3397)(cid:1848) (cid:4670)1.1(cid:4671)
(cid:3047) (cid:3031) (cid:3032)
For interaction between spherical gas bubble and spherical particle of radii R and R , the
1 2
dispersion energy (V ) is described by the following expression
d
(cid:1848) (cid:3404) (cid:3398)
(cid:3002)(cid:3117)(cid:3119)(cid:3118)(cid:3019)(cid:3117)(cid:3019)(cid:3118)
(cid:4674)1(cid:3398)
(cid:2869)(cid:2878)(cid:2870)(cid:3029)(cid:3039)
(cid:4675) [1.2]
(cid:3031)
(cid:2874)(cid:3009)(cid:4666)(cid:3019)(cid:3117)(cid:2878)(cid:3019)(cid:3118)(cid:4667) (cid:2869)(cid:2878)(cid:3029)(cid:3030)/(cid:3009)
where A is the Hamaker constant of particle (1) and bubble (2) interaction in the aqueous
132
medium (3), and H is the separation distance. The second part of the equation represents the
correction factor due to the retardation effect in presence of the electrolyte in solution. Since the
Hamaker constant of water (A ) is less than that of the solid (A ) but greater than that of the air
33 11
(A ), A is negative according to the combining rule15. Thereby, the dispersion energy between
22 132
the gas bubble and the particle is always repulsive.
1 |
Virginia Tech | Fig 1.2. The “dimpled” shape means that the film thickness at the center is larger than that
at its periphery. Thus once a “dimple” is formed, the liquid inside dimple is entrapped by a
thinner “barrier ring”. Fisher et al.37 explained that the “dimple” phenomenon is due to
viscous liquid drag. The liquid at the barrier ring drains faster than those at the center, and
thereby a convex “dimpled” shape is formed. Also the convex shape introduces the positive
Laplace pressure that resists the drainage of the liquid inside the dimple.
Figure 1.2 Schematic diagram of a “dimple” formation as the gas bubble approaches the flat solid surface38
Scheludko39 first derived an expression as follows to predict the rate of thinning of the film
using Reynolds equation, based on the assumption that thin liquid films are plane-parallel.
(cid:3398)(cid:1856)(cid:1860) 2(cid:1860)(cid:2871)Δ(cid:1842)
(cid:1848) (cid:3404) (cid:3404) (cid:4670)1.12(cid:4671)
(cid:3045)(cid:3032) (cid:1856)(cid:1872) 3(cid:2015)(cid:1844)(cid:2870)
where R is the radius of the film, (cid:2015) is the viscosity of liquid and Δ(cid:1842) is the driving pressure for
film thinning. In Scheludko’s cell, the driving pressure Δ(cid:1842) is a sum of the capillary pressure (cid:1842)
(cid:3097)
and disjoining pressure Π. Thus the equilibrium film thickness could be predicted at a finite film
thickness where the capillary pressure equates to disjoining pressure.
The derivation of equation 1.12 is based on Reynolds lubrication approximation which
assumes that the liquid flows between plane parallel surfaces and the film surfaces are
tangentially immobile. Frankel and Mysels35 developed a purely hydrodynamic theory of the
evolution of dimples, assuming that the rate of flow through the barrier ring was independent of
the radius. They proposed that the film thickness at the center of dimple h and film thickness at
o
the barrier ring h as a function of time t is given by Equation [1.13] and Equation [1.14].
b
5 |
Virginia Tech | 0.0096(cid:1870)(cid:2874)(cid:2015)(cid:1866)(cid:2870) (cid:2869)/(cid:2872)
(cid:2868)
(cid:1860) (cid:3404) (cid:4678) (cid:4679) (cid:4670)1.13(cid:4671)
(cid:2868) (cid:2011)(cid:1844)(cid:1872)
0.006(cid:1866)(cid:2870)(cid:1844)(cid:2873)(cid:2025)(cid:1859)(cid:2015) (cid:2869)/(cid:2870)
(cid:1860) (cid:3404) (cid:4678) (cid:4679) (cid:4670)1.14(cid:4671)
(cid:3029) (cid:2011)(cid:3047)(cid:1872)
Frankel-Mysels model is corresponding to experimental observation done by Platikanov34, but
slightly overestimates the “dimple” shape.
After Frankel-Mysels’ initial work on the theoretical modeling of draining dimpled film,
numerous investigators have developed models describing the thinning of dimpled films.
Hartland and Robinson40 presented a model for an axisymmetric dimpled draining film assuming
the film was parabolic. They suggested that pressure is constant at the center of film and falls to
zero just outside the barrier ring. Dimitrov and Ivanov41 derived an equation for the rate of
thinning of dimpled films by matching the asymptotic coordinate expansions. It is shown that the
film could be considered as practically plane-parallel at a short film thickness. Jain and Ivanov38
later presented a simplified model for the thinning of dimpled films, assuming that all the energy
dissipation takes places in the barrier. They suggested that the presence of a dimple increases the
velocity of thinning. Lin and Slattery42 developed a hydrodynamic model for drainage of thin
liquid film which agrees with the experimental results. Also, they suggested that the surface
viscosities have little effect upon the drainage rate in presence of surfactants. Chen and Slattery43,
44 later extended the model for drainage of film by considering the van der Waals forces and
electrostatic double-layer forces. Recently, Ralson et al.45, 46 studied the drainage of aqueous thin
film as the gas bubble approach the flat solid surface using scanning interferometry. The
experimental equilibrium film thickness agrees with the theoretical prediction by the DLVO
theory.
1.2.3 Bubble-Solid Interaction
The colloidal interaction between air bubble and particle was studied for decades by TFB47,
SFA48, 49 and AFM50, 51, primarily because it is of utmost importance for flotation. It is well
known that the wetting film is stabilized on the hydrophilic surface and ruptures spontaneously on
hydrophobic surfaces. Tchaliovska et al.52 studied the interaction between air bubble and a mica
surface immersed in dodecylammonium chloride (DAC) solution. The results showed that
hydrophobic attraction forces played a vital role in thin-film lifetimes and rates of expansion of
the meniscus perimeter. Yoon and Yordan53 measured the critical thickness of film on
hydrophobic surfaces by using TFB technique. It was shown that critical rupture thickness
increases with an increasing degree of methylation. The rupture thickness also reaches the
maximum, when pH of the amine solution is 9-9.5 where the adsorption is most favored. The
following papers23, 54-56 by Yoon et al. suggested that thinning and rupture of thin water film
intervened by hydrophobic surfaces must include the influence of hydrophobic attractive forces.
The mechanism of attachment between air bubble and hydrophobic particle, however, is
still discussed to illustrate the existence of so called “hydrophobic forces”. Alexandrova et al.57
6 |
Virginia Tech | studied the stability of an aqueous thin film containing tetradecyltrimethylammonium bromide
(C TAB) between air bubbles and silica substrates. They explained that the spontaneous rupture
14
of thin aqueous film was interpreted in terms of electrostatic mechanism, also named as
“Heterocoagulation mechanism”. At relatively low surfactant concentration, the gas/liquid
interface was net positive charged while the liquid/solid interface remained negative charged,
which led to an attractive electrostatic forces between gas bubble and solid.
Schulze et al.,58, 59 suggested that gas bubble at heterogeneous solid surfaces was
responsible for the rupture of wetting film on methylated silica without considering any long-
range hydrophobic attraction. Mahnke et al.60 observed a hole in the dimpled wetting film on
hydrophobic glass surfaces coated with fatty acid Langmuir-Blodgett layers, and they indicated
that nucleation of the air bubble is the reason for the high rupture thickness for hydrophobic
surface coated with L-B layers. In other communication, Mahnke et al.,61 also suggested that the
rupture of thin wetting films on methylated glass surfaces was explained by hole formation.
Stckelhuber et al.,62, 63recently proposed that nanobubbles on the hydrophobic solid surface could
be the cause of rupture of wetting films without considering any surface forces acting on the
interface.
Vinogradova64-67 proposed that the observed long-range hydrophobic forces measured in
drainage technique may occur due to slippage. It is a possibility that application of Reynolds
equation theory may led to overestimating the hydrophobic forces. Thereby, the magnitude of
hydrophobic forces correlated to slip lengths of liquid over solid or vapor, which is probably
linked to a decrease in viscosity in the vicinity of a solid.
1.3 Research Objective
The objective of present work is to measure the thinning kinetics of wetting film between
the gas bubble and the hydrophobic solid surface by Thin Film Balance technique. The
hydrophobic forces constant K is calculated using the Reynolds equation, assuming that
132
Reynolds equation could apply to predict the thinning kinetics of thin water film at the “barrier
ring”. The obtained K in wetting film on the hydrophobic surface is used to compare with
132
surface forces data reported in previous literature to determine the validity of the combining rule.
Chapter 2 discusses the drainage of wetting film on the gold surface treated by potassium
amyl xanthate (PAX). Chapter 3 describes the drainage of wetting film on the polished quartz
surface in octadecyltrimethylammonium chloride (C TACl) solution.
18
1.4 References
1. Fuerstenau, D. W., A Century of Developments in the Chemistry of Flotation processing.
In Froth flotation: a century of innovation, M.C. Fuerstenau, G. J., Roe‐Hoan Yoon, Ed. SME: 2007.
2. A.J. Lynch, J. S. W., J.A. Finch, G.E. Harbort, History of Flotation Technology. In Froth
flotation: a century of innovation, M.C. Fuerstenau, G. J., Roe‐Hoan Yoon, Ed. SME: 2007.
7 |
Virginia Tech | Chapter 2
Hydrophobic Forces in Wetting Films Between Air Bubbles and
Hydrophobized Gold Surfaces
Abstract
The kinetics of thinning for the wetting films of water formed on hydrophobic gold substrates
has been studied using the thin film pressure balance (TFBC) technique. The changes in film
thickness have been monitored by recording the profiles of the dimpled films as a function of
time using a high-speed video camera. It was found that the film thinning kinetics as measured at
the barrier rims of a film of water formed on a hydrophilic silica surface can be predicted using
the Reynolds lubrication approximation with the non-slip boundary condition. The results
obtained using the wetting films of water formed on hydrophobized gold substrates showed that
the kinetics increases with increasing hydrophobicity. The kinetics data obtained at different
hydrophobicities have been fitted to the Reynolds approximation to determine the hydrophobic
force constants (K ) of a power law. It was found that K increases with increasing contact
132 132
angle and decreases with electrolyte (NaCl) concentration. It was found also that the K values
132
can be predicted from the hydrophobic force constants (K ) obtained for the interaction between
131
hydrophobic surfaces and the same (K ) for the foam films using the geometric mean combining
232
rule that is frequently used to predict asymmetric molecular forces from symmetric ones.
12 |
Virginia Tech | 2.1 Introduction
Properties of the thin liquid films between particles, bubbles and drops control the behavior of
their suspensions and interactions with each other. In flotation, air bubbles collide with particles
and create wetting films between them. If the films are unstable, particles break the films, attach
themselves to the bubbles, and float. If the films are stable, no flotation would occur. Thus,
control of the stability of wetting films is of critical importance in flotation. The key parameter
controlling the stability of wetting films is the hydrophobicity of particles. Flotation is a rate
process, and its kinetics increases with particle hydrophobicity.1, 2 Various reagents are used to
render selected minerals hydrophobic. For sulfide minerals and precious metals, short-chain alkyl
xanthates and thionocarbamates are commonly used as hydrophobizing agents (collectors). For
the flotation of non-metallic minerals such as silica and iron oxides, long-chain high HLB
surfactants are used for hydrophobization. Air bubbles have been in use for minerals flotation
since 1905 when the process was first patented,3 and yet the basic mechanisms involved in the
rupture of wetting films are not well understood.
When an air bubble is pressed against a hydrophilic plate such as mica and quartz in water4-7,
the intervening liquid drains until a stable film is formed. The stability of the wetting film arises
from the disjoining (or ‘wedging-apart’) pressure, which was considered to arise from double-
layer force, van der Waals-dispersion force, and structural force.8 The first two were classical
colloidal forces, while the third was attributed to the hydrogen bonding between the solid and
water molecules in the film. At film thicknesses above 20 nm, double-layer force dominates,
while at thicknesses below approximately 10-15 nm dispersion force also contributes to
stabilizing the wetting films.9, 10 The film thickness decreases with electrolyte concentration and
the valence of electrolyte.11 It has also been reported that the wetting films on quartz rupture
when the charge of the substrate was reversed by Al3+ ions12 or by a cationic surfactant13. Thus,
the wetting films on hydrophilic surfaces behave just like a typical colloidal film, for which the
DLVO theory may be useful.
Blake and Kitchener9 used the bubble-against-plate technique to study the wetting films
formed on both hydrophilic and hydrophobic silica plates. The hydrophobic silica was prepared
by coating the surface with trichloromethylsilane (TMCS). The thicknesses of the aqueous films
formed on both substrates were approximately the same, which was attributed to the observation
that the methylation did not significantly change the ζ-potentials of quartz. When the film
thickness was reduced by KCl addition, however, the film on the hydrophobic surface ruptured
spontaneously at a thickness of 64 nm, which was attributed to the presence of hydrophobic force
in the wetting film. Tchaliovska et al.14 studied the wetting films on mica hydrophobized with
dodecylammonium hydrochloride (DAH) and suggested that both hydrophobic force and
attractive electrostatic force are important in determining the film lifetime and the rates of
expansion of the meniscus perimeter. More recently, Mahnke et al.15 modeled the rupture of the
wetting films formed on methylated glass plates using a long-range hydrophobic force with a
decay length of 13 nm, while Churaev16 discussed the role of hydrophobic force in the rupture of
wetting films.
13 |
Virginia Tech | The first direct measurement of hydrophobic force was reported by Israelachvili and
Pashley.17 The measurements were made using the surface force apparatus (SFA) in
cetyltrimethyl-ammonium bromide (CTAB) solutions. Many investigators18 conducted follow-up
experiments using SFA and atomic force microscope (AFM) with surfaces coated with various
hydrophobizing agents and reported much longer-ranged and stronger hydrophobic forces than
reported by Israelachvili and Pashley. However, the origin of the hydrophobic force is not yet
known, and many investigators suggested various possible mechanisms. These include
electrostatic interaction between the charged domains of adsorbed surfactants,19 cavition, 20, 21
nanobubbles,22, 23 and others.24-26 Of these, the possibility of nanobubbles causing the long-range
attractions has received much attention in recent years. It has been shown, however, that long-
range attractions were also observed in degassed solutions,27, 28 although the attraction becomes
stronger in the presence of dissolved gases. Further, recent thermodynamic studies showed that
macroscopic hydrophobic interactions entail entropy decrease, contrary to the case of molecular-
scale hydrophobic interactions.29, 30 This finding suggests that the long-range hydrophobic force
originates from the structural changes in the water present in the confined spaces between
hydrophobic surfaces.
The possibility of nanobubbles playing a role in the rupture of wetting films has been explored
by some investigators. Stockelhuber et al.31 suggested that the thin liquid films formed between
the nanobubbles nucleating on a hydrophobic surface and the air/water interface of a wetting film
act like foam films, in which attractive van der Waals force can cause the rupture by the capillary
wave mechanism.32, 33 According to Laskowski et al. 34, there are no attractive forces in wetting
films; therefore, its rupture cannot be accounted by the capillary wave mechanism.
Platikanov35 monitored the kinetics of thinning of the wetting films on hydrophilic glass
plates. They found that the films formed dimples initially and produced flat films at equilibrium
as predicted by Frankel and Mysels.36 The dimples disappeared, however, when film radii became
small. The author showed that the kinetics measured at 0.1 M KCl can be described by Reynolds
lubrication approximation with the non-slip boundary condition for both the solid/water and
air/water interfaces. Thus, the author concluded that the dynamic method of using the Reynolds
approximation can be used to determine the disjoining pressure if liquid films are flat and plane-
parallel. Wang and Yoon37 also used the Reynolds approximation to determine the contributions
from the hydrophobic force to the disjoining pressures in single foam films. They found that air
bubbles are hydrophobic and that the hydrophobic force in foam films decreases with increasing
surfactant and NaCl concentrations.
Schulze and his co-workers10 measured the critical rupture thicknesses of the wetting films
formed on hydrophobic surfaces and compared the results with the film thinning kinetics
predicted using the Reynolds equation. Without recognizing the presence of hydrophobic force,
the authors suggested that the kinetics of film thinning should follow the same theoretical curve,
because the surface charge did not change after hydrophobization with hexamethyldisilazane
(HMDS). It was found that the critical rupture thicknesses plotted vs. film lifetime were randomly
distributed around the theoretical thinning curve, which lead to their conclusion that films rupture
due to the presence of gas nuclei formed on heterogeneous surfaces and the hole formation
14 |
Virginia Tech | mechanism suggested by Sharma and Ruckenstein.38 More recent work of Sharma39 suggested,
however, that the hole formation is due to hydrophobic attraction.
In the present work, we monitored the kinetics of thinning of wetting films using the TFPB
technique. Water films were formed on gold plates hydrophobized with potassium amyl xanthate
(PAX). The kinetics was monitored by means of a high-speed video camera, which allowed
accurate measurements of film thicknesses changing with time at any point of a dimpled film.
The results were analyzed using the Reynolds approximation with the non-slip boundary
condition to determine the disjoining pressures. It was found that the kinetics of film thinning
increases with increasing hydrophobicity, which was attributed to the increase in hydrophobic
force in wetting films. The magnitudes of the hydrophobic forces measured in the wetting films
were compared with those measured in foam films and in the films between hydrophobic solid
surfaces.
2.2 Theoretical Approach
When an air bubble is pressed against a flat solid surface in a horizontal orientation, the
air/water interface deforms to produce initially a plane-parallel wetting film. The change in
curvature associated with the deformation creates a pressure difference between the liquid in the
film and the bulk and causes the film to thin. As the thinning continues, the film becomes a
convex lens (or “dimple”) with inverted curvature. The torus-shaped water film surrounding a
dimple is referred to “barrier rim”. The dimpled film is no longer plane parallel, but many
investigators40, 41 modeled the thinning process using the Reynolds lubrication approximation,
which has been derived using the boundary conditions that the two interfaces are parallel to each
other and that the liquid velocity at the two interfaces are zero, i.e., the film thins under no slip
conditions. The radii of the barrier rings observed in the present work were larger than 0.08 mm,
while we were monitoring the thinning rate at film thicknesses below approximately 300 nm. The
large difference in the length scales should satisfy the first boundary condition. Maali et al.42
showed that the non-slip boundary condition is appropriate for the water flow on the hydrophilic
surface. Also, Lin and Slattery43 showed that the air/water interfaces are immobile in the presence
of a trace of surfactant. Platikanov35 and Frankel et al.36 showed that the Reynolds lubrication
approximation can be used to model the thinning of wetting films at the barrier ring.
The Reynolds lubrication approximation is usually presented in the following form:44
(cid:1856)(cid:1860) 2(cid:1860)(cid:2871)∆(cid:1842)
(cid:3404) (cid:3398) (cid:4670)1(cid:4671)
(cid:1856)(cid:1872) 3(cid:2020)(cid:1844)(cid:2870)
(cid:3033)
where h is film thickness, t drainage time, μ the liquid viscosity, R the film radius, and ΔP is the
f
pressure difference causing a film to thin. In dimpled wetting films, the ΔP may be expressed as
follows,45
2(cid:2026) (cid:2011) (cid:2034) (cid:2034)(cid:1860)
Δ(cid:1842) (cid:3404) (cid:3398)Π(cid:3398) (cid:3436)(cid:1870) (cid:3440) (cid:4670)2(cid:4671)
(cid:1844) (cid:1870)(cid:2034)(cid:1870) (cid:2034)(cid:1870)
15 |
Virginia Tech | Hydrophobic forces measured in experiments are usually represented in single- or double-
exponential functions. It has been shown that a power law of the following form can also be
used,47, 48
(cid:1837)
(cid:2869)(cid:2871)(cid:2870)
Π (cid:3404) (cid:3398) (cid:4670)7(cid:4671)
(cid:3035) 6(cid:2024)(cid:1860)(cid:2871)
where K is a constant representing the magnitude of the hydrophobic force in a wetting film of
132
thickness h. Eq. [7] is of the same form as Eq. [5], which makes it easier to compare K directly
132
with A .
132
2.3 Experimental Details
Materials
Polished fused quartz plates (Technical Glass Inc.) and gold-coated glass plates (CA134, EMF)
were used as substrates for wetting films. Potassium amyl xanthate (PAX, >90%, TCI, America)
was purified twice by dissolution in acetone (HPLC, Fisher Sci.) and recrystallization in diethyl
ether (99.999%, Sigma-Alrich). Sodium chloride (99.999%, Sigma-Aldrich) was roasted at 600°C
for 6 hours to remove organic impurities. All solutions were prepared using the Millipore water
of >18.2 MΩ/cm, which was obtained using a Direct-Q3 water purification system.
Procedure
Both the fused quartz and gold plates were cleaned by boiling in piranha solutions (7:3 by
volume of H SO :H O ) for 30 minutes, followed by rinsing with the Millipore water and drying
2 4 2 2
in a nitrogen gas stream. After the cleaning, the gold plates were hydrophobized by immersing
them in freshly prepared PAX solutions. The hydrophobicity was controlled by varying the
concentration and immersion time. The treated surfaces were washed with Millipore water and
dried by blowing high-purity nitrogen gas on the surface. The quartz was used as substrate for
wetting films without hydrophobization.
The kinetics of film thinning was measured using the TFPB technique, which was originally
designed to study foam films.32 A flat substrate was placed on top of a film holder (2.0 mm
radius) immersed in water. After making sure that no air bubbles were adhering on the surface,
the assembly was placed on an inverted microscopic stage (Olympus IX51) to monitor the
changes in film thickness with time. A halogen lamp (100W, Osram) was used as a light source
with a band-pass filter (NT46-053, Edmund Optics) to produce monochromatic green light source
(λ=526 nm).
Initially, the thickness of a film was reduced by pulling the water out of the film holder by
means of a piston. Once interference patterns (Newton rings) began to appear in the microscopic
field of view, the film was allowed to thin spontaneously while recording the images by means of
a high-speed CCD camera (Fastcam 512PCI, Photron) at a speed of 60 frames per second. The
17 |
Virginia Tech | camera was capable of taking the images at much higher speeds, but 50 FPS was found to be
adequate. The interference patterns recorded were used to obtain timed profiles of a wetting film
across the entire film holder, which in turn were used to monitor the changes in film thicknesses
(h) as a function of time at any point of the film. The film thicknesses were calculated as
described by Nedyalkov et al.49
The surface tensions of the solutions used in the present work were measured using the
Wilhelmy plate method. Advancing and receding contact angles of the hydrophobized gold plates
were measured using the sessile drop technique by means of a goniometer (Ramé-Hart, Inc.).
The ξ-potentials of spherical gold power (1.5-3.0 µm, Alfa Aesar) were measured by dynamic
light scattering (Zetasizer Nano). The gold power (99.96%) was hydrophobized in the same
manner as the gold plate used for the film thinning kinetics measurements. It was assumed that
the ξ-potentials of the gold plate and powder were the same.
Figure 1. Schematics of the TFBC apparatus used for the study of wetting films
2.4 Results
Figure 2 compares the timed profiles of the wetting films formed on gold substrates of two
different hydrophobicities. Figure 2a shows the profiles of the film formed on a freshly-cleaned
18 |
Virginia Tech | gold plate that had not been hydrophobized. Its equilibrium contact angle (θ ) was 42o with 60o
e
advancing (θ ) and 17o receding (θ) angles. Initially, the film showed dimpled profiles. As the
a r
drainage continued, the film became flat and reached an equilibrium thickness (h ) of 80 nm in 20
e
s, when the capillary pressure was equal to the disjoining pressure. Figure 2b shows the timed
profiles of the wetting film formed on a gold plate hydrophobized in a 5x10-6 M PAX solution for
60 min to obtain θ = 79°. The drainage rate became substantially faster than observed with the
r
untreated gold plate; the film thickness decreased from 300 to 80 nm in 1.26 s and ruptured. The
thickness at which the film fails catastrophically is referred to as critical rupture thickness (h ).
cr
Note that the dimpling effect was more significant with the hydrophobic gold, indicating that the
wetting film formed on the hydrophobic surface thins faster at the barrier rim than at the center.
Note also that the curvature of the air/water interface on either side of the barrier rim was about
the same, which would allow one to ignore the hydrodynamic pressure term of Eq. [2] during the
last stages of the film thing process.
Fig. 3 shows the changes in thickness of a wetting film formed on a hydrophilic fused quartz
plate at its center and at the barrier rim. The measurements were conducted in a 0.1 M NaCl
solution so that the Π term of Eq. [4] can be ignored. Since the substrate was hydrophilic,
(cid:3032)
Π was also ignored. Eq. [3] can then be reduced to,
(cid:3035)
2(cid:2011) 2(cid:2011) (cid:1827)
(cid:2869)(cid:2871)(cid:2870)
Δ(cid:1842) (cid:3404) (cid:3398)Π (cid:3404) (cid:3398) (cid:4670)8(cid:4671)
(cid:1844) (cid:3031) (cid:1844) 6(cid:2024)(cid:1860)(cid:2871)
By using the values of γ = 72.4 mN/m at 0.1 M NaCl and A = -1.13x10-20 J,50 the kinetics curve
132
obtained at the barrier rim was fitted to Eq. [1]. The fit was excellent indicating that one can
safely use the Reynolds approximation to describe the kinetics of wetting films. The results
showed that the non-slip condition applies to both the surfactant-free quartz-water interface and
the free air-water interface. The kinetics curve obtained at the center of the dimple thins slower
than at the rim. The results presented in Figure 3 are similar to the work of Platikanov. 35
Figure 4 compares the kinetics of film thinning on gold substrates with and without
hydrophobization. After one hour of immersion time in a 5x10-6 M PAX at open circuit and
natural pH (= 7.3), the gold plate was rendered hydrophobic with θ = 79o. The untreated gold
r
plate showed only a slight hydrophobicity with θ = 17o possibly due to contaminants. When a
r
gold surface was freshly cleaned in a piranha solution, the contact angle was zero. But the angle
increased considerably after a short exposure to the atmosphere, during which time contaminants
could adsorb on the surface from the air owing to the large Hamaker constant of gold. Despite the
apparent but a low-level hydrophobicity, its kinetics was much slower than on the treated surface,
as shown in Figure 4, and the film did not rupture. On the contrary, the wetting film on formed on
the treated gold surface ruptured at h = 80 nm and thinned substantially faster, which may be
cr
attributed to the presence of hydrophobic force in the wetting film.
In general, xanthate-coated mineral surfaces are negatively charged in water, and the ζ-
potentials do not change substantially at low concentrations.51 Also, xanthate adsorption should
not change the Hamaker constant of the gold plate (A ) significantly. For the experiment
131
conducted with the hydrophobic gold, the substrate was hydrophobized prior to forming a wetting
film with pure water. Therefore, the chemistry of the air/water interface should be the same as
19 |
Virginia Tech | that of the experiment conducted with untreated gold, that is, the ζ-potential and the Hamaker
constant of the air bubble (A ) should essentially be the same as those of the untreated gold
232
surface. It is, therefore, suggested that the fast kinetics of the film thinning on the hydrophobic
surface was due to the hydrophobic force. We estimated the magnitude of the hydrophobic force
by fitting the kinetics curves to the Reynolds equation (Eq. [1]). ΔP was calculated using Eqs.
[3]-[7]. The values of the various parameters used for the fit are given in Figure 4. In calculating
the contribution from the hydrophobic force (Π ), it was necessary to use the values of K =
(cid:3035) 132
2.0x10-17 J for the hydrophobized gold substrate and K = 0 for the untreated substrate. Note
132
here that K was positive and that its magnitude was much larger than that of the Hamaker
132
constant (A = -2.02x10-20 J), which was negative.
132
Figure 5 shows the results of the kinetics studies conducted with gold-coated silica plates by
varying the immersion time in a 10-5 M PAX solution. The film thicknesses were measured at the
barrier rims of the timed film profiles. It was found that the film thinning kinetics was the fastest
after 10 minutes of immersion time and became slower at longer contact times. The results
obtained after the 10-min contact time was fitted to the Reynolds equation with K = 2.0x10-17 J.
132
After a 120 min immersion time, K decreased to 7.0x10-18 J and the drainage rate decreased,
132
which may be attributed to a multilayer formation. It is well known that xanthate adsorption on
sulfide minerals and precious metals results in the formation of a multilayer.52-54 Xanthate
adsorption results in the formation of chemisorbed xanthates in the first monolayer, followed by
the adsorption of metal xanthates and/or dixanthogen on the top at higher electrochemical
potentials.55, 56 Evidences for the multilayer formation in the gold-PAX system after long contact
times and higher concentrations have been presented in another communication.57 Note that θ
r
was 78o after 10 min contact time, which decreased to 75o after the 120 min contact time. Albeit
small, the decrease in contact angle may be a reflection of the fact that the species adsorbing in
the multilayer expose the head groups (-OCSSAu) toward the aqueous phase. Although the head
group should be less hydrophobic than the end group (CH ) of the chemisorbed xanthate in the
3
monolayer, it may be substantially less polar than those of the high HLB number surfactants,
providing an explanation for the relatively small decrease in θ observed in the present work. The
r
decrease in K with increasing contact time may thus be attributed to the decrease in the
132
hydrophobicity of the gold substrate. This finding is consistent with the results of the AFM force
measurements conducted between xanthate-coated gold sphere and gold plate.57 It is also possible
that the multilayer formation increases the roughness of the xanthate-coated gold plate, which
should also contribute to the decrease in the drainage rate and hence the K values estimated
132
using the Reynolds approximation.
After the 10 min contact time, which was considered short enough to prevent the multilayer
formation, a set of timed film profiles have been obtained at different PAX concentrations (10-6 to
10-4 M), and the changes in film thickness at the barrier rims have been monitored and plotted in
Figure 6. The receding contact angle (θ) increased from 65o at 10-6 M to 80o at 10-5 M PAX. As
r
expected, drainage rate increased with increasing PAX concentration and θ, suggesting that the
r
increased kinetics was due to the increased hydrophobic force in the wetting films. Figure 7
shows the K values obtained by fitting the kinetics data to Eq. [1] plotted vs. the PAX
132
concentration. As shown, K increased with increasing concentration and reached a maximum at
132
approximately 3x10-5 M. The data presented in Figures 6 and 7 provides strong evidence that
20 |
Virginia Tech | hydrophobic force exists in the wetting films formed on hydrophobic surfaces and serves as the
major driving force for the film drainage and rupture.
Figure 8 shows the effect of electrolyte (NaCl) on the kinetics of film thinning on hydrophobic
surfaces. The experiments were conducted with gold plates treated in 5x10-6 M PAX solutions for
60 minutes. Also shown for comparison is the result (dashed line) obtained with a wetting film
formed with pure water on the surface of an untreated gold plate. In the absence of NaCl, the
wetting film formed on hydrophobized gold plate thinned substantially faster than that formed on
untreated hydrophilic plate, which can be attributed to the presence of a strong hydrophobic force
in the former. The kinetics curve obtained with the hydrophobic gold in the absence of NaCl can
be fitted to the Reynolds equation with K = 2.0x10-17 J. In the presence of 10-5 M NaCl, K
132 132
decreased to 1.6x10-17 J, causing a decrease in the kinetics of film thinning. At 10-4 and 10-3 M
NaCl, K decreased further to 6.0x10-18 and 5.5x10-18 J, respectively, with a further decrease in
132
the kinetics. Table 1 shows the various parameters used to fit the data presented in Figure 8 to the
Reynolds equation.
That the hydrophobic force in wetting films decreased in the presence of NaCl was consistent
with the results from the AFM58, 59 and foam film60, 61 studies that the hydrophobic forces in the
thin films confined between solid surfaces and between air bubbles decrease in the presence of
electrolytes. It was suggested that electrolytes can break the hydrogen bonds between water
molecules and, hence, cause a decrease in cohesive energy (W ) and hydrophobic force.60 The
c
observation that the drainage rate of the wetting films formed on the hydrophobic surfaces
decreased in the presence of NaCl appeared to be contrary to the DLVO theory, according to
which the kinetics should actually increase due to double-layer compression. This apparent
discrepancy simply indicates that the decrease in the attractive hydrophobic force was greater
than the decrease in the repulsive electrostatic force due to double layer compression.
2.5 Discussion
We have shown that the wetting films formed on horizontal, planar surfaces begin to thin due
to the capillary forces created by the changes in curvature. Its drainage rate can be predicted by
the Reynolds lubrication approximation, with the capillary pressure serving as the driving force.
As the film continues to thin, the solid/water and air/water interfaces interact with each other and
create a disjoining pressure (Π), which also begins to affect the kinetics of film thinning.
Apart from the kinetics of film thinning, whether a film ruptures or not is determined by
thermodynamics. When a wetting film ruptures, a new interface, i.e., solid-gas interface, is
created at the expense of solid-liquid and solid-gas interfaces. Thus, the Gibbs free energy change
(ΔG) associated with the rupture can be given by the following relation:
Δ(cid:1833) (cid:3404) (cid:2011) (cid:3398)(cid:2011) (cid:3398)(cid:2011) (cid:3407) 0 (cid:4670)9(cid:4671)
(cid:2869)(cid:2870) (cid:2869)(cid:2871) (cid:2870)(cid:2871)
where (cid:2011) , (cid:2011) , and (cid:2011) represent the free energies at the interfaces between solid 1, air 2, and
(cid:2869)(cid:2870) (cid:2869)(cid:2871) (cid:2870)(cid:2871)
water 3 phases. Combining Eq. [9] with the Young’s equation
21 |
Virginia Tech | hydrophobizing agents. Control of surface forces, e.g., double-layer forces by control of pH and
coagulant addition, is sufficient.
Even if film rupture is thermodynamically favorable, the process can be kinetically hindered,
e.g., by increasing Π, creating surface roughness to slow down drainage rate, increasing film
elasticity, etc. We have considered that Π consists of dispersion (Π ), electrostatic (Π ), and
d e
hydrophobic (Π ) components, as shown in Eq. [4]. Π is repulsive as A is negative in wetting
h d 132
films, while Π is also negative in the gold-xanthate system studied here and in many other
e
systems. We found that the kinetics of film thinning increases with increasing xanthate
concentration and contact angle (θ), which suggests that hydrophobic force is present in the
r
wetting films formed on hydrophobic surfaces. Using the Reynolds lubrication approximation, we
calculated the values of K representing the magnitudes of Π (and of hydrophobic forces),
132 h
which have been found to increase with θ. We found also that by recognizing the presence of
r
hydrophobic force in wetting films, it was not necessary to invoke the capillary wave models.32, 33
The wetting films ruptured spontaneously at h . In the gold-PAX system, Π was the only
cr h
negative component of Π and hence could bring the film thickness to h in a short time frame.
cr
Likewise, Manica et al.45 found that when Π is strongly negative, there was no need to invoke
e
the capillary wave model to predict the thinning and rupture of the water films between mica and
mercury.
That hydrophobic force is present in wetting films may be more readily acceptable if one can
recognize that air bubbles in water are hydrophobic. In this case, the thinning and rupture of the
wetting films formed on hydrophobic surfaces may be viewed as one of asymmetric hydrophobic
interaction. van Oss et al.64 suggested that the air side of the air-water interface is the most
hydrophobic surface known and is about 30% more hydrophobic than octane and Teflon. The
basis of this argument is that the tension at the air/water interface (72 mN/m) is substantially
higher than those (~50 mN/m) at the hydrocarbon/water interfaces. The vibrational sum
frequency (VSF) spectra of the water molecules straddling at the hydrophobic surface/water
interfaces show sharp peaks at 3600-3700 cm-1, which represent the characteristic non-hydrogen-
bonded (free) OH stretch vibrations.65, 66 The high interfacial tensions at the hydrophobic
surface/water interfaces are due to the presence of the free OH groups at these interfaces.
Interestingly, the free OH peaks observed at the CCl /water and hexane/water interfaces are
4
observed at 3669±1 cm-1, while the same is observed at ~3700 cm-1. The red shift of the
characteristic peak shows an attractive interaction between the free OH groups and the organic
molecules at the interface. In fact, the binding energy for the CCl -H O dimer has been reported
4 2
to be -1.4 kcal/mol.66 These reports are consistent with the fact that the dispersion components of
W at the hydrophobic liquid/water interfaces are ~20 mJ/m2, while it should be zero at the
a
air/water interface.
We used the Reynolds lubrication approximation to estimate the K values for hydrophobic
132
disjoining pressure (Eq. [7]). There are several questions to be raised in this approach. First, the
approximation is useful for a film of fluid between nearly plane-parallel surfaces. Although we
monitored the rate of film thinning at the barrier rim, which is a curved surface, the film thickness
(h) was much smaller than the radius of curvature. Therefore, we have measured the thinning
rates effectively between plane-parallel surfaces. Second, the roughness of the surface should
23 |
Virginia Tech | affect drainage rate and, hence, the K values obtained using the Reynolds approximation. In the
132
present work, we found that drainage rate decreased with increasing contact time in PAX
solutions due to decreased hydrophobicity and the surface roughness created by multilayer
formation.
Perhaps the most important question to be raised would be the question regarding the non-slip
boundary condition employed in deriving the Reynolds approximation. There seems to be no
doubt that it applies for the flows around hydrophilic surfaces.42 For the flows around
hydrophobic surfaces, however, some found that the non-slip condition may not be applicable. On
the other hand, the correlation between slip lengths and contact angle was very poor.67 As for the
flow around air bubbles (or air/water interface), it was found that the liquid/gas interface becomes
nearly immobile even with only a trace of surfactant present.68 Although we have conducted
experiments in the absence of surfactants, there is a possibility that trace of gold xanthates be
present at the air/water interface, making it possible to use the non-slip condition. If not, the K
132
values determined in the present work may have been overestimated and require appropriate
corrections in future work. Nevertheless, it is highly unlikely that the correction would relinquish
the possibility that hydrophobic force is present in the wetting films formed on hydrophobic
surfaces. It should be noted also that Horn and his co-workers found that the non-slip boundary
condition applies for the surfactant-free mica-water-mercury systems.45 Further, the drainage
experiments conducted in the present work (see Figure 3) and by Platikanov35 with wetting films
produced with surfactant-free air/water interface could be fitted to the Reynolds approximation
with non-slip boundary conditions.
The geometric mean combining rule is used to predict the Hamaker constants for van der
Waals interactions between unlike surfaces from those between like ones. It is based on the
Berthelot relation derived originally for molecular interactions.69 It has been reported previously
that asymmetric hydrophobic force constants (K ) can be predicted using the geometric mean
132
combining rule as follows:48
(cid:1837) (cid:3404) (cid:3493)(cid:1837) (cid:1837) (cid:4670)16(cid:4671)
(cid:2869)(cid:2871)(cid:2870) (cid:2869)(cid:2871)(cid:2869) (cid:2870)(cid:2871)(cid:2870)
in which K is the force constant for symmetric hydrophobic interaction between two surfaces of
132
identical contact angle of θ , and K is the same for contact angle θ .
1 232 2
In Figure 9, the values of K obtained using the Reynolds approximation have been plotted
132
vs. the values of K in logarithmic scales. According to Eq. [16], the slope should be ½. From
131
the intercept obtained numerically, we found that K = 5.3x10-17 J. This value is close to that
232
estimated by extrapolating the K values for the foam films stabilized at different concentrations
232
of ionic and non-ionic surfactants.60 The K values used in the plot shown in Figure 9 include
131
those by Wang and Yoon57 and those measured specifically for the present work using the AFM
force measurements conducted with PAX-coated gold surfaces in pure water. It is interesting that
Eq. [16] applies for the hydrophobic interactions between solid/liquid, gas/gas, and gas-solid
interactions. This finding suggests that hydrophobic force may be a molecular force representing
the properties of the thin liquid films confined between hydrophobic surfaces, regardless of
whether the interacting surfaces are solid, liquid, or gas. This finding is consistent with the results
24 |
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