Understanding the Effects of Polymer Extrusion Filter

Layering Configurations Using Simulation-based

Optimization

K.R. Fowler,1∗ E.W. Jenkins,2 S.M. LaLonde1

1 Department of Mathematics, Clarkson University, Potsdam, NY2 Department of Mathematical Sciences, Clemson University, Clemson, SC

Abstract

A challenge in polymer processing is the removal of debris, via filtration, fromthe polymer melt during the extrusion process. We propose measures of filter per-formance and an optimization strategy to indentify parameters to maximize thelifetime of the filter while maintianing product quality. We analyze the benefitsof using extrusion filters with more than one layer and evaluate the feasbility ofcombining competing objectives into a single functional. Our optimization uses athree-dimensional simulation tool for the filtration process as a black-box; thus, gra-dient information is unavailable and a derivative-free method, the implicit filteringalgorithm, is used. We present numerical results that verify that multi-layered filtersare more effective than single-layer filters, and that optimal designs have smallerpore diameters in the bottom layer than in the top layer. In addition, the resultsindicate that our single functional approach is reasonable.Keywords: filtration, derivative-free optimization, implicit filtering, barrier ap-proaches

1 Introduction

Researchers at the Center for Advanced Engineering Fibers and Films (CAEFF) assistindustrial partners in the design and production of polymer products. Initial fund-ing for CAEFF has been provided through the Engineering Research Centers Program

∗Corresponding author. email: kfowler@clarkson.edu

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of the National Science Foundation, and CAEFF is transitioning to a self-supportedfiber and film research center. More information can be found at the CAEFF websitehttp://www.clemson.edu/caeff.

Part of the research effort at CAEFF has been to develop a computational simulator forthe various stages of a fiber-spinning process [[2]]. A schematic of a fiber-spinning processis given in Figure 1. The metering pump pushes molten polymer through an extrusionfilter before the polymer enters the spinneret. The spinneret has a large number of verysmall holes and produces a set of thin fibers that are then converged into the final fiberproduct. Unfiltered debris particles can obstruct holes in the spinneret, resulting in aninconsistent product as a reduced number of thin fibers are converged. Filtration is alsoused to prevent unmolten gel particles from entering the finished fiber. Inclusion of theseparticles would also produce a fiber that did not meet quality specifications.

. . .

Godets

Convergence Guide

Air Quench

FilterMetering Pump

Spinneret

Spin Bobbin

Polymer Melt

Figure 1: Fiber melt-spinning process

The diameter of the pores in the filter is small enough to capture particles a fewmicrons in diameter. As these debris particles accumulate inside the filter, the pressuredrop across the filter (i.e, the difference in pressure values at the entrance and exit ofthe filter) must increase to ensure a constant mass flow rate is maintained throughoutthe system. The pumping mechanism can be damaged if the pressure drop is allowed toexceed a threshold value. Effective system management requires replacement of the filterbefore this threshold is reached.

The CAEFF simulation tool contains a three-dimensional numerical model of an extru-sion filter [[13]]. This numerical model was developed to help process engineers understandthe mechanisms behind debris removal, and previous work indicates that the simulated

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values for pressure drop are in agreement with empirical data [[13]]. We are using thefilter simulator in an optimization context to find design parameters that will improvefilter performance.

Our filter performance measures are based on two objectives: effective removal ofdebris particles and prolonged filter lifetime. Filter replacement is costly, often requiringthat the entire spinline be taken out of production. In-line filters are being introduced,but analysts believe that the trend will be to combine these filtration methods withconventional filters located at the spinneret [[8]]. We are thus faced with competingobjectives, as extending the filter lifetime could be achieved by simply reducing the amountof trapped debris.

Previous work attempted to optimize the performance of the filter using a multi-objective genetic algorithm [[6]], which requires the use of two competing objective func-tions. In this work, we combine the competing objectives into a single functional usingbarrier methods. This allows us to minimize the functional using single-search methodsas opposed to multi-objective algorithms. The barrier methods incorporate an additiveterm into the objective function that we use to place a limit on the maximum amount ofdebris allowed to escape the filter. This limit can be easily adjusted for any productionrequirement.

Our objective function is defined in terms of output from the simulator. The modelfor the filtration process includes a system of coupled partial differential equations forthe polymer flow through the filter along with empirical tools that describe the amountof debris in the polymer and the probability that a debris particle is retained by thefilter. Thus, we lack a continuous function that describes the process and would providegradient information for calculus-based minimization techniques. We have chosen to usean implementation of the implicit filtering algorithm [[9, 10]] which uses objective functionvalues generated by the simulator to guide the optimization.

The data obtained from this design space sampling will guide us towards optimaldesign parameters and provide insight into the effects of such parameters on the behaviorof the filter. An advantage of this approach is that it provides a framework for a varietyof simulation methods, including the three dimensional algorithm we apply here.

The paper is structured as follows. An overview of the filtration model and the simu-lation tool is given in Section 2. In Section 3, we outline the theoretical approaches takenand the derivations of three objective functions which measure the filter performance.We describe the implicit filtering algorithm used for optimization and provide numericalresults in Section 4. Conclusions and future work are given in Section 5.

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2 Filtration model

The model for the filtration code (i.e., the “simulator”) was originally developed in [4]in a one-dimensional setting. The work was extended to three spatial dimensions in [13],which lead directly to the computational tool we use in this paper.

The material entering the filter consists of molten polymer, unmelted polymer gel par-ticles (formed when the polymer does not completely melt in the heating/mixing stage),and other debris such as metal particles. We assume that the density of the debris isnegligible in comparison to the polymer density, and the that the filter thickness is smallenough to allow gravitational forces to be ignored [[4, 13]].

The flow through the filter is governed by the continuity equation (mass conservation)and Darcy’s law, modified to account for the non-Newtonian behavior of the polymermelt. The debris entering the filter is specified in terms of a mass fraction relative to themass of the mixture. The distribution of debris particles is characterized statistically, asin [8], using either truncated or standard normal probability density functions [[6]]. Weuse three material types in the simulator, each characterized by a different probabilitydistribution. The means and standard deviations of the distributions are input variables,and a user may choose to either a truncated or full normal distributions to describe thedebris.

Debris is tracked as it travels through each computational cell. Fluxes calculatedalong each cell edge determine the amount of debris that enters the cell. Particles witha larger diameter than the pore diameter associated with the cell are eligible for capture.The percentage of these particles retained by the filter is determined by an empiricalretention function associated with the filter [[13]]. The retention function is dependenton the initial pore diameter, and the parameters for the retention function change asthe optimizer selects new points to evaluate. The choice of parameters was based on anextension of empirical data for a variety of filters.

All particles not captured within a computational cell are transported to downstreamcells. The porosity of the filter and the average pore diameter of the filter are updated atthe end of every time step using the information on the mass of the deposited debris.

The permeability of a porous medium is related to the ability of the medium to trans-mit fluids. The filter becomes less efficient at fluid transport as debris is deposited; thus,any function that models permeability must decrease as porosity decreases. The perme-ability parameter k is currently modeled using a Blake-Kozeny relationship that dependson the average pore size, dp, and the current filter porosity, η. This relationship for k,

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derived in [1] and used in [4, 13], is

k =d2

pη3

150(1 − η)2.

The permeability is calculated at the end of every time step, using the updated values fordp and η. The Darcy relationship states that the fluid’s (averaged) velocity is proportionalto the pressure gradient, or pressure drop, across the filter. The proportionality constantdepends, in part, on the permeability coefficient. As k decreases, the pressure drop mustincrease to maintain a constant mass flow rate throughout the filter.

The large values of pressure required at the inlet may lead to damage of the meteringpump, so the filter must be replaced once the pressure drop reaches a threshold value.The threshold value is set at 35e6 Pa in the simulator, which is comparable to existingindustry thresholds [[13]].

Filters can be made from sand, sintered metal, compressed with sufficient force toproduce a cake material, or layers of wired mesh, with mesh spacings small enough totrap particles a few microns in diameter. A filter pack often contains multiple layers,each of which is characterized by a porosity, an average pore diameter, and a retentionfunction. The pore diameter of each layer usually increases from the top down, withthe pore diameter of the bottom layer often matching that of the top layer [[12]]. Thesimulator can handle up to five distinct filter layers, but we consider only one- and two-layer filters in this work.

More detailed information on the model can be found in [3, 6], and [13].

3 Measures of filter performance

We formulate the optimization problem as

minimize F (x), x ∈ Ω, (1)

where F (x) measures the performance of the filter, x is a vector of decision variables, andΩ defines acceptable bound constraints on those variables. We consider three problemformulations, each using a different measure of filter performance. We choose the porosityη and average pore diameter dp in each layer as decision variables. Thus, for a one-layermodel, x = (η1, dp1), and, for a two-layer model, x = (η1, dp1, η2, dp2).

The primary goal in all of the formulations of the objective function is to maximizethe lifetime of the filter. The secondary, competing goal is to minimize the amount ofdebris that escapes the filter. We handle the secondary objective using a barrier method.

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This allows a production manager to place an acceptable bound on the amount of debrisleaving the filter. The barrier, or penalty, function is defined as follows. Let b representthe acceptable bound on escaped debris, and let ξ(x) represent the total mass of debristhat escapes over the lifetime of the filter. We construct a barrier function that keeps ξ(x)bounded away from b. The function should also attain its minimum at ξ(x) = 0. Theserequirements lead to the function

B(ξ) =ξ(x)

b − ξ(x), (2)

which is defined on the half-open interval [0, b). We can see that B(ξ) → ∞ as ξ → b−.Likewise, B(ξ) → 0 as ξ → 0+. Thus B(ξ) is minimized when ξ = 0 and grows withoutbound as ξ approaches b. We considered the percentage of escaped debris as a measureof filter peformance as well in [6]. Minimizing the total mass of escaped debris, however,seems to be a more restrictive, and reasonable, measure.

The objective functions described below measure filter lifetimes in different manners.The penalty function B (ξ) is used in all three formulations.

3.1 F1(x): Filter lifetime

The first objective function we consider uses a simple measure for filter lifetime. Let t(x)represent the lifetime of the filter in hours. As we wish to maximize t(x), we will minimizeits negative. Thus we define

F1(x) = −t(x) + ρ1B(ξ), (3)

where ρ1 is a constant with units of hours. Note that F1(x) is not defined for values ofx : ξ(x) ≥ b. We handle these cases computationally by setting F1(x) to an unreason-ably large value.

3.2 F2(x) : Pressure drop across filter

The second formulation F2(x) is motivated by the pressure drop curves. We wish toprolong the time at which the filter must be replaced by minimizing the maximum rate ofchange of the pressure drop over a single time step. This should cause the pressure dropcurve to be less steep and thus indirectly extend the lifetime of the filter. We retain thesecondary objective of minimizing the total mass of escaped debris.

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Let pn denote the difference in pressure (in pascals) across the filter at the nth timesteptn. The change in pressure between two successive timesteps can be written as

∆pn =pn − pn−1

tn − tn−1

, (4)

which has units of pascals per hour. If P = ∆p1, ∆p2, . . . , ∆pk, where k is the totalnumber of time steps, then we can define αmin = max P . There is a unique αmin associatedwith each set of decision variables x, so we can write αmin = αmin(x).

Incorporating the barrier function B (ξ) gives the second objective function

F2(x) = αmin(x) + ρ2B(ξ), (5)

where ρ2 is a constant with units of pascals per hour. As with F1(x), F2(x) is definedonly on the set x : ξ(x) < b. The computational treatment of F2(x) is the same as thatof F1(x).

3.3 F3(x) : Overall pressure change

Another approach to “flattening” the pressure drop curve is to use a linear estimate of thetotal change in pressure over the lifetime of the filter. Suppose the lifetime of the filterconsists of N time steps. The total change in pressure can be modeled by the line passingthrough the points (t0, p0) and (tN , pN). A reasonable objective would be to minimize theslope of this line, which can be expressed as

m =pN − p0

tN − t0, (6)

where m has units of pascals per hour. Since t0 = 0, the filter lifetime t is measured bytN , and t = t(x). We also note that pN = pN(x), and we can define the third objectivefunction

F3(x) =pN(x) − p0

t(x)+ ρ3B(ξ). (7)

The constant ρ3 has units pascals per hour. Thus F3(x) is defined for values x : ξ(x) < band for values t(x) 6= 0. The first case is handled as it was previously; the second can beneglected, since t(x) = 0 is impractical.

4 Numerical results

We solve the optimization problem given by (1) using a MATLAB implementation of theimplicit filtering algorithm developed by C.T. Kelley at North Carolina State University

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[[9]]. Implicit filtering is entirely derivative-free and is based on a quasi-Newton approach,which uses the objective function to build approximate gradients of F (x) and a secant-like method for the Newton-step [[7, 11]]. The advantage of implicit filtering is that thesize of the finite-difference interval on which the gradient is approximated is decreasedas the optimization progresses. This makes implicit filtering particularly effective forminimizing “noisy” functions; that is, functions that have a large number of local minima,which are not necessarily known in advance. By using large difference increments earlyin the optimization, the algorithm may avoid local extrema and close in on the globalminimum. The algorithm requires a means to compute the objective function, a feasibleinitial iterate, and termination criteria either in the form of a function evaluation budgetor a fixed number of times that the finite-difference stencil is reduced.

We consider the optimal values of (ηi, dpi) for both a one-layer model (i = 1) and atwo-layer model (i = 1, 2). Extensive numerical experiments were performed on all of theformulations of the objective function, using a variety of values for ρ. For the results givenhere, we use ρ = 10 for F1(x) and ρ = 104 for F2(x) and F3(x). The allowable debris, b,was 8 × 10−5 kg. We present and discuss selected results from these tests.

4.1 One-layer Model

We consider a one-layer, 1 cm thick filter and seek the values of (η, dp) using (8) as thebound constraints, where

Ω1 = (ηi, dpi) : 0.4 ≤ ηi ≤ 0.65, 23 µ ≤ dpi ≤ 40 µ (8)

. We used an initial iterate of x0 = (0.5, 25 µ) and a budget of 60 evaluations of F (x).Table 1 contains results from the problem formulations. The data in the second and thirdcolumns are the optimal values of (η, dp) for each objective function. The data in theremaining columns provide the corresponding lifetime of the filter (t), αmin as describedin Section 3.2, the mass of the escaped debris (ξ), and the slope of the secant line for theoverall pressure drop (m). A version of these results appears in [5], where the retentionfunction was kept constant throughout the optimization. The results in Table 1 differslightly, but are comparable to the earlier data.

All formulations result in identical values for porosity which lie on the upper boundconstraint, but the values of pore diameter differ. In particular, F2(x) has the smallestpore diameter and offers the lowest values for both lifetime and escaped debris, F3(x)offers the highest values for both, with F1(x) being comparable to F3(x). It is apparentthat F2(x) is the most conservative of the three in terms of controlling the escaped debrisdue to the low value of dp. The expense is a significantly shorter lifetime than that ofF1(x) or F3(x).

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Table 1: A comparison of filter performance for F1(x), F2(x), F3(x).Fi η dp t αmin ξ m

(µ) (hours) (104 Pa/hr) (10−5 kg) (105 Pa/hr)F1 0.65 29.9205 78.1 7.7934 3.9468 2.0200F2 0.65 23.0000 65.2 7.2276 2.1475 2.3083F3 0.65 30.0850 81.4 7.9401 4.7075 2.0933

The requirement that pressure drops be minimal across a single time step places theemphasis on the escaped debris rather than the filter lifetime. The simulator changesthe pore diameter associated with the filter as debris accumulates in the interior. Thus,it appears that in order to keep the pressure drop across a time step minimal, the filtermust trap almost all of the debris early in the simulation. More investigation is needed,however, to better understand these results. Clearly, the choice of the best functionalto use in this case would need to be made by a production manager, who would havea predefined goals for the filtration process. Future work will evaluate the feasibility ofcombining all of the objectives into a single, weighted objective function.

In Figure 2 we show F1(x) as it varies with the pore diameter and porosity value pairssampled by the optimizer and we examine the corresponding lifetimes of the filter for thevarious values of F1(x). The figure on the left shows choices of (η, dp) sampled by theoptimizer. The porosity is clustered about η = 0.65 while the pore diameter varies morein an attempt to find the minimum. As a majority of the design points are clustered nearthe best point found, the optimizer likely converged early within the function evaluationbudget.

The figure on the right clearly shows that F1(x) is strongly dependent on the lifetimeof the filter. The relationship between the two is nearly linear until the barrier termbegins to dominate near the end of the simulation. When the filter lasts longer than 80hours, we see the effects of the penalty placed on the function by the amount of debristhat escapes from the filter.

The same data is displayed for F2 in Figure 3. In this case, the optimizer sampled awider range of values for the porosity and the pore diameter prior to convergence, but westill see a cluster of points close to the minimum. While the plot on the right shows thatF2 tends to decrease overall as the filter lifetime increases, the nonlinear behavior of thedata indicates that this objective is less connected to the filter lifetime and may not bethe ideal performance measure if maximizing the lifetime is the primary objective.

Finally, we consider the results for F3. The results in Table 1 imply that the behaviorof F3 will be similar to that of F1. The left graph in Figure 4, however, shows that the

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1.3x 10

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Obj

ectiv

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F2(

x) (

Pa/

hr)

(b)

Figure 3: Scatter plots of the value of F2(x) versus(η, dp)(a) and lifetime (b)

optimizer sampled a wider range of design points in both η and dp. The second graph inFigure 4 shows that the slope defined in (6) is minimized when the lifetime is large, whichis what we wanted.

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F3(

x) (

Pa/

hr)

(b)

Figure 4: Scatter plots of the value of F3(x) versus(η, dp)(a) and lifetime (b)

4.2 Two-layer Model

We now consider a 1 cm filter consisting of two distinct layers that are 0.5 cm each. Herewe seek x = (η1, dp1, η2, dp2) using the same three objectives described above. Since theimplicit filtering algorithm depends on bound constraints for the decision varaiables andon an initial iterate, we consider different optimization parameters to better understandthe complexity of the design space.

From an engineering perspective, a multi-layer filter should capture larger particlesin the top layer and smaller particles in the bottom layer [[12]]. For this reason, weuse x0 = (0.5, 40µ, 0.5, 40µ, ) as an initial iterate. We also consider Ω2 for our boundconstraints on x that differs in the pore diameter in each layer and has a larger upperbound for the porosity. We use

Ω2 = (ηi, dpi), i = 1, 2 : 0.4 ≤ ηi ≤ 0.7, 38 µ ≤ dp1 ≤ 40 µ, 23 µ ≤ dp2 ≤ 40 µ. (9)

We allow a function evaluation budget of 65. Table 2 contains the results for these opti-mization parameters and the two-layer configuration for all three performance measures.

The two layer configurations given by F1 and F3 result in a lifetime that is as muchas 23% longer than a one-layer model. Again, F2 is the most conservative formulationand results in a design with a lower filter lifetime that is comparable to the one-layerdesign, and with a lower amount of escaped debris. Note that the difference between theF2 configuration and the other two is primarily the smaller porosity in the first layer.These results demonstrate the balance required between pore diameter and porosity toextend filter lifetimes. The F1 and F3 configurations have larger porosities in the first

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Table 2: Two-layer results using an engineering perspective for optimization parametersFi dp1 η1 dp2 η2 t m αmin ξ

(µ) (µ) (hours) (105 Pa/hr) (104 Pa/hr) (10−5 kg)F1 38 0.65 23 0.6433 96.4 1.7905 9.0344 5.1664F2 38 0.51 23 0.64 57.3 2.621 8.3123 4.0229F3 32.57 0.682 23 0.605 94.6 1.8419 8.7536 4.2888

layer than the F2 configuration, which leads to a nearly 70% increase in lifetime. The F3

configuration also has a smaller pore diameter in the first layer than F1, which leads to acomparable lifetime but a smaller amount of escaped debris.

In both layers, the pore diameter tends toward the lower bound constraint for F1 andF2 while for all three objectives, the porosity is towards the upper bound, but not lying onthat constraint. The small pore diameter ensures the entrapment of debris while a largeporosity makes sense, as the primary concern in formulating the objective functions wasto extend the lifetime of the filter. Filter lifetimes are maximized if the filter is completelyopen and hence traps nothing.

Figure 5 shows the iteration histories for these optimization runs. For F1 and F2, theflat line of function values for a large number of function calls implies that the optimizer istemporarily stuck in a local minimum before finding a new descent direction. The valuesof F3 decrease more steadily implying that this formulation may be a more well-posedproblem. Although this is a two layer model, this coincides with implications from theleft scatter plot in Figure 4 that the optimizer is aggressively searching the design space.

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Figure 5: Iteration histories for data in Table 2

Since implicit filtering is considered a local optimizer and can depend significantlyon a good initial guess, we also start the optimization with parameters that describe a

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Table 3: Two-layer results using a one-layer initial configuration and expanded designspaceObjective dp1 η1 dp2 η2 t m αmin ξ

Function (µ) (µ) (hours) (105 Pa/hr) (104 Pa/hr) (10−5 kg)F1 28.71 0.70 23 0.413 92 1.8840 8.4924 3.4298F2 23 0.625 23 0.4478 58.9 2.5511 7.3992 2.0578F3 38.53 0.65 23 0.646 97.5 1.7748 9.0895 5.2819

one-layer model. For this investigation, we use x0 = (0.6, 35 µ, 0.6, 35 µ). Morever, sinceour solutions above tend to lie on the bound constraints for dp, we alter Ω2 slightly byincreasing the upper bound on the constraint for the pore diameter:

Ω2 = (ηi, dpi), i = 1, 2 : 0.4 ≤ ηi ≤ 0.7, 23 µ ≤ dpi ≤ 55 µ. (10)

The results are given in Table 3.Both the F1 and F3 objectives return filter configurations that have smaller pore

diameters in the second layer than in the first. In all cases, the pore diameter for thesecond layer lies on the lower bound constraint, indicating that the purpose of this layeris to ensure that the filter captures the maximum amount of debris. The configurationfor F3 leads to a slightly longer lifetime than that of F1. The larger pore diameter in thefirst layer for F3, coupled with the larger porosity in the second layer, delays the time ittakes to surpass the pressure drop threshold. The results for F2 seem to indicate that asingle-layer design is suboptimal in terms of extended lifetimes. The pressure drop curvesfor all three objective functions are given in Figure 6. The slope for the pressure dropcurve associated with the parameters from the F3 objective is lower than the slopes forthe other two curves. We appear to have achieved our goal with this objective.

Plots of the debris deposition, in kilograms, are given in Figure 7. The plot on theleft is the debris deposition associated with the optimal parameters for F2(x) from Table3, and the plot on the right is the debris deposition for the F3(x) parameters from thesame table. The deposition is primarily contained in the top layer using the optimalparameters for F2(x), while the deposition is more evenly distributed over the entire filterfor the optimal F3(x) parameters. Thus, F3(x) clearly provides a parameter set thatmakes more efficient use of the entire filter.

We also considered using Ω2 with an unreasonable initial iterate, x0 = (35 µ, 0.6, 42 µ, 0.4),that would tend to trap smaller particles in the top layer. This was done to see if theimplicit filtering algorithm would recover a two-layer model as in Tables 2 and 3 for F1

or F3. The results are given in Table 4.

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0 10 20 30 40 50 60 70 80 90 1001.8

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ress

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Figure 6: Pressure drop curves for data in Table 3

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Figure 7: Debris deposition plots for F2(x) (a) and F3(x)(b) from Table 3

In this case, F3 is the only objective that returned a two-layer design comparable tothe first set of results. Both F1 and F2 returned suboptimal designs, although the filterlifetime associated with the F1 configuration is surprisingly competetive.

The result for the temporal objective function given the bad initial guess needs furtherexploration. The parameter set gives a competitive lifetime for what is essentially a 0.5cm thick filter. A graph of the debris deposition shows that no debris is deposited inthe second layer, which led us to examine a one-layer, 1 cm thick filter with the samepore diameter and porosity as the first layer. The lifetimes for both the one-layer andtwo-layer models are the same, as is the amount of debris that escapes the filter. Theseresults lead us to believe that the depth of the filter is not very important, relative tothis objective, and that the filter lifetime is not our best measure for filter performance.

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Table 4: Two-layer results using a bad initial guessFi dp1 η1 dp2 η2 t m αmin ξ

(µ) (µ) (hours) (105 Pa/hr) (104 Pa/hr) (10−5 kg)F1 27.715 0.70 42.736 0.669 90.1 1.9178 8.4175 3.537F2 23 0.652 34.987 0.40 65.7 2.29113 7.2358 2.1536F3 34.35 0.673 23 0.40 95.4 1.8264 8.8603 4.617

We used two additional optimization runs with different initial guesses to further evaluatethis point. We obtained the same point given in Table 4 using an initial guess of x0 =[0.6, 35 µ, 0.4, 40 µ]. Using x0 = [0.5, 35 µ, 0.5, 40 µ], we obtain again a similarresult, with a lifetime of 93 hours. This analysis indicates that the optimization is verysensitive to the initial guess and bound constraints on porosity. This is not surprising,as our primary focus in this study was to maximize the lifetime of the filter. This willnecessarily place the emphasis on porosity rather than pore diameter.

5 Conclusions

Our goal is to better understand the measures of filter performance by investigating newobjective functions and incorporating more decision variables. In addition, we would liketo associate a cost with filter performance, which will require the use of a single objectivefunctional. There should be a cost associated with filter replacement, and there should bea cost associated with loss of finished product due to a debris inclusion. This work showsthat we have devised three reasonable methods for evaluating filter lifetime and that ouruse of a barrier approach to combine our competing objectives into a single functionalis reasonable. The derivative-free sampling method we chose for optimization gives usintermediate function values that can be used to analyze the behavior of the objectivefunction. It also allows us to choose from a range of simulation methods which can beused in a black-box fashion.

Previous work with the simulator indicates there is a significant benefit to using multi-layered filters, but we know of no organized study that has fully explored the designspace associated with these more complicated structures. Our results show significantlyextended lifetimes for our optimal two-layer filters when compared with our optimal one-layer filters. In addition, we show that even when starting with a single-layer filter design,the optimizer moves the parameter set to a two-layer model for two of the three cases.However, the analysis also indicates that the optimizer can return questionable resultswhen the initial guess is more random.

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Future work will include the use of population-based, heuristic, global optimizationmethods that do not rely on a single initial iterate. However, a drawback of these methodsis that they require a large number of function evaluations. For this work one simlua-tion can take up to three hours. More investigation is needed to choose an appropriateoptimizer to accomodate this computational burden. Parallel methods are particularlyattractive.

Our plans are to extend this work to consider more layers, the length of each layer,and production costs. We also need to evaluate the method by which we chose param-eters for the retention functions. These parameters are dependent on the pore diameterassociated with the filter layer. We have empirical data for a select number of filters,and we extrapolated this data in order to obtain parameters for any point chosen by theoptimizer. These functions have a significant impact on the ability of a filter to trapdebris, and thus any improvement in finding these parameters will enhance the simulator.We believe that our efforts in understanding filter design through optimization will giveindustrial partners an effective tool with which to reduce the operating costs associatedwith one of the components of fiber production.

6 Acknowledgements

This work was partially supported by the ERC program of the National Science Founda-tion under Award Number EEC-9731680. We thank Dr. Chris Cox of Clemson Universityfor many fruitful discussions on filtration. We also thank IBM for the use of an IBM work-station with a Intel Dual Core Xeon 5260 processor, which significantly reduced the timerequired to obtain our results.

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