optimisation and graphical representation of multi-stage(1)

Journal of Membrane Science 211 (2003) 59–70

Optimisation and graphical representation of multi-stagemembrane plants

Ken R. Morison∗, Xin SheDepartment of Chemical and Process Engineering, University of Canterbury, Private Bag 4800, Christchurch, New Zealand

Received 4 May 2002; accepted 7 August 2002

Abstract

A procedure is given for the design and optimisation of continuous multi-stage membrane plants using the example of wheyultrafiltration. During the design stage, the number of stages, area per stage, diafiltration water ratios and possible pressurescan be varied to minimise capital and operating costs. The concept of an optimal or ideal plant was developed to providea basis for comparison for the designs. To interpret the results, and check the validity of the optimisation results, a varietyof graphical representations were developed. Graphical profiles of purity versus total solids, and of protein concentrationversus lactose concentration were effective when interpreting design results and a graph of component mass flux helpedidentify yield losses. These and other graphs provided insight into the designs and often enable sensible improvements tothem.© 2002 Elsevier Science B.V. All rights reserved.

Keywords:Multi-stage; Graphical; Design; Membrane; Optimisation

1. Introduction

The design of batch and multi-stage membraneplants is well established in the commercial worldand equations for the design of plants have been welldocumented[1–3]. Some efforts have been madeto analyse and optimise multi-stage plants. Niemiand Palosaari[4] simulated multi-stage reverse os-mosis and ultrafiltration plants. Qi and Henson[5]gave a method for optimisation of gas separationin multi-stage spiral-wound plants by mixed integernon-linear programming (MINLP). A similar tech-nique[6] was used for the design of reverse osmosisnetworks.

∗ Corresponding author. Tel.:+64-3-364-2578;fax: +64-3-364-2063.E-mail address:[email protected] (K.R. Morison).

Although much of a design can be theoretical, itis very dependent on the equation for permeate fluxwhich is normally obtained empirically for the particu-lar solution, membrane and operating conditions used.As a result optimality cannot be theoretically provenfor a general case and results from one design are notnecessarily true for another.

The development of spreadsheets, such as Mi-crosoft Excel, with built in optimisation programs,such as Solver, bring to the engineer’s desktop thecapability for fast and accurate calculation of plantdesigns. The result of such calculations, however,can be a mass of numbers and a design, of whichthe validity and optimality is uncertain. The un-certainty is likely to increase with the number ofvariables, e.g. when there are many stages and whendiafiltration is used to achieve high purity in theproduct.

0376-7388/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S0376-7388(02)00375-7

60 K.R. Morison, X. She / Journal of Membrane Science 211 (2003) 59–70

Nomenclature

A membrane area (m2)C concentration as mass fractionJ flux (kg m−2 h−1)k constantm mass (kg)

Greek symbolsρ density (kg m−3)σ retention coefficientφ diafiltration ratio

Subscriptsdes designD diafiltrationF feedj componentjP permeateR retentatespec specified

Two additional methods of analysis are proposed forthe design of membrane plants. One is an assessmentof an ideal design. The ideal design can be one that isproduced to meet a specified objective with only theconstraints of the separation and possibly without anyeconomic optimisation. In membrane design, one idealdesign could be the plant with the minimum possiblemembrane area for the specified separation. There areanalogies in chemical engineering, such as the idealtubular reactor which gives the lowest volume for areactor, and a distillation column with perfect tray ef-ficiency and total reflux that gives the maximum pos-sible separation of volatile mixtures[7].

The other feature proposed is the graphical repre-sentation of the results of the design to enable visualassessment of the design and comparison betweendesigns. There are numerous examples in chemicalengineering, such as the McCabe–Thiele distilla-tion diagram[8] and the graphical analysis of heatexchanger networks[9].

In this paper, some of the equations for design arereviewed and methods of calculation are formally pre-sented. Some concepts of optimal design are devel-oped and a number of methods for the representationof these designs are given.

2. Design procedures

2.1. Design objective

The objective of the design depends on the separa-tion required, but a number of features are likely to becommon. A membrane separation is likely to be spec-ified by mass or volume throughput, feed compositionand required product concentration, yield and/or pu-rity. As examples, the products from the ultrafiltrationof whey are usually specified in terms of concentra-tion and purity of the retentate, whereas the productsof ultrafiltration of a hydrolysate may be specified asyield in the permeate only.

The design objective then becomes one of design-ing a process to obtain the specified separation usingmembranes with known flux and retention behavioursuch that the overall profit is maximised. The over-all profit depends on a number of variables but manysuch as labour and maintenance costs are not directlyrelated to the design. The variable costs influenced bythe design include:

• membrane purchase and replacement;• stagewise capital costs;• pumping energy cost;• loss of yield;• plant water costs;• CIP costs (plant volume based);• downstream processing costs.

To optimise the process many design variables andoperating can be changed. The process temperaturestrongly influences the design but it is likely to beset to the maximum allowed by either the membraneor product so as to maximise the flux. Other designvariables include:

• number of stages;• arrangement of stages within a plant;• arrangement of vessels within a stage;• area per stage;• diafiltration ratio to each stage;• plant pressures.

Within the design process there are also some ad-ditional objectives. The designer may want to:

• see if the design is sensible;• obtain insight for innovation;

K.R. Morison, X. She / Journal of Membrane Science 211 (2003) 59–70 61

• obtain a measure of the sensitivity of the design toprocess parameters;

• determine the feasible operating envelope of the de-sign;

• differentiate between different designs with similaroptima;

• maximise the operating flexibility of the plant whenbuilt.

2.2. Equations

The basic layout for a general membrane stage withstage recirculation is shown inFig. 1. It is assumedthat the rate of recirculation is sufficiently high thatthe composition within the recirculation loop is uni-form and equal to the retentate composition. A sys-tem of equations can be defined for the steady stateoperation of a single ultrafiltration stage, as shown inFig. 1, which includes the overall mass balance (1),the retention relation (2), the component mass bal-ances (3), the permeate flow relationship (4) and thediafiltration ratio (5):

FF = FR + FP − FD (1)

CP,j = CR,j (1 − σj ), for each componentj (2)

CF,j = ρRFRCR,j + ρPFPCP,j

ρFFF,

for each componentj (3)

FP = f (CR,j , T )A (4)

Fig. 1. One stage of a membrane plant with retentate recirculation.

FD = φFP (5)

whereσ is the retention coefficient,ρ the density (bothof which may be a function of the concentrations),Fthe volumetric flow rate,C the mass fraction,A themembrane area,φ the stage diafiltration ratio, sub-scripts F, R, P, D refer to feed, retentate, permeate anddiafiltration, and subscriptj refers to componentj. Thisset of equations can be used for each of many stages.

The equations from this apply to most batch andcontinuous membrane plants except plants that do nothave any recirculation, e.g. single-pass reverse osmo-sis. A difficulty with these equations is that they de-pend on the retentate composition and only implicitlyon the feed composition. Thus, to solve them, the re-tentate composition must be known or guessed, butthis problem is easily overcome by using the equationsolving capabilities of an optimisation program, suchas Solver in Microsoft Excel.

The flux relationship (4) must be obtained fromplant tests but it is well known that the function de-pends strongly on pressure, temperature, compositionand fouling, and is not at all straightforward to ob-tain. However, for design purposes, only an estimateof the worst case is required, probably an equation re-lating flux to composition in a fouled plant operatingat the lowest acceptable flow rate and at a nominalplant pressure.

As indicated byEq. (5), a diafiltration flow canbe added to any stage to aid the flushing out of per-meable components thus giving a higher purity ofnon-permeable components in the retentate, or a highyield of the permeable components in the permeate.


The diafiltration fluid is normally the solvent (usuallywater) added fresh to each stage as cross-current[10].Very often the diafiltration fluid is pure demineralisedwater and will be referred to as diafiltration waterhere. If the diafiltration fluid is not pure solvent,Eq. (3) must be modified to include the componentsin the diafiltration flow.

The overall diafiltration ratio,φoverall, is expressedas the ratio of the total diafiltration flow to the feedflow:

φoverall =∑

stages 1 tonFD,i

FF(6)

Often a low overall diafiltration ratio is desired as thisgives a lower cost for the demineralisation of the di-afiltration water and for downstream processes for thepermeate.

To solve the system, the temperature and functionsfor retention coefficients, density and flux need to bespecified. Normally variables, such as the feed flowrate, feed composition, retentate total solids, retentatepurity, diafiltration ratio and area might be specifiedand the remaining variables that must be solved for areother flow rates and mass fractions. A more specificexample is given later in the algorithm.

3. Design preliminaries

A number of different plant configurations are pro-posed by various authors[10] but for most continu-ous multi-stage processes there is little option but tohave a number of stages in series. Stages in parallelcan generally be considered as just one large singlestage. As stated above, diafiltration can be arranged ascross-current or counter-current. It is difficult to opti-mise the plant structure as this requires a mixed integernon-linear programming approach[5] that is difficultto solve. If various plant and diafiltration configura-tions seem possible, each may need to be optimisedand then optima compared, in which case analysis ofthe sensitivity and flexibility of the designs is veryuseful.

An optimisation of the transmembrane pressure ispossible but often it is constrained by the maximumcross-flow pressure drop and by the maximum trans-membrane pressure drop of the membrane used. Atthe maximum cross-flow pressure drop, the cross-flow

velocity is highest so the flux is maximised and thefouling minimised. If there is sufficient informationon the effect of cross-flow velocity and transmem-brane pressure on fouling and flux, together with equa-tions for cross-flow pressure drops[11], the pressures,and hence cross-flow velocity and power consumption,can be optimised[12]. However, such information isdifficult to obtain and it is likely that the cross-flowpressure drops will be constrained at their maximumor taken from experience. Thus, cross-flow pressuredrop, cross-flow velocity and power consumption donot form part of the design optimisation in this paper.

A related issue, in the case of spiral-wound mem-branes, is the number of membrane elements to beplaced in series in a vessel[13]. The optimisation ofthis again depends on the flux characteristics of themembrane and the product, and does not form part ofthis paper.

One of the design problems is the determinationof the flow rate of diafiltration water to add to eachstage. Conventional wisdom is that the amount of wa-ter is minimised if it is added where the retentate con-centration is high. However, high concentrations alsogive lower fluxes so more area may be required to re-move the diafiltration water at these low fluxes. Fora general case, an overall optimisation is required tobalance lower fluxes with the lower amount of waterused. In the optimisation a particular pattern of diafil-tration should not be assumed.

3.1. Objective function

The objective function,J, for the optimisation canbe a combination of the costs that can be influencedby design, in this case a combination of membranearea, diafiltration costs and product losses:

objective function= k1A + k2VD + k3mlosses (7)

whereA is the membrane area,VD the annual volumeof diafiltration water,mlossesthe annual mass of lostproduct, andk1, k2 andk3 the constants.

Wagner[13] gives the replacement cost per squaremetre of polysulfone UF membranes at US$ 25–50,RO membranes US$ 15–25 and NF membranes US$20–40. Typically the lifetime of the membrane is about2 years giving an annual cost of up to US$ 25 m−2.

The cost of diafiltration comes from both the supplycost and downstream processing costs. If the permeate


is concentrated by evaporation or reverse osmosis,there is an associated energy and capital cost. The in-stalled capital cost of a falling film evaporator is givenby Fincham and Jebson[14] as about 20,000F0.46 inUS$, whereF is the evaporation flow in kg h−1. Theenergy cost for a typical evaporator will be in the or-der of US$ 10 t−1 of steam with an evaporation effi-ciency of 10 t of evaporation per tonne of steam givingUS$ 1 t−1 of evaporation. If say, the permeate flowis 50,000 kg h−1 without diafiltration, the incrementalcapital (discounted over say 7 years) and energy costof adding diafiltration water can be calculated to beabout US$ 2 m−3 of diafiltration water. Some prelim-inary calculations show that this could be reduced toUS$ 0.50 m−3 with appropriate preconcentration byRO. In addition, the cost of supply of diafiltration wa-ter might be US$ 0.50 m−3.

The cost of mass losses can be very difficult toquantify with certainty. It is dependent on the reten-tion coefficient of the desired product. In the caseof protein ultrafiltration, lost protein is that whichpasses into the permeate and much of it is likely to beprotein fragments. Accurate estimation of losses forthe purpose of design optimisation depends on accu-rate retention data. Unless there is sufficient reliabledata yield losses should not be added to the objec-tive function. Otherwise the conclusions may be arte-facts of incorrect assumptions rather than of the realbehaviour.

For this study, the values used fork1 andk2 wereUS$ 25 m−2 and 1.50 m−3, respectively, but the valueof k3 was set to zero. At all stages in the optimi-sation these values were just used as a basis fromwhich to obtain insight into the optimisation of thedesign.

For the purposes of obtaining a spreadsheet solu-tion, constraints were added to the optimisation. Theyare that the feed and retentate compositions, and thefeed flow rate is the specified throughput of the plant.It was found that these non-linear constraints were bestapplied as a penalty function added to the objectivefunction. Thus, the objective function became:

objective function

= k1A + k2FD + k4

n∑j=1

(CF,des,j − CF,spec,j

CF,spec,j

)2

+k5(FF,des− FF,spec)2 (8)

where subscripts des and spec refer to those from thedesign and specified, and the sum is overn compo-nents in the feed. The penalty function weightings,k4 and k5, were set to about 106 but varied as nec-essary to obtain solutions that are sufficiently closeto the constraints. The variables of the optimisationwere the area of each membrane, the diafiltration ra-tio for each stage (Eq. (5)), the retentate flow rate andcomposition.

An algorithm was developed to set up and solve themodel.

1. Define the feed composition and flow rate, and therequired retentate total solids and purity.

2. Obtain equations or constants to describe flux (4),retention coefficients (2) and densities.

3. Set the number of stages and guess the optimisationvariables: area per stage and the stage diafiltrationratios.

4. Guess the solution variables: retentate flow rate andcomposition. A good guess can be obtained fromthe desired retentate total solids content and purity.

5. For each stage from the last (using the compositionand flow in step 4) to the first:5.1. CalculateCP for each component using (2),

FP using (4),FD using (5),FF using (1) andCF for each component using (3).

5.2. Calculate the total solids (TS) and purity ofthe retentate from the stage.

5.3. SetCR andFR from the next upstream stageto the calculated values ofCF andFF.

6. Define and calculate an objective function to min-imise, e.g. a weighted sum of area and total diafil-tration flow rate (6).

7. Define and calculate a penalty function for the de-viation of compositions and flow rates from thedesired values, e.g. a weighted sum of normalisedsquared deviations. Add this to the objective func-tion (7).

8. Minimise the total objective function (e.g. usingExcel Solver) by changing the stage areas, diafil-tration ratios, and guessed retentate flow rate andcomposition.

While results are relatively easy to obtain, the va-lidity of those results and innovation based on themis difficult. The optimisation may have over 20 vari-ables and in the feasible domain many local optimaare possible.


3.2. Ideal design

A first useful step to determine the validity of adesign is the development of an “ideal” design againstwhich the others can be compared.

There is an analogy between membrane plants andreactors. A batch reactor gives the highest rate ofreaction of species of any type of reactor, while acontinuously stirred tank reactor has a slower rate ofreaction because the concentration in the reactor is aslow as that in the product and the rate of reaction iscorrespondingly low[15]. The conversion of a batchreactor can be achieved using a theoretical plug flowtubular reactor which, for a given flow rate, has theminimum volume. It can be shown that an infiniteseries of infinitely small stirred tank reactors achievesthe same conversion in the same volume as a tubularreactor. A finite series of batch reactors has a lowerconversion for the same volume.

In a batch membrane plant, the membrane area re-quired for a given time-averaged throughput is lessthan any continuous plant because the concentration isas low as it can be at any stage of the process and fluxis thus a maximum. In contrast, in a continuous mem-brane plant with high recycle (Fig. 1) the concentra-tion is the same as that required in the product and theflux and throughput are correspondingly low. Higherthroughput for a given membrane area can be obtainedby using a series of membrane stages to increase theconcentration stagewise. The theoretical efficiency ofa batch plant can be matched by an ideal continuousplant with an infinite number of small stages each witha high cross-velocity. With such a plant the concen-tration in any stage is as low as it can be and the fluxis maximised. Exceptions to this are those membraneprocesses in which the flux does not continuously de-crease as concentration increases.

For design purposes, an ideal plant can be closelyapproximated by a continuous plant with a finite butlarge number of stages.

3.3. Case study

For the purpose of demonstrating some of the ideaspresented here a case study of the ultrafiltration of anidealised whey was examined. A feed of 50,000 kg h−1

of ideal whey contains 5% lactose and 0.65% protein.A retentate product is required with a purity of 95%,

and 25% total solids. (With only two components ahigh purity is possible.) The retention coefficients ofthe protein and lactose are 0.98 and 0.10, respectively.With the maximum acceptable amount of fouling theflux, J, in l m−2 h−1 is given by:

J = 10 log10

(0.33

CR,prot

)(9)

whereCR,prot is the mass fraction of protein in the re-tentate. To prevent high viscosity the retentate proteinfraction should not exceed 25%. The maximum stagediafiltration ratio is limited to 80%.

4. Graphical representation of the designs

To show features of the designs a number of differ-ent graphical representations are proposed. The first isa plot of the product purity versus total solids contentof the retentate.Fig. 2 shows this profile for an idealdesign for about 240 stages of 10 m2 each, for casestudy one, and also includes the curve for a plant withno diafiltration. We see from this that without diafil-tration the desired product purity cannot be achievedand in this case 71% diafiltration in each stage (seeEq. (2)) was found to be optimal.

The same data is also represented byFig. 3showingthe protein concentration versus the lactose concentra-tion. This more clearly shows the distinction betweenultrafiltration (UF) and diafiltration (DF).

Fig. 2. Profiles of purity vs. total solids in the ideal plant withand without diafiltration.


Fig. 3. Profile of composition in the ideal plant with and withoutdiafiltration.

However, neither graph offers an assurance that theoverall diafiltration strategy found is optimal. Exam-ination of different scenarios shows that there is atrade-off between the cost of area and diafiltration wa-ter in the optimisation. This is shown inFig. 4.

From Fig. 4 it can be seen that, for the case study,the membrane area cost is only weakly related to theoverall diafiltration ratio and that the most variablecost is that of diafiltration water. In a real design, asafety margin of perhaps 10% might be added to theminimum membrane area offering a plant that could

Fig. 4. The effect of overall diafiltration ratio on costs.

be operated over a range of diafiltration ratios and/or ata pressure lower than the pressure used for the designflux. In this case and in others tested, a small amountof extra area provides a large amount of freedom forthe diafiltration ratios used in plant operation.

A real design must have a practical number ofstages, the number of which can be optimised. At thispoint we should add to the objective function a costfor the equipment required in each stage. Typically asanitary pump, isolating valves, a permeate flow me-ter, a diafiltration water control valve, some additionalautomation and some plumbing are required. The costof this is at least US$ 10,000 per stage which canbe annualised by using a capital cost recovery factoror a payback time. Here a simple payback time of 5years is used and thus the additional annualised costof each stage is about US$ 2000. The cost of vesselsand processing equipment for the feed and retentate isunaffected by the number of stages. If the stages costsare added to the objective function, the optimisationbecomes a mixed integer non-linear program but be-cause of the difficulty of solving these, this is bestavoided. It is perhaps better to state the optimisationas: find the maximum number of stages at which thereduction in the overall cost achieved by adding onemore stage is less than the cost of that stage.

The design can be repeated for a number of differentstages and the cost calculated with typical results asshown inFig. 5.


Fig. 5. Optimisation of the number of stages.

The cost saving when the number of stages is in-creased from 9 to 10 is US$ 2400 but from 10 to 11stages is only about US$ 850 per year which is lessthan the cost of US$ 2000 per year for an extra stage.It would seem, for this example, that the optimal num-ber of stages is about 10.

When the solution for a practical plant is opti-mised, the area for each stage is calculated. Againone has the problem of determining whether ornot the solution obtained is sensible and critical.Firstly, the solution can be shown as compositionprofiles as inFigs. 6 and 7for plants with 4 and

Fig. 6. Profile of purity vs. total solids for an ideal plant, a 4-stageplant, and a 10-stage plant.

10 stages. This shows that with 10 stages, the pro-file is better able to approach the profile of the idealplant.

It is also proposed that the membrane area of eachstage be shown graphically. A graph, of which the inte-gral is area, would give a visual impression. One possi-bility is a graph of the inverse of flux versus cumulativepermeate flow through the stages, the integral of whichis area (P. Dejmek, personal communication). Whendiafiltration is used the cumulative permeate flow isaffected by the diafiltration flow and interpretation ismore difficult. Instead, one could plot the flow rates

Fig. 7. Composition profiles for an ideal plant, a 4-stage plant,and a 10-stage plant.


Fig. 8. Graphical representation of the area required to achieve a change in purity for an ideal plant (curve) and a four-stage plant (bars).

and flux of the permeate free of diafiltration water giv-ing a graph of 1/J(1−φ) versus cumulativeFP(1−φ),the integral of which is also area. The current au-thors were unable to gain insight from either of thesegraphs.

Another possibility is a graph of dA/d(purity) versuspurity, the integral of which is area. For plants withstages of finite size, the graph isA/�(purity) for eachstage versus purity.Fig. 8 shows such a graph for theideal plant and a crudely designed four-stage plant.

Fig. 9. A 10-stage plant is much closer to the optimal plant.

Visually we can see that when the purity is either lowor high more area is required to obtain a given changein the purity. By comparing the area under the graphin each section, it can be seen that the last stage ofthe four-stage design requires a lot more area than theideal design. More stages are required to obtain areassimilar to the optimal plant.Fig. 9 shows the samegraph for a plant with 10 stages and it seems that thedesign appears sensible as the areas of the graphs arevery similar.


Fig. 10. Graphical representation of the area required to increase total solids content vs. total solids content.

In this example, there is also the objective of con-centrating the solution. The same approach can beused to construct a graph ofA/�C versusC (Fig. 10)but this is less clear when there is diafiltration. It doesshow though that the area required to concentrate theproduct after the introduction of diafiltration is muchgreater for the 10-stage plant than for the ideal plant.This suggests that a larger number of small stages inthe diafiltration section may be beneficial, but whentried (not shown) this had little effect. The second peakthat occurs when diafiltration water addition is started

Fig. 11. Component fluxes of protein and lactose in a 10-stage plant.

is the result of negligible change in concentration. Thisis best seen inFig. 2which shows that the total solidscontent actually drops at the start of diafiltration. Themembrane area at this point in the plant is thus usedsolely for purification rather than concentration.

The combination of theFigs. 9 and 10gives someconfidence that the 10-stage design is reasonable.

Other variations in this approach are possible. Foranother application in which permeate yield was im-portant[16], a graph ofA/�(yield) versus yield gaveadditional insight.


4.1. Mass flux

The concept of component mass flux or flow canadd another view of the process. For a particular com-ponent the mass flux,Jj , of that component is givenby Eq. (10):

Jj = ρCP,jFP

A(10)

As concentration increases, the flux generally declines,but the change in the component mass flux depends onthe component and membrane. The component massfluxes of protein and lactose from a 10-stage plantfor case study one is shown inFig. 11. The overallyield loss was calculated to be 14%. In this graph,the assumption of the constant retention coefficient forprotein of 0.98 is almost certainly incorrect, and thisgraph will be very strongly affected by the assumption.Even without accurate data, this graph encourages oneto think about loss reduction. With cheaper membranecosts, one could consider using a tighter membrane ifthe protein flux was reduced to a greater extent than thelactose flux. A graph such likeFig. 11using real plantdata could indeed influence membrane replacementdecisions.

5. Discussion

The procedure and graphs shown here have beenvery useful when developing plant designs. The idealplant design provided a basis for comparison and inFig. 4 showed the sensitivity of the design costs tothe overall diafiltration ratio. This graph was useful inshowing the relative insensitivity of cost to diafiltra-tion ratio, and it prevented an unnecessary search fora very precise optimum.Fig. 5 showed the diminish-ing returns from extra stages. This graph is quite sen-sitive to membrane and water costs and would havelooked quite different if drawn using costs from 20years ago.

Figs. 6 and 7show how the number of stages influ-ences the composition profile in the plant, but they donot quantify the effect of following a profile differ-ent from that of the ideal plant. However,Figs. 8–10graphically show, from the graph area, the membranearea of the plant. The contrast between the 4- and10-stage plants is much more obvious inFigs. 8 and

9 than in Figs. 6 and 7. Fig. 8 shows the regions ofthe plant that would be more effective with smallerstages, in this case the start and end, rather than themiddle.Fig. 10was less useful for the case study pre-sented but it did provide ideas, albeit of no benefit, fora modification to the design. For other plants involv-ing only concentration without diafiltrationFig. 10may be more useful.

In many cases the results given by the optimisationwere not globally optimal. The results indicated thatthere were numerous local optima. Graphical compar-ison of the results with other plant designs and theideal design, quickly indicated that better optima werelikely to exist.

One of the features shown inFig. 2 is that diafil-tration can cause a decrease in the total solids levelfrom one stage to the next within the plant. This isknown as “backup” and is not considered desirable ina real plant. The difficulty often is that the operator canmonitor only the total solids within the plant, perhapsusing a refractometer, as analysis of the compositiontakes longer. The operator must ensure that there isnot potential for the total solids to exceed a maximumpotentially causing high viscosity and blockage of themembranes. Operators find it easier to operate a plantwith no potential for backup.

Most of the graphs shown can be used to monitorthe operation of an existing plant to gain a better un-derstanding of variations in the plant and opportunitiesfor improvement.

References

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